Research Article On the Generalized Lognormal Distributiondownloads.hindawi.com/journals/jps/2013/432642.pdf · On the Generalized Lognormal Distribution ThomasL.TouliasandChristosP.Kitsos

Hindawi Publishing CorporationJournal of Probability and StatisticsVolume 2013 Article ID 432642 15 pageshttpdxdoiorg1011552013432642

Research ArticleOn the Generalized Lognormal Distribution

Thomas L Toulias and Christos P Kitsos

Technological Educational Institute of Athens Department of Mathematics Ag Spyridonos amp Palikaridi Street12210 Egaleo Athens Greece

Correspondence should be addressed toThomas L Toulias ttouliasteiathgr

Received 11 April 2013 Revised 4 June 2013 Accepted 5 June 2013

Academic Editor Mohammad Fraiwan Al-Saleh

Copyright copy 2013 T L Toulias and C P KitsosThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper introduces investigates and discusses the 120574-order generalized lognormal distribution (120574-GLD) Under certain values ofthe extra shape parameter 120574 the usual lognormal log-Laplace and log-uniform distribution are obtained as well as the degenerateDirac distributionThe shape of all themembers of the 120574-GLD family is extensively discussedThe cumulative distribution functionis evaluated through the generalized error function while series expansion forms are derived Moreover the moments for the 120574-GLD are also studied

1 Introduction

Lognormal distribution has been widely applied in manydifferent aspects of life sciences including biology ecologygeology and meteorology as well as in economics financeand risk analysis see [1] Also it plays an important role inAstrophysics and Cosmology see [2ndash4] among others whilefor Lognormal expansions see [5]

In principle the lognormal distribution is defined asthe distribution of a random variable whose logarithmis normally distributed and usually it is formulated withtwo parameters Furthermore log-uniform and log-laplacedistributions can be similarly defined with applications infinance see [6 7] Specifically the power-tail phenomenonof the Log-Laplace distributions [8] attracts attention quiteoften in environmental sciences physics economics andfinance as well as in longitudinal studies [9] RecentlyLog-Laplace distributions have been proposed for modelinggrowth rates as stock prices [10] and currency exchangerates [7]

In this paper a generalized form of Lognormal distri-bution is introduced involving a third shape parameterWith this generalization a family of distributions is emergedwhich combines theoretically all the properties of LognormalLog-Uniform and Log-Laplace distributions depending onthe value of this third parameter

Thegeneralized120574-order Lognormal distribution (120574-GLD)is the distribution of a random vector whose logarithm fol-lows the 120574-order normal distribution an exponential powergeneralization of the usual normal distribution introducedby [11 12] This family of 119901-dimensional generalized normaldistributions denoted by N119901

120574(120583 Σ) is equipped with an

extra shape parameter 120574 and constructed to play the role ofnormal distribution for the generalized Fisherrsquos entropy typeof information see also [13 14]

The density function 119891119883 of a 119901-variate 120574-order normallydistributed random variable 119884 sim N119901

120574(120583 Σ) with location

vector 120583 isin R119901 positive definite scale matrix Σ isin R119901times119901 andshape parameter 120574 isin R [0 1] is given by [11]

119891119884 (119910) = 119891119884 (119910 120583 Σ 120574)

= 119862119901

120574|detΣ|minus12 expminus

120574 minus 1

120574119876120579(119910)

1205742(120574minus1)

119910 isin R119901

(1)

where 119876120579 is the quadratic form 119876120579(119910) = (119910 minus 120583)TΣminus1(119910 minus 120583)

120579 = (120583 Σ) while 119862119901120574being the normalizing factor

119862119901

120574= 120587minus1199012 Γ (1199012 + 1)

Γ (119901 ((120574 minus 1) 120574))(120574 minus 1

120574)

119901((120574minus1)120574)minus1

(2)

2 Journal of Probability and Statistics

From (1) notice that the second-ordered normal is theknown multivariate normal distribution that is N119901

2(120583 Σ) =

N119901(120583 Σ) see also [13 15]In Section 2 a generalized form of the Lognormal distri-

bution is introduced which is derived from the univariatefamily of N120574(120583 120590

2) = N1

120574(120583 1205902) distributions denoted by

LN120574(120583 120590) and includes the Log-Laplace distribution as wellas the Log-UniformdistributionThe shape of theLN120574(120583 120590)members is extensively discussed while it is connected to thetailing behavior of LN120574 through the study of the cdf InSection 3 an investigation of the moments of the generalizedLognormal distribution as well as the special cases of Log-Uniform and Log-Laplace distributions is presented

The generalized error function that is briefly pro-vided here plays an important role in the development ofLN120574(120583 120590) see Section 2 The generalized error functiondenoted by Erf119886 and the generalized complementary errorfunction Erfc119886 = 1minusErf119886 119886 ge 0 [16] are defined respectivelyas

Erf119886 (119909) =Γ (119886 + 1)

radic120587int

119909

0

119890minus119905119886

119889119905 119909 isin R (3)

The generalized error function can be expressed (changing to119905119886 variable) through the lower incomplete gamma function120574(119886 119909) or the upper (complementary) incomplete gammafunction Γ(119886 119909) = Γ(119886) minus 120574(119886 119909) in the form

Erf119886 (119909) =Γ (119886)

radic120587120574 (

1

119886 119909119886) =

Γ (119886)

radic120587[Γ (

1

119886) minus Γ (

1

119886 119909119886)]

119909 isin R

(4)

see [16] Moreover adopting the series expansion form of thelower incomplete gamma function

120574 (119886 119909) = int

119909

0

119905119886minus1

119890minus119905119889119905 =

infin

sum

119896=0

(minus1)119896

119896 (119886 + 119896)119909119886+119896

119909 119886 isin R+

(5)

a series expansion form of the generalized error function isextracted

Erf119886 (119909) =Γ (119886 + 1)

radic120587

infin

sum

119896=0

(minus1)119896

119896 (119896119886 + 1)119909119896119886+1

119909 119886 isin R+ (6)

Notice that Erf2 is the known error function erf that isErf2(119909) = erf(119909) while Erf0 is the function of a straightline through the origin with slope (119890radic120587)minus1 Applying 119886 = 2the known incomplete gamma function identities such as120574(12 119909) = radic120587 erf radic119909 and Γ(12 119909) = radic120587(1 minus erf radic119909) =

radic120587 erfcradic119909 119909 ge 0 are obtained Moreover Erf119886 0 = 0 for all119886 isin R+ and

lim119909rarrplusmninfin

Erf119886119909 = plusmn1

radic120587Γ (119886) Γ (

1

119886) 119886 isin R+ (7)

as 120574(119886 119909) rarr Γ(119886) when 119909 rarr +infin

2 The 120574-Order Lognormal Distribution

Thegeneralized univariate Lognormal distribution is definedthrough the univariate generalized 120574-order normal distribu-tion as follows

Definition 1 When the logarithm of a random variable 119883follows the univariate 120574-order normal distribution that islog119883 sim N120574(120583 120590

2) then 119883 is said to follow the generalized

Lognormal distribution denoted byLN120574(120583 120590) that is 119883 sim

LN120574(120583 120590)

TheLN120574(120583 120590) is referred to as the (generalized) 120574-orderLognormal distribution (120574-GLD) Like the usual Lognormaldistribution the parameter 120583 isin R is considered to belog-scaled while the non log-scaled 120583 (ie 119890120583 when 120583 isassumed log-scaled) is referred to as the location parameter ofLN120574(120583 120590) Hence if119883 sim LN120574(120583 120590) then log119883 is a 120574-ordernormally distributed variable that is log119883 sim N120574(120583 120590

2)

Therefore the location parameter 120583 isin R of 119883 is in fact themean of119883rsquos natural logarithm that is E[log119883] = 120583 while

Var [log119883] = (120574

120574 minus 1)

2((120574minus1)120574)Γ (3 ((120574 minus 1) 120574))

Γ (((120574 minus 1) 120574))1205902 (8)

Kurt [log119883] =Γ (120574 minus 1120574) Γ (5 (120574 minus 1120574))

Γ2 (3 ((120574 minus 1)120574)) (9)

see [15] for details onN120574Let 119884 = log119883 sim N120574(120583 120590

2) with density function as in

(1) and 119883 = 119892(119884) = 119890119884 Then the density function 119891119883 of

119883 sim LN120574(120583 120590) can be written through (1) as

119891119883 (119909) = 119891119883 (119909 120583 120590 120574) = 119891119884 (119892minus1(119909))

10038161003816100381610038161003816100381610038161003816

119889

119889119909119892minus1(119909)

10038161003816100381610038161003816100381610038161003816

= 119891119884 (log119909)1

119909

=

exp minus ((120574 minus 1) 120574) 1003816100381610038161003816(log119909 minus 120583) 1205901003816100381610038161003816120574(120574minus1)

2120590 ((120574 minus 1)120574)1120574Γ ((120574 minus 1) 120574) 119909

(10)

The probability density function 119891119883 as in (10) is defined inRlowast+= R+0 that isLN120574(120583 120590) has zero thresholdTherefore

the following definition extends Definition 1

Definition 2 When the logarithm of a random variable 119883 +

120599 follows the univariate 120574-order normal distribution that islog(119883 + 120599) sim N120574(120583 120590

2) then 119883 is said to follow the genera-

lized Lognormal distribution with threshold 120599 isin R that is119883 sim LN120574(120583 120590 120599)

It is clear that when 119883 sim LN120574(120583 120590 120599 ) log(119883 minus 120599) is a120574-order normally distributed variable that is log(119883 minus 120599) sim

N120574(120583 1205902) and thus 120583 is the mean of (119883 minus 120599)rsquos natural

logarithm while Var[log119883] is the same as in (8)Let 119884 = log(119883 + 120599) sim N120574(120583 120590

2) The density function of

119883 = 119890119884minus 120599 sim LN120574(120583 120590 120599) is given by 119891119883(119909) = 119891119883(119909 minus 120599)

119909 gt 0

Journal of Probability and Statistics 3

Let 119911 = (log(119909 minus 120599) minus 120583)120590 Then the limiting thresholddensity value of 119891119883(119909) with 119909 rarr 120599

+ implies that

lim119909rarr120599+

119891119883 (119909)

= 120590minus11198621

120574lim119911rarrminusinfin

expminus119911(120590 +120583

119911minus120574 minus 1

120574|119911|1(120574minus1)

)

= 120590minus11198621

120574119890(sgn 120574)(minusinfin)

(11)

and therefore

lim119909rarr120599+

119891119883 (119909) = 0 120574 isin (1 +infin)

+infin 120574 isin (minusinfin 0) (12)

that is the 119891119883rsquos defining domain for the positive-orderedLognormal random variable 119883 can be extended to includethreshold point 120599 by letting 119891119883(120599) = 0

The generalized Lognormal family of distributionsLN120574is a wide range family bridging the Log-Uniform LULognormal LN and Log-Laplace LL distributions aswell as the degenerate Dirac D distributions We have thefollowing

Theorem 3 The generalized Lognormal distributionLN120574(120583120590) for order values of 120574 = 0 1 2 plusmninfin is reduced to

LN120574 (120583 120590) =

D (119890120583) 120574 = 0

LU (119890120583minus120590

119890120583+120590

) 120574 = 1

LN (120583 120590) 120574 = 2

LL(1198901205831

1205901

120590) 120574 = plusmninfin

(13)

Proof From definition (1) of N120574 the order 120574 value is areal number outside the closed interval [0 1] Let 119883120574 sim

LN120574(120583 120590) with density function 119891119883120574

as in (10) We considerthe following cases

(i) The limiting case 120574 = 1 let 119909 isin Rlowast+such that

| log119909 minus 120583| le 1 Using the gamma function additiveidentity Γ(119911 + 1) = 119911Γ(119911) 119911 isin R+ in (10) we haveLN1(120583 120590) = lim120574rarr1+LN120574(120583 120590) with

1198911198831

(119909) = lim120574rarr1+

119891119883120574

(119909)

=

1

2120590119909 119909 isin [119890

120583minus120590 119890120583+120590

]

0 119909 isin (minusinfin 119890120583minus120590

) cup (119890120583+120590

+infin)

(14)

which is the density function of the Log-Uniform dis-tributionLU(119886 119887) 0 lt 119886 lt 119887 with 119886 = 119890

120583minus120590 and 119887 =119890120583+120590 that is 120583 = (12) log(119886119887) and 120590 = (12) log(119887119886)Therefore first-ordered Lognormal distribution is infact the Log-Uniform distribution with vanishingthreshold density 119891119883

1

(0) = lim119909rarr0+1198911198831

(119909) = 0For the purposes of statistical application the Log-Uniform moments are not the same as the modelparameters that is although 120583119883 = 120583 120590119883 = 120590radic3

(ii) The ldquonormalrdquo case 120574 = 2 it is clear thatLN2(120583 120590) =LN(120583 120590) as 119891119883

2

coincides with the Lognormaldensity function and therefore the second-orderedLognormal distribution is in fact the usual Lognormaldistribution

(iii) The limiting case 120574 = plusmninfin we have LNplusmninfin(120583 120590) =lim120574rarrplusmninfinLN120574(120583 120590) with

119891119883plusmninfin

(119909) = lim120574rarrplusmninfin

119891119883120574

(119909) =1

2120590119909expminus

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

=

119890minus120583120590

2120590119909(1minus120590)120590

119909 isin (0 119890120583]

119890120583120590

2120590119909minus(120590+1)120590

119909 gt 119890120583

(15)

which coincides with the density function of theknown Log-Laplace distribution (symmetric log-exponential distribution) LL(120583

1015840 120572 120573) with 1205831015840 = 119890

120583

and 120572 = 120573 = 1120590 see [8] Therefore the infinite-ordered log-normal distribution is in fact the Log-Laplace distribution with threshold density


(0) = lim119909rarr0+


(119909) =

0 120590 lt 1

1 120590 = 1

+infin 120590 gt 1

(16)

For the purposes of statistical application the Log-Laplace moments are not the same as the modelparameters that is although 120583119883 = 120583 120590119883 = radic2120590

(iv) The limiting case 120574 = 0 we have

lim120574rarr0minus

119891119883120574

(119909) = lim119896=[(120574minus1)120574]rarrinfin

119891119883120574

(119909) 119909 isin Rlowast

+ (17)

where [119886] is the integer value of 119886 isin R For the value119909 = 119890120583 the pdf as in (10) implies

lim120574rarr0minus

119891119883120574

(119890120583) =

1

2120590119890120583( lim119896rarrinfin

119896119896

119896) sdot 1198900= +infin (18)

through Stirlingrsquos asymptotic formula 119896 asymp radic2120587119896(119896

119890)119896 as 119896 rarr infin Assuming now 119909 = 119890

120583 (10) through(18) implies

1198911198830

(119909) = lim120574rarr0minus

119891119883120574

(119909) =

1003816100381610038161003816log119909 minus 1205831003816100381610038161003816

2radic21205871205902119909sdot 0 sdot

1

119890= 0 (19)

that is LN0(120583 120590) = lim120574rarr0minusLN120574(120583 120590) = D(119890120583)

as 1198911198830

coincides with the Dirac density functionwith the (non-log-scaled) location parameter 119890120583 ofLN0(120583 120590) being the singular (infinity) point There-fore the zero-ordered Lognormal distribution LN0is in fact the degenerateDirac distributionwith pole atthe location parameter of LN120574rarr0minus (with vanishingthreshold density 119891119883

0

(0) = lim119909rarr0+1198911198830

(119909) = 0)

From the above limiting cases (i) (iii) and (iv) the definingdomain R [0 1] of the order values 120574 used in (1) is safely


extended to include the values 120574 = 0 1 plusmninfin that is 120574 cannow be defined outside the open interval (0 1) Eventuallythe family of the 120574-order normals can include the Log-Uniform Lognormal Log-Laplace and the degenerate Diracdistributions as (13) holds

FromTheorem 3 (12) and (15) the domain of the densityfunctions 119891119883

120574

(119909) 119909 gt 0 can also be extended to include thethreshold point 119909 = 0 by setting 119891119883

120574

(0) = 0 for all non-negative-ordered Lognormals that is for all 120574 isin 0 cup [1 +infin)while for the Log-Laplace case of 120574 = +infin with 120590 = 1 bysetting 119891119883

+infin

(0) = (12)119890minus120583

From the fact that N0(120583 sdot) = D(120583) see [15] one cansay that the degenerate log-Dirac distribution say LD(120583)equals LN0(120583 sdot) and hence through Theorem 3 we canwriteLD(120583) = D(119890

120583)

Proposition 4 The mode of the positive-ordered Lognormalrandom variable 119883120574 sim LN120574(120583 120590) 119883 lt 119890

120583120574 isin (1 +infin) is

given by

Mode119883120574 = 119890120583minus120590120574

(20)

with corresponding maximum density value

max119891119883120574

= 119891119883120574

(Mode119883120574)

=exp (120590120574) 120574 minus 120583

2120590((120574 minus 1)120574)1120574Γ ((120574 minus 1) 120574)

(21)

Proof Recall the density function of 119883120574 sim LN120574(120583 120590) as in(10) and let 119898 = Mode 119883120574 gt 0 Then it holds that (119889119889119909)119891119883120574

(119898 120583 120590 120574) = 0 that is

(1

120590

1003816100381610038161003816log119898 minus 1205831003816100381610038161003816)

1(120574minus1)119889

119889119909

1003816100381610038161003816log119909 minus 1205831003816100381610038161003816119909=119898 = minus

120590

119898 (22)

From (119889119889119909)|119909| = sgn119909 119909 isin R we have

119898 = 119890120583minus120590120574

(23)

provided that 119909 lt 119890120583 Otherwise (23) holds trivially as (22)

implies 120590 = 0 that is 119898 = 119890120583 Moreover (119889119889119909)119891119883

120574

(119909) gt 0

when

1 + sgn (log119909 minus 120583) 120590120574(1minus120574)1003816100381610038161003816log119909 minus 12058310038161003816100381610038161(120574minus1)

gt 0 119909 gt 0

(24)

and thus 119891119883120574

is a strictly ascending density function on(0 119890120583minus120590120574

) when 120574 gt 1 and also on (119890120583minus120590120574

119890120583) when 120574 lt 0

Similarly with (119889119889119909)119891119883120574

(119909) lt 0 119891119883 is a strictly descendingdensity function on (119890

120583minus120590120574

+infin) when 120574 gt 1 and also on(0 119890120583minus120590120574

) cup (119890120583 +infin) when 120574 lt 0 Specifically for 120574 lt 0 the

point 119890120583 is a nonsmooth point of 119891119883 as

lim119909rarr119890120583

119889

119889119909119891119883120574

(119909) = minus

1198621

120574

(120590119890120583)2[120590 + lim119909rarr119890120583

sgn (log119909 minus 120583)

times

10038161003816100381610038161003816100381610038161003816

120590

log119909 minus 120583

10038161003816100381610038161003816100381610038161003816

1(1minus120574)

] = +infin

(25)

Therefore the positive-ordered Lognormals are formed bya unimodal density function with mode as in (20) andcorresponding maximum density as in (21) see Figures 1(a1)1(a2) and 1(a3)

Proposition 5 The global mode point of the negative-orderedLognormal random variable 119883120574 sim LN120574(120583 120590) 120574 isin (minusinfin 0)is (in limit) the threshold 0 which is an infinite (probability)density point Moreover the location parameter (ie 119890120583) ofLN120574(120583 120590) is a nonsmooth (local) mode point for all119883120574lt0 thatcorresponds to locally maximum density

119891119883120574

(119890120583) =

1

2120590((120574 minus 1)120574)1120574Γ ((120574 minus 1) 120574) 119890120583

(26)

while 119890120583minus120590120574

is a local minimum (probability) density point withcorresponding locally minimum density 119891119883

120574

(119890120583minus120590120574

)

Proof The negative-ordered Lognormals are formed by den-sity functions admitting threshold 0 (in limit) for theirglobal mode point (of infinite density) as shown in (12)Moreover from the previously discussedmonotonicity of119891119883

120574

in Proposition 4 all the negative-ordered Lognormals admitalso 119890

120583 as a local nonsmooth mode point and exp120583 minus 120590120574

as a local minimum density point with densities as in (26)and 119891119883

120574

(119890120583minus120590120574

) respectively see Figures 1(b1) 1(b2) and1(b3)

Furthermore Proposition 5 holds (in limit) for randomvariables1198831 and119883plusmninfin which provide Log-Uniform and Log-Laplace distributions respectively Indeed for given 119886 119887 isin

Rlowast+ 1198831 sim LU(119886 119887) = LN1(120583 120590) with 120583 = (12) log(119886119887)

and 120590 = (12) log(119887119886) we get through (20) and (21) thatMode1198831 = Mode119883120574rarr1+ = 119890

120583minus120590= 119886 with corresponding

maximum density (ie the maximum value of the densityfunction)

max1198911198831

= 1198911198831

(119886) =119890120590minus120583

2120590lim120574rarr1+

((120574 minus 1) 120574)(120574minus1)120574

Γ (((120574 minus 1) 120574) + 1)

=1

119886 log (119887119886)

(27)

Moreover the nonzerominimumdensity (ie theminimumbut not zero value of the density function) is obtained at119909 = 119887withmin119891119883

1

= 1198911198831

(119887) = 12120590119887 = 1(119887 log(119887119886))Theseresults are in accordance with the Log-Uniform densityfunction in (14)

For 119883plusmninfin sim LNplusmninfin(log120583 1120590) = LL(120583 120590 120590) weevaluate through (20) and (21) that

Mode119883+infin =

0 120590 lt 1

120583

119890120590 = 1

120583 120590 gt 1

(28)

with the corresponding maximum density value being infi-nite that is max119891119883

120574

= 119891119883120574

(0) = +infin provided 120590 lt 1 and


eminus23 e23

0

05

1

15

x

0 1 2 3

X1 simℒ119984(eminus23 e23)

2X sim ℒ119977(0 23)

0

05

1

15

x

0 1 2 3 4

eminus1 e1

X1 sim ℒ119984(eminus1 e1)

X2 sim ℒ119977(0 1)X+infin sim ℒℒ(1 1 1)

X+infin sim ℒℒ(1 32 32)

(b3)

(b2)

(b1)

(a3)

(a2)

(a1)

x

0 1 20

05

1

2

15

eminus32


0

05

1

15

x

0 1 2 31e

Xminusinfin sim ℒℒ(1 1 1)

x

0 1 2 30

05

1

15


x

01 2 30

05

1

15


X1112 19

X1112 19

X3410

X3410

X1112 19

X3410

Xminus10minus9minus1


Xminus10minus9minus1Xminus09minus08minus01



120574ge1(023)

fX120574sim119977

120574lt0(023)

fX120574sim119977

120574ge1(01)

fX120574sim119977

120574ge1(032)

fX120574sim119977

120574lt0(01)

fX120574sim119977

120574lt0(032)

fX120574sim119977

119984(eminus32 e32)X1 sim ℒ

(0 32)X2 sim ℒ119977

Figure 1 Graphs of the density functions 119891119883120574

119883120574 sim LN120574(0 120590) for 120590 = 23 1 32 and various positive (left subfigures) and negative (rightsubfigures) 120574 values


max119891119883+infin

= 120590(2120583) provided 120590 ge 1 The same result canalso be derived through (26) as 120574 rarr minusinfin These results arein accordance with the Log-Laplace density function in (15)although for 120590 = 1 Mode119883plusmninfin can be defined through (15)for any value inside the interval (0 120583]

The above discussion on behavior of the modes withrespect to shape parameter 120574 is formed in the followingpropositions

Proposition 6 Consider the positive-ordered Lognormal fam-ily of distributionsLN120574(120583 120590) with fixed parameters 120583 120590 and120574 ge 1When 120574 rises that is when onemoves fromLog-Uniformto Log-Laplace distribution inside the LN120574 family the modepoints ofLN120574 are

(i) strictly increasing from 119890120583minus120590 (Log-Uniform case) to

119890120583 (Log-Laplace case) provided that 120590 lt 1 (withtheir corresponding maximum density values movingsmoothly from (12120590)119890

120590minus120583 to +infin)(ii) fixed at 119890120583minus1 for all LN120574ge1(120583 120590 = 1) (with the cor-

responding maximum density values moving smoothlyfrom (12)119890

1minus120583 to (12)119890minus120583)(iii) strictly decreasing from 119890

120583minus120590 (Log-Uniform case) tothreshold 0 (Log-Laplace case) provided that 120590 gt 1

(with their corresponding maximum density valuesmoving smoothly from (12120590)119890

120590minus120583 to (12120590)119890minus120583)

Proof Let 119883120574 sim LN120574(120583 120590) Mode119883120574 is a smooth mono-tonous function of 120574 isin (minusinfin 0) cup (1 +infin) for positive-andnegative-ordered119883120574 as

119889

119889120574Mode119883120574 = minus120590

120574 log (120590) 119890120583minus120590120574

(29)

For119883plusmninfin sim LNplusmninfin(120583 120590) we evaluate through (20) and (21)that

Mode119883+infin =

119890120583 120590 lt 1

119890120583minus1

120590 = 1

0 120590 gt 1

(30)


120574

= 119891119883120574

(0) = +infin provided 120590 gt 1 andmax119891119883

+infin

= 1(2120590119890120583) provided 120590 le 1

Assume that 120574 ge 1 Considering (27) (30) with (29) andProposition 4 the results for the positive-ordered Lognor-mals hold

Proposition 7 For the negative-ordered Lognormal family ofdistributionsLN120574(120583 120590)with 120574 lt 0 when 120574 rises that is whenone moves from Log-Laplace to degenerate Dirac distributioninside the LN120574 family the local minimum (probability)density points ofLN120574 are

(i) strictly increasing from threshold 0 (Log-Laplace case)to 119890120583minus1 (Dirac case) provided that 120590 lt 1

(ii) fixed at 119890120583minus1 for allLN120574 (120583 120590 = 1)

(iii) strictly decreasing from 119890120583 (Log-Laplace case) to 119890120583minus1

(Dirac case) provided that 120590 gt 1

Proof Assume now that 120574 lt 0 From (29) we have (119889119889120574)(119890120583minus120590120574

) lt 0 when 120590 gt 1 and (119889119889120574)(119890120583minus120590120574

) gt 0 when 120590 lt 1Therefore the local minimum density point 119890

120583minus120590120574

(seeProposition 5) for 120590 gt 1 is decreasing from 119890

120583minus120590120574

|120574rarrminusinfin = 119890120583

to Mode1198830 = 119890120583minus1 through (20) When 120590 = 1 119890120583minus120590

120574

=

119890120583minus1 for all 120574 lt 0 while for 120590 lt 1 119890120583minus120590

120574

increases from119890120583minus120590120574

|120574rarrminusinfin = 0 to Mode1198830 = 119890120583minus1 through (20)

It is easy to see that for the Log-Laplace caseLL(120583 120590 120590)the local minimum density point 119890120583minus120590

120574

of 119883120574lt0 with 120590 gt 1

coincides (in limit) with the local nonsmooth mode point 119890120583of 119883120574 see Figure 1(b3) Also notice that the local minimumdensity point 119890120583minus120590

120574

120574 lt 0 for the Dirac case D(119890120583) is

the limiting point 119890120583minus1 although the (probability) density inD(119890120583) case vanishes everywhere except at the infinite pole 119890120583

Figure 1 illustrates the probability density functions 119891119883120574

curves for scale parameters 120590 = 23 1 32 of the positive-ordered lognormally distributed 119883120574 sim LN120574ge1(0 120590) inFigures 1(a1)ndash1(a3) respectively while the pdf of negative-ordered lognormally distributed 119883120574 sim LN120574lt0(0 120590) aredepicted in Figures 1(b1)ndash1(b3) respectively Moreover thedensity points 119890120583minus120590

120574

on 119891119883120574

are also depicted (small circlesover pdf curves with their corresponding ticks on 119909-axis)According to Proposition 7 in Figures 1(a1)ndash1(a3) that isfor positive-ordered 119883120574ge1 these density points represent themode points on 119891119883

120574

while in Figures 1(b1)ndash1(b3) that is fornegative-ordered119883120574lt0 represent the local minimum densitypoints on 119891119883

120574

curvesFor the evaluation of the cumulative distribution function

(cdf) of the generalized Lognormal distribution the follow-ing theorem is stated and proved

Theorem 8 The cdf 119865119883120574

of a 120574-order Lognormal randomvariable119883120574 sim LN120574(120583 120590) is given by

119865119883120574

(119909)

=1

2+

radic120587

2Γ ((120574 minus 1) 120574) Γ (120574 (120574 minus 1))

times Erf 120574(120574minus1) (120574 minus 1

120574)

(120574minus1)120574 log119909 minus 120583120590

(31)

= 1 minus1

2Γ ((120574 minus 1) 120574)Γ(

120574 minus 1

120574120574 minus 1

120574(log119909 minus 120583

120590)

120574(120574minus1)

)

119909 isin Rlowast

+

(32)

Proof From density function 119891119883120574 as in (10) we have

119865119883120574

(119909) = 119865119883120574

(119909 120583 120590 120574) = int

119909

0

119891119883120574

(119905) 119889119905

= 120590minus11198621

120574int

119909

0

119905minus1 expminus

120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log 119905 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119905

(33)


Applying the transformation 119908 = (log 119905 minus 120583)120590 119905 gt 0 theabove cdf is reduced to

119865119883120574

(119909) = 1198621

120574int

(log 119909minus120583)120590

minusinfin

expminus120574 minus 1

120574|119908|120574(120574minus1)

119889119908

= Φ119885120574

(log119909 minus 120583

120590)

(34)

where Φ119885120574

is the cdf of the standardized 120574-order normaldistribution 119885120574 = (1120590)(log119883120574 minus 120583) sim N120574(0 1) MoreoverΦ119885120574

can be expressed in terms of the generalized errorfunction In particular

Φ119885120574

(119911) = 1198621

120574int

119911

minusinfin


120574|119908|120574(120574minus1)

119889119908

= Φ119885120574

(0) + 1198621

120574int

119911

0


120574|119908|120574(120574minus1)

119889119908

(35)

and as 119891119885120574

is a symmetric density function around zero wehave

Φ119885120574

(119911) =1

2+ 1198621

120574int

119911

0


120574|119908|120574(120574minus1)

119889119908

=1

2+ 1198621

120574int

119911

0

exp

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(120574 minus 1

120574)

(120574minus1)120574

119908

1003816100381610038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119908

(36)

and thus

Φ119885120574

(119911) =1

2+ 1198621

120574(

120574

120574 minus 1)

(120574minus1)120574

times int

((120574minus1)120574)(120574minus1)120574

119911

0

exp minus119906120574(120574minus1) 119889119906

(37)

Substituting the normalizing factor as in (2) and using (3)we obtain

Φ119885120574

(119911) =1

2+

radic120587

2Γ (((120574 minus 1) 120574) + 1) Γ ((2120574 minus 1) (120574 minus 1))

times Erf120574(120574minus1) (120574 minus 1

120574)

(120574minus1)120574

119911 119911 isin R

(38)

and finally through (34) we derive (31) which forms (32)through (4)

It is essential for numeric calculations to express (31)considering positive arguments for Erf Indeed through (37)we have

119865119883120574

(119909) =1

2+

sgn (log119909 minus 120583)radic1205872Γ ((120574 minus 1) 120574) Γ (120574 (120574 minus 1))


120574)

(120574minus1)120574 10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

(39)

while applying (4) into (39) it is obtained that

119865119883120574

(119909) =1 + sgn (log119909 minus 120583)

2minussgn (log119909 minus 120583)2Γ ((120574 minus 1) 120574)

times Γ(120574 minus 1

120574120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

)

(40)

As the generalized error function Erf119886 is defined in (4)through the upper incomplete gamma function Γ(119886

minus1 sdot)

series expansions can be used for a more ldquonumerical-orientedrdquo form of (4) Here some expansions of the cdf ofthe generalized Lognormal distribution are presented

Corollary 9 The cdf 119865119883120574

can be expressed in the seriesexpansion form

119865119883120574

(119909) =1

2+

((120574 minus 1)120574)(120574minus1)120574

(2120574) Γ ((120574 minus 1) 120574)(log119909 minus 120583

120590)

times

infin

sum

119896=0

(((1 minus 120574)120574)1003816100381610038161003816(log119909 minus 120583)120590

1003816100381610038161003816120574(120574minus1)

)119896

119896 [(119896 + 1) 120574 minus 1]

119909 isin Rlowast

+

(41)

Proof Substituting the series expansion form of (6) into (39)and expressing the infinite series using the integer powers 119896the series expansion as in (41) is derived

Corollary 10 For the negative-ordered lognormally dis-tributed random variable 119883120574 with 120574 = 1(1 minus 119899) isin Rminus 119899 isin N119899 ge 2 the finite expansion is obtained as

119865119883120574

(119909)=1

2+1

2sgn (log119909 minus 120583) minus

sgn (log119909 minus 120583)

2 exp 1198991003816100381610038161003816(log119909 minus 120583)12059010038161003816100381610038161119899

times

119899minus1

sum

119896=0

119899119896

119896

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

119896119899

(42)

Proof Applying the following finite expansion form of theupper incomplete gamma function

Γ (119899 119909) = (119899 minus 1)119890minus119909119899minus1

sum

119896=0

119909119896

119896 119909 isin R 119899 isin N

lowast= N 0

(43)

into (40) we readily get (42)

Example 11 For the (minus1)-ordered lognormally distributed119883minus1 (ie for 119899 = 2) we have

119865119883minus1

(119909) =1

2+1

2sgn (log119909 minus 120583) minus sgn (log119909 minus 120583)

times

1 + 2radic1003816100381610038161003816(log119909 minus 120583) 120590

1003816100381610038161003816

2 exp 2radic1003816100381610038161003816(log119909 minus 120583) 1205901003816100381610038161003816

(44)


while for the (minus12)-ordered lognormally distributed 119883minus12

(ie for 119899 = 3) we have

119865119883minus12

(119909) =1

2+1


times

1 + 33radic1003816100381610038161003816(log119909 minus 120583) 120590

1003816100381610038161003816 + 93

radic((log119909 minus 120583)120590)2

2 exp 3 3radic1003816100381610038161003816(log119909 minus 120583) 1205901003816100381610038161003816

(45)

Example 12 For the second-ordered Lognormal randomvariable 1198832 sim LN2(120583 120590) we immediately derive from (31)that

1198651198832

(119909) = Φ1198832

(log119909 minus 120583

120590) =

1

2+1

2Erf2 (

log119909 minus 120583radic2120590

)

=1

2+1

2erf (

log119909 minus 120583radic2120590

)

(46)

that is the cdf of the usual Lognormal is derived as it isexpected due toLN2 = LN see Theorem 3

Example 13 For the infinite-ordered Lognormal 119883plusmninfin sim

LNplusmninfin(120583 120590) setting (120574 minus 1)120574 = 1 we obtain through (41)and the exponential series expansion that


(119909)

=1

2+1

2radic120587Erf1 (

log119909 minus 120583120590

)

=1

2minus1

2sgn (log119909 minus 120583)

infin

sum

119896=0

1

(119896 + 1)(minus

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816)

119896+1

=1

2+1

2sgn (log119909 minus 120583) minus 1

2sgn (log119909 minus 120583)

times expminus10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

(47)

and hence


(119909) =

1

2119890120583120590

1199091120590 119909 isin (0 119890

120583]

1 minus119890120583120590

21199091120590 119909 isin (119890

120583 +infin)

(48)

which is the cdf of the Log-Laplace distribution as in (15)This is expected as LNplusmninfin(120583 120590) = LL(119890

120583 1120590 1120590) see

Theorem 3

It is interesting to mention here that the same result canalso be derived through (42) as this finite expansion can beextended for 119899 = 1 which provides (in limit) the cdf of theinfinite-ordered Lognormal distribution

Example 14 Similarly for the first-ordered random variable1198831 sim LN1(120583 120590) the expansion (41) can be written as

119865119883120574

(119909)

=1

2+

((120574 minus 1)120574)(120574minus1)120574

2Γ (((120574 minus 1) 120574) + 1)(log119909 minus 120583

120590)

times[[

[

1 + (120574 minus 1)

infin

sum

119896=1

(((1 minus 120574)120574)1003816100381610038161003816(log119909 minus 120583)120590

1003816100381610038161003816120574(120574minus1)

)119896

119896 [(119896 + 1) 120574 minus 1]

]]

]

(49)

and provided that (log119909 minus 120583)120590 le 1 we obtain

1198651198831

(119909) = lim120574rarr1+

119865119883120574

(119909) =1

2+log119909 minus 120583

2120590(1 + 0)

=log119909 minus 120583 + 120590

2120590

(50)

with 1198651198831

(119890120583minus120590

) = 0 and 1198651198831

(119890120583+120590

) = 1 Therefore

1198651198831

(119909) =

0 119909 isin (0 119890120583minus120590

)

1

2120590(log119909 minus 120583 + 120590) 119909 isin [119890

120583minus120590 119890120583+120590

]

1 119909 isin (119890120583+120590

+infin)

(51)

coincides with the cdf of the Log-Uniform distributionLU (119886 = 119890

120583minus120590 119887 = 119890

120583+120590) as in (15) This is expected as

LN1(120583 120590) = LU(119890120583minus120590

119890120583+120590

) see Theorem 3

Table 1 provides the probability values 1198751205741 = Pr119883120574 le 119894119894 = 12 1 2 5 for various 119883120574 sim LN120574(0 1) Notice that1198751205741 = 12 for all 120574 values due to the fact that 1 = 119890

120583|120583=0 =

Med119883120574 (see Theorem 8) that is the point 1 coincides withthe 120574-invariant median of the LN120574(0 1) family discussedpreviously Moreover the last two columns provide also the1st and 3rd quartile points 1199021120574 and 1199023120574 of119883120574 that is Pr119883120574 le119902119896120574 = 1198964 119896 = 1 3 for various 120574 values These quartiles areevaluated using the quantile function Q119883

120574

of rv119883120574 that is

Q119883120574

(119875) = inf 119909 isin Rlowast

+| 119865119883

120574

(119909) ge 119875

= exp sgn (2119875 minus 1) 120590

times [120574

120574 minus 1Γminus1(120574 minus 1

120574 |2119875 minus 1|)]

(120574minus1)120574

119875 isin (0 1)

(52)

for 119875 = 14 34 that is derived through (40) The values ofthe inverse upper incomplete gamma function Γminus1((120574minus1)120574 sdot)were numerically calculated

Figure 2 illustrates the cdf 119865119883120574

curves as in (39) for cer-tain rv 119883120574 sim LN120574(0 120590) and for scale parameters 120590 = 23


Table 1 Probabilitymass values119875120574119894 = Pr119883120574 le 119894 119894 = 12 1 2 5 and the 1st and 3rd quartiles 1199021120574 1199023120574 for various generalized lognormallydistributed119883120574 sim LN120574(0 1)

120574 11987512057412 1198751205741 1198751205742 1198751205743 1198751205744 1198751205745 1199021120574 1199023120574

minus50 02501 05000 07499 08326 08739 08987 04998 20008minus10 02505 05000 07495 08297 08698 08940 04990 20038minus5 02508 05000 07492 08264 08652 08887 04982 20071minus2 02515 05000 07485 08187 08539 08756 04964 20145minus1 02521 05000 07479 08097 08408 08601 04945 20223minus12 02524 05000 07476 07989 08248 08410 04925 20303minus110 02528 05000 07482 07757 07895 07984 04986 204261 01534 05000 08466 10000 10000 10000 06065 1648732 02381 05000 07619 08848 09437 09721 05172 193342 02441 05000 07559 08640 09172 09462 05094 196303 02472 05000 07528 08505 08989 09267 05049 198044 02481 05000 07519 08452 08917 09188 05034 198675 02486 05000 07514 08425 08878 09145 05025 1989910 02494 05000 07506 08375 08810 09068 05011 1995450 02499 05000 07501 08341 08761 09013 05002 19992plusmninfin 02500 05000 07500 08333 08750 09000 05000 20000

1 32 in the 3 subfigures respectively Moreover the 1st and3rd quartile points Q119883

120574

(14) and Q119883120574

(34) are also depicted(small circles over cdf curves with their corresponding tickson 119909-axis)

Theorem 15 The (non-log-scaled) location parameter 119890120583 is infact the geometricmean aswell as themedian for all generalizedlognormally distributed 119883120574 sim LN120574(120583 120590) Moreover thismedian is also characterized by vanishing median absolutedeviation

Proof Considering (39) and the fact that Erf1198860 = 0 119886 isin Rlowast+

it holds that Med119883120574 = 119865minus1

119883120574

(12) = 119890120583 For the geometric

mean (120583119892)119883120574

= 119890E[log119883

120574] we readily obtain (120583119892)119883

120574

= 119890120583 as

log119883120574 sim N120574(120583 1205902) with E[119883120574] = 120583 A dispersion measure

for the median is the so-called median absolute deviation orMAD defined by MAD(119883120574) = Med | 119883120574 minus Med119883120574 | For119883120574 sim LN120574(120583 120590) we have 119883120574 minus Med119883120574 = 119883120574 minus 119890

120583sim

LN120574(120583 120590 minus119890120583) that is 119883120574 minus 119890

120583 follows the generalizedLognormal distributionwith thresholdminus119890120583 Furthermore |119884|is the ldquofolded distributionrdquo case of 119884 = 119883120574 minus 119890

120583 which isdistributed through pdf of the form

119891|119884| (119909) = 119891119884 (minus119909) + 119891119884 (119909) 119909 isin R+ (53)

where 119891119884 is the pdf of 119884 For example see [17] on thefolded normal distribution However the density function119891119884is defined in (minus119890120583 +infin) due to thresholdminus119890120583 while it vanisheselsewhere that is

119891|119884| (119909) = 119891119884 (minus119909) + 119891119884 (119909) 0 le 119909 le 119890

120583

119891119884 (119909) 119909 gt 119890120583

(54)

Therefore the cdf of |119884| is given by

119865|119884| (119909) = int

119909

0

119891|119884| (119905) 119889119905 = int

119890120583

0

119891119884 (minus119905) 119889119905 + int

119890120583

0

119891119884 (119905) 119889119905

+ int

119909

119890120583119891119884 (119905) 119889119905 119909 isin R+

(55)

Applying the transformation 119908 = (1120590) [log(119890120583 minus 119905) minus 120583] 119905 lt119890120583 into the first integral of (55) and 119911 = (1120590) [log(119905+119890120583)minus120583]119905 gt minus119890

120583 into the other two integrals we obtain

119865|119884| (119909) = 1198621

120574int

0

minusinfin


120574|119908|120574(120574minus1)

119889119908 + 1198621

120574

times int

(1120590) log 2

0


120574|119911|120574(120574minus1)

119889119911

+ 1198621

120574int

119892(119909)

(1120590) log 2expminus

120574 minus 1

120574|119911|120574(120574minus1)

119889119911

119892 (119909) =log (119909 + 119890120583) minus 120583

120590 119909 isin R+

(56)

and hence

119865|119884| (119909) = Φ119885 (0) + [Φ119885 (log 2120590

) minus Φ119885 (0)]

+ [Φ119885 (119892 (119909)) minus Φ119885 (log 2120590

)]

= Φ119885(log (119909 + 119890120583) minus 120583

120590)

(57)


x

0 1 2 3

X2simℒ119977(0 23)


sim119977

120574(023)

eminus23 e23

0

01

02

03

04

05

06

07

08

09

1

X1112 19

X3410



FX120574



(a)

120574(01)

0

01

02

03

04

05

06

07

08

09

1

X2simℒ119977(0 1)



x

0 1 2 3

X1 sim ℒ119984(eminus1 e1)Xminus10minus9minus1Xminus09minus08minus01

X1112 19

X3410

eminus1 e1

FX120574sim119977

(b)

00

01

02

03

04

05

06

07

08

09

1

x

1 2 3

eminus32


X2simℒ119977(0 32)

X1112 19

X3410


119977120574(032)

FX120574sim119977



(c)

Figure 2 Graphs of the cdf 119865119883120574

119883120574 sim LN120574(0 120590) for 120590 = 23 1 32 and various 120574 values

withΦ119885 being the cdf of the standardized rv119885 sim N120574(0 1)From (38) and the fact that Erf1198860 = 0 119886 isin Rlowast

+ it is clear that

(57) implies MAD119883120574 = Med|119883120574 minus 119890120583| = 119865minus1

|119883120574minus119890120583|

(12) = 0 forevery 119883120574 sim LN120574(120583 120590) and the theorem has been proved

3 Moments of the 120574-OrderLognormal Distribution

For the evaluation of themoments of the generalized Lognor-mal distribution the following holds


Proposition 16 The 119905th raw moment 120583(119905)119883

of a generalizedlognormally distributed random variable 119883 sim LN120574(120583 120590) isgiven by

120583(119905)

119883=

119890119905120583

Γ ((120574 minus 1) 120574)

infin

sum

119899=0

(119905120590)2119899

(2119899)(

120574

120574 minus 1)

2119899((120574minus1)120574)

times Γ((2119899 + 1)120574 minus 1

120574)

(58)

and coincides with the moment generating function of the 120574-order normally distributed log119883 that is119872log119883(119905) = 120583

(119905)

119883

Proof From the definition of the 119905th raw moment 120583(119905)119883 we

have

120583(119905)

119883= E [119883119905] = int

R+

119909119905119891119883 (119909) 119889119909

=1

1205901198621

120574intR+


120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119909

(59)

and applying the transformation 119911 = ((120574 minus 1)120574)(120574minus1)120574

(1120590)

(log119909 minus 120583) 119909 gt 0 we get

120583(119905)

119883= 1198621

120574(

120574

120574 minus 1)

(120574minus1)120574

intR

exp119905120583 + 119899(120574

120574 minus 1)

(120574minus1)120574

120590119911

times exp minus|119911|120574(120574minus1) 119889119911(60)

Through the exponential series expansion

exp119905(120574

120574 minus 1)

(120574minus1)120574

120590119911 =

infin

sum

119899=0

(119905120590)119899

119899(

120574

120574 minus 1)

119899((120574minus1)120574)

119911119899

(61)

it is obtained that

120583(119905)

119883= 21198621

120574(

120574

120574 minus 1)

(120574minus1)120574

119890119905120583infin

sum

119899=0

(119905120590)2119899

(2119899)(

120574

120574 minus 1)

2119899((120574minus1)120574)

times intR+

1199112119899 exp minus119911120574(120574minus1) 119889119911

(62)

Finally substituting the normalizing factor 1198621120574as in (2) into

(62) and utilizing the known integral [16]

intR+

119909119898119890minus119887119909119899

119889119909 =Γ ((119898 + 1) 119899)

119899119887(119898+1)119899 119899 119898 119887 isin R

lowast

+ (63)

we obtain (58)Moreover for 119884 = log119883 sim N120574(120583 120590

2) we have 119872119884(119905) =

E[119890119905119884] = E[119883119905] = 120583(119905)

119883 and the proposition has been proved

Example 17 For the second-ordered lognormally distributed119883 sim LN2(120583 120590) (58) implies

120583(119905)

119883=

119890119905120583

radic120587

infin

sum

119899=0

(211990521205902)119899

(2119899)Γ (119899 +

1

2) 119905 isin R+ (64)

and through the gamma function identity

Γ (119899 +1

2) =

(2119899)

22119899119899radic120587 119899 isin N (65)

we have

120583(119905)

119883= 119890119905120583infin

sum

119899=0

(119905120590)2119899

2119899119899= 119890119905120583infin

sum

119899=0

1

119899(1

211990521205902)

119899

= 119890119905120583+(12)(119905120590)

2

119905 isin R+

(66)

which is the 119905th raw moment of the usual lognormallydistributed 119883 sim LN(120583 120590) with mean 120583119883 = 120583

(1)

119883= E[119883] =

exp120583 + (12)1205902 This is true as119872log119883(119905) = 120583

(119905)

119883= exp119905120583 +

(12)(119905120590)2 is the known moment-generating function of the

normally distributed log119883 sim N(120583 1205902)

Theorem 18 The 119896th central moment (about the mean) 120583(119905)119883

of a generalized lognormally distributed random variable 119883 sim

LN120574(120583 120590) is given by

120583(119896)

119883=

119890119896120583

Γ ((120574 minus 1) 120574)

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899

119878119896minus119899 119896 isin N (67)

where

119878119896 =

infin

sum

119898=0

(119896120590)2119898

(2119898)(

120574

120574 minus 1)

2119898((120574minus1)120574)

Γ((2119898 + 1)120574 minus 1

120574)

119896 isin N

(68)

Proof From the definition of the 119896th central moment 120583(119896)119883

wehave

120583(119896)

119883= E [(119883 minus 120583119883)

119896] = int

R+

(119909 minus 120583119883)119896119891119883 (119909 120583 120590 120574) 119889119909

(69)

while using the binomial identity we get

120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus120583119883)

119899intR+

119909119896minus119899

119891119883 (119909) 119889119909

=

119896

sum

119899=0

(119896

119899) (minus120583119883)

119899120583(119896minus119899)

119883

(70)

Applying Proposition 16 (70) implies that

120583(119896)

119883=

119890119896120583

Γ ((120574 minus 1) 120574)

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898)

times (120574

120574 minus 1)

2119898((120574minus1)120574)

Γ ((2119898 + 1) ((120574 minus 1) 120574))

(71)


while taking the summation index 119899 until 119896 minus 1 we finallyobtain (67) and the theorem has been proved

Example 19 Recall Example 17 Substituting (66) and themean 120583119883 = 119890

120583+(12)1205902

into (70) the second-ordered lognor-mally distributed119883 sim LN2(120583 120590) provides

120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus1)

119899119890119896120583+(12)[119899+(119896minus119899)

2

]1205902

119896 isin N (72)

while

1205902

119883= Var [119883] = 120583

(2)

119883= 1198902120583+1205902

(1198901205902

minus 1) (73)

which are the 119896th central moment and the variance respec-tively of the usual lognormally distributed 119883 sim LN(120583 120590)The same result can be derived directly through (67) for 120574 = 2

and the use of the known gamma function identity as in (65)

Theorem20 Themean 120583119883 = E [119883] variance 1205902119883= Var[119883]

coefficient of variation 119862119881119883 skewness 120582119883 and kurtosis 120581119883 ofthe generalized lognormally distributed 119883 sim LN120574(120583 120590) arerespectively given by

120583119883 =119890120583

Γ ((120574 minus 1) 120574)1198781 (74)

1205902

119883= minus 120583

2

119883+

1198902120583

Γ ((120574 minus 1) 120574)1198782 (75)

1198621198812

119883= Γ(

120574 minus 1

120574)1198782

11987821

minus 1 (76)

120582119883 = minus 119862119881minus3

119883minus 119862119881minus1

119883+

1198903120583

1205903119883Γ ((120574 minus 1) 120574)

1198783 (77)

120581119883 = minus 119862119881minus4

119883minus 6119862119881

minus2

119883minus 4

120582119883

119862119881119883

+1198904120583

1205904119883Γ ((120574 minus 1) 120574)

1198784

(78)

where the sums 119878119894 119894 = 1 4 are given by (68)

Proof From Proposition 16 we easily obtain (74) as 120583119883 =

120583(1)

119883 FromTheorem 18 we have

1205902

119883= 120583(2)

119883= 1205832

119883+ [Γ(

120574 minus 1

120574)]

minus1

(11989021205831198782 minus 2119890

1205831205831198831198781) (79)

Hence substituting 1198781 from (74) (75) holds Moreover thesquared coefficient of variation is readily obtained via (75)and (74) By definition skewness 120582119883 is the standardizedthird (central) moment that is 120582119883 = Skew[119883] = 120583

(3)

1198831205903

119883

Theorem 18 provides that

120582119883 = minus119862119881minus3

119883+ [1205903

119883Γ(

120574 minus 1

120574)]

minus1

times (11989031205831198783 minus 3119890

21205831205831198831198782 + 3119890

1205831205831198831198781)

(80)

Substituting 1198781 and 1198782 from (74) and (75) we obtain (77)Finally kurtosis 120581119883 is (by definition) the standardized fourth(central) moment that is 120581119883 = Kurt[119883] = 120583

(4)

1198831205904

119883 which

provides throughTheorem 18 that

120581119883 = 119862119881minus4

119883+ [1205904

119883Γ(

120574 minus 1

120574)]

minus1

times (11989041205831198784 minus 4119890

31205831205831198831198783 + 6119890

21205831205832

1198831198782 minus 4119890

1205831205833

1198831198781)

(81)

Substituting 119878119894 119894 = 1 2 3 from (74) (75) and (77) we obtain(78)

Example 21 For the second-ordered lognormally distributed119883 sim LN2(120583 120590) utilizing (65) into (68) we get 119878119899 =

radic120587119890(1198992

1205902

)2 119899 isin Nlowast Applying this to Theorem 20 we derive(after some algebra)

120583119883 = 119890120583+(12)120590

2

1205902

119883= 1198902120583+1205902

(1198901205902

minus 1)

119862119881119883 =radic1198901205902

minus 1

120582119883 = (1198901205902

+ 2)radic1198901205902

minus 1 120581119883 = 11989041205902

+ 211989031205902

+ 311989021205902

minus 3

(82)

which are the mean variance coefficient of variation skew-ness and kurtosis respectively of usual lognormally dis-tributed119883 sim LN(120583 120590)

For the usual lognormally distributed random variable119883 sim LN it is known that Mode119883 lt Med119883 lt 120583119883 Thefollowing corollary examines this inequality for the LN120574family of distributions

Corollary 22 For the 120574-ordered lognormally distributed119883120574 simLN120574(120583 120590) it is true that Mode119883120574 le Med119883120574 = (120583119892)119883

120574

le

120583119883120574

The first equality holds for the Log-Laplace distributed119883+infin with 120590 lt 1 as well as for all the negative-ordered 119883120574lt0where Mode119883120574 is considered to be the local (nonsmooth)mode point of119883120574 The second equality holds for the degenerateDirac case of1198830

Proof From (74) andTheorem 15 we have

Med119883120574 = (120583119892)119883120574

= 119890120583lt 120583119883

120574

(83)

for every 119883120574 sim LN120574(120583 120590) The above inequality becomesequality for the limiting Dirac case of 1198830 For the relationbetween the mode and the median of119883120574 the following casesare considered

(i) The positive-ordered Lognormal case 120574 gt 1 from(20) we have

Mode119883120574 = 119890120583minus120590120574

lt 119890120583= Med119883120574 (84)

For the Log-Laplace case of119883+infin it holds

Mode119883120574 = 119890120583= Med119883120574 (85)


provided that 120590 lt 1 while for 120590 ge 1 we have

Mode119883120574 = 0 lt 119890120583= Med119883120574 (86)

For 120590 = 1 the inequality (84) clearly holds(ii) The negative-ordered Lognormal case 120574 lt 0 from

Proposition 4 the inequality as in (86) holds More-over if Mode119883120574 is considered as the nonsmoothlocal mode point of the negative-ordered119883120574 then theequality as in (85) holds

From the above cases and (83) the corollary holds true

Corollary 23 The raw and central moments of a Log-Uni-formly distributed random variable119883 sim LU(119886 119887) 0 lt 119886 lt 119887are given by

120583(119905)

119883=

119887119905minus 119886119905

119905 log (119887119886) 119905 isin R (87)

120583(119896)

119883=

(119886 minus 119887)119896

log119896 (119887119886)+

1

log (119887119886)

times

119896minus1

sum

119899=0

(119896

119899)(119886 minus 119887)

119899(119887119896minus119899

minus 119886119896minus119899

)

(119896 minus 119899) log119899 (119887119886) 119896 isin N

(88)

respectively while the mean variance coefficient of variationskewness and kurtosis of 119883 are given respectively by

120583119883 =119887 minus 119886

log (119887119886) (89)

1205902

119883=

(119887 minus 119886)2

log2 (119887119886)+(119887 minus 119886) (119887 + 119886)

2 log (119887119886) (90)

119862119881119883 = radic1 +119887 + 119886

2 (119887 minus 119886)log 119887

119886 (91)

120582119883 =1

1205903119883

[1198873minus 1198863

3 log (119887119886)minus 3

(119887 minus 119886)2(119887 + 119886)

2 log2 (119887119886)+ 2

(119887 minus 119886)3

log3 (119887119886)]

(92)

120581119883 =1

1205904119883

[1198874minus 1198864

4 log (119887119886)minus 4

(119887 minus 119886) (1198873minus 1198863)

3 log2 (119887119886)

+3(119887 minus 119886)

3(119887 + 119886)

log3 (119887119886)minus 3

(119887 minus 119886)4

log4 (119887119886)]

(93)

Proof Recall Proposition 16 with119883120574 sim LN120574(120583 120590) Throughthe gamma function additive identity (58) can be written as

120583(119905)

119883120574

=119890119905120583

119905120590Γ ((120574 minus 1) 120574 + 1)

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)(

120574

120574 minus 1)

2119898((120574minus1)120574)

times Γ((2119898 + 1)120574 minus 1

120574+ 1)

(94)

Thus letting 119883 = 1198831 sim LN1(120583 120590) = LU(119886 119887) with120583 = (12) log(119886119887) and 120590 = (12) log(119887119886) it holds (recall theexponential odd series expansion) that

120583(119905)

119883= lim120574rarr1+

120583(119905)

119883120574

=119890119905120583

119905120590

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)=119890119905(120583+120590)

minus 119890119905(120583minus120590)

2119905120590

119905 isin R

(95)

and hence (87) holds Moreover 120583119883 = 120583(1)

119883= E[119883] = (12120590)

(119890120583+120590

minus 119890120583minus120590

) and therefore (89) holdsWorking similarly (67) implies

120583(119896)

119883= lim120574rarr1+

120583(119896)

119883120574

= 119890119896120583119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898 + 1)

119896 isin N

(96)

Using the exponential odd series expansion the above expan-sion becomes

120583(119896)

119883=119890119896120583

2120590

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 119890(119896minus119899)120590

minus 119890minus(119896minus119899)120590

(119896 minus 119899) 119896 isin N

(97)

and through (89) we obtain (88) Moreover for 119896 = 2 1205902119883=

Var[119883] = 120583(2)

119883minus 1205832

119883implies (90) and hence (91) also holds

For 119896 = 3 and 119896 = 4 through 120583(3)119883

and 120583(4)119883 we obtain (92) and

(93) respectively

Corollary 24 The raw and central moments of a Log-Laplacedistributed random variable119883 sim LL(120583 120590 120590) are given by

120583(119905)

119883=

1205831199051205902

1205902 minus 1199052gt 120583119905 120590 gt 119905 119905 isin R (98)

120583(119896)

119883= 120583119896119896

sum

119899=0

(119896

119899)

1205902(119899+1)

(1 minus 1205902)119899[1205902 minus (119896 minus 119899)

2] 120590 gt 119896 119896 isin N

(99)

The mean variance coefficient of variation skewness andkurtosis of 119883 are given respectively by

120583119883 =1205831205902

1205902 minus 1gt 120583 120590 gt 1 (100)

1205902

119883=

12058321205902(21205902+ 1)

(1205902 minus 4) (1205902 minus 1)2 120590 gt 2 (101)

119862119881119883 =1

120590

radic21205902+ 1

1205902 minus 4 120590 gt 2 (102)

120582119883 =2 (15120590

4+ 71205902+ 2)

120590 (1205902 minus 9)radic

1205902minus 4

(21205902 + 1)3 120590 gt 3 (103)


120581119883 =3 (81205908+ 212120590

6+ 95120590

4+ 33120590

2+ 12) (120590

2minus 4)

(1205902 minus 16) (1205902 minus 9) (21205902 + 1)2

120590 gt 4

(104)

Proof Let 119883120574 sim LL120574(120583 120590 120590) = LN120574(log 120583 1120590 1120590) For120574 = plusmninfin that is 120574(120574 minus 1) = 1 the raw moments as in (58)provide

120583(119905)

119883= 120583(119905)

119883plusmninfin

= 120583119905infin

sum

119896=0

(119905

120590)

2119896

119905 isin R (105)

as119883 = 119883plusmninfin while through the even geometric series expan-sion it is

120583(119905)

119883plusmninfin

=1

2120583119905[

infin

sum

119896=0

(119905

120590)

119896

+

infin

sum

119896=0

(minus119905

120590)

119896

]

=1

2120583119905(

120590

120590 minus 119905+

120590

120590 + 119905)

(106)

provided that 120590 gt 119905 and hence (98) holds Moreover 120583119883 =120583(1)

119883= E[119883] and hence (100) holdsWorking similarly (67) implies

120583(119896)

119883= 1205831198961205902119896

sum

119899=0

(119896

119899)

(minus120583119883120583)119899

1205902 minus (119896 minus 119899)2 119896 isin N (107)

provided 120590 gt 119896 and hence through (100) the centralmoments (99) are obtained

Moreover for 119896 = 2 and due to 1205902119883= Var[119883] = 120583

(2)

119883minus 1205832

119883

(101) holds true while for 119896 = 3 and 119896 = 4 we derive through120583(3)

119883and 120583(4)

119883 (103) and (104) respectively

Example 25 For a uniformly distributed rv 119880 sim U(119886 119887) =

N1(120583 120590) with 119886 = 120583 minus 120590 and 119887 = 120583 + 120590 it holds thatLU = 119890

119880sim LU(119890

120583minus120590 119890120583+120590

) due to Theorem 3 and thereforeLU is a Log-Uniform distributed rv as LU sim LU(119890

119886 119890119887)

Applying (87) the known moment-generating function ofthe uniformly distributed 119880 sim U(119886 119887) is derived that is119872119880(119905) = E[119890119905119880] = 120583

(119905)

LU = (119890119905119887minus 119890119905119886)(1119905(119887 minus 119886))

Similarly for a Laplace distributed rv 119871 sim L(120583 120590) =

Nplusmninfin(120583 120590) it holds that LL = 119890119871sim LL(119890

120583 1120590 1120590) due to

Theorem 3 and therefore LL is a Log-Laplace distributed ran-dom variable Applying (98) we derive the known moment-generating function of the Laplace distributed 119871 sim L(120583 120590)that is119872119871(119905) = E[119890119905119871] = 120583

(119905)

LL = 119890119905120590(1 minus 11990521205902)minus1

4 Conclusion

The family of the 120574-order Lognormal distributions wasintroduced which under certain values of 120574 includes theLog-Uniform Lognormal and Log-Laplace distributions aswell as the degenerate Dirac distribution The shape of thesedistributions for positive and negative shape parameters 120574 aswell as the cumulative distribution functions was extensivelydiscussed and evaluated through corresponding tables andfigures Moreover a thorough study of moments was carried

out in which nonclosed forms as well as approximationswereobtained and investigated in various examples This general-ized family of distributions derived through the family of the120574-order normal distribution is based on a strong theoreticalbackground as the logarithmic Sobolev inequalities provideFurther examinations and calculations can be producedwhilean application to real data is upcoming

Acknowledgment

The authors would like to thank the referee for his valuablecomments that helped improve the quality of this paper

References

[1] E L Crow and K Shimizu Lognormal Distributions MarcelDekker New York NY USA 1988

[2] A Parravano N Sanchez and E J Alfaro ldquoThe Dependence ofprestellar core mass distributions on the structure of theparental cloudrdquoTheAstrophysical Journal vol 754 no 2 article150 2012

[3] F Bernardeau and L Kofman ldquoProperties of the cosmologicaldensity distribution functionrdquo Monthly Notices of the RoyalAstronomical Society vol 443 pp 479ndash498 1995

[4] P Blasi S Burles and A V Olinto ldquoCosmological magneticfield limits in an inhomogeneous Universerdquo The AstrophysicalJournal Letters vol 514 no 2 pp L79ndashL82 1999

[5] F S Kitaura ldquoNon-Gaussian gravitational clustering field statis-ticsrdquoMonthly Notices of the Royal Astronomical Society vol 420no 4 pp 2737ndash2755 2012

[6] G Yan and F B Hanson ldquoOption pricing for a stochastic-volatility jump-diffusion model with log-uniform jump-amplitudesrdquo in Proceedings of the American Control Conference2006

[7] T J Kozubowski and K Podgorski ldquoAsymmetric LaplacedistributionsrdquoTheMathematical Scientist vol 25 no 1 pp 37ndash46 2000

[8] T J Kozubowski and K Podgorski ldquoAsymmetric Laplace lawsand modeling financial datardquo Mathematical and ComputerModelling vol 34 no 9ndash11 pp 1003ndash1021 2001

[9] M Geraci and M Bottai ldquoQuantile regression for longitudinaldata using the asymmetric Laplace distributionrdquo Biostatisticsvol 8 no 1 pp 140ndash154 2007

[10] D BMadan ldquoThe variance gamma process and option pricingrdquoThe European Financial Review vol 2 pp 79ndash105 1998

[11] C P Kitsos and N K Tavoularis ldquoLogarithmic Sobolevinequalities for information measuresrdquo IEEE Transactions onInformation Theory vol 55 no 6 pp 2554ndash2561 2009

[12] C P Kitsos and N K Tavoularis ldquoNew entropy type infor-mation measuresrdquo in Proceedings of the Information TechnologyInterfaces (ITI rsquo09) Cavtat Croatia June 2009

[13] C P Kitsos and T L Toulias ldquoNew information measures forthe generalized normal distributionrdquo Information vol 1 no 1pp 13ndash27 2010

[14] C P Kitsos and T L Toulias ldquoEvaluating informationmeasuresfor the -order Multivariate Gaussianrdquo in Proceedings by IEEE ofthe 14th Panhellenic Conference on Informatics (PCI rsquo10) pp 153ndash157 Tripoli Greece September 2010

[15] C P Kitsos T L Toulias and P C Trandafir ldquoOn the mul-tivariate 120574-ordered normal distributionrdquo Far East Journal ofTheoretical Statistics vol 38 no 1 pp 49ndash73 2012


[16] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Elsevier 2007

[17] F C Leone L SNelson andR BNottingham ldquoThe foldednor-mal distributionrdquo Technometrics vol 3 pp 543ndash550 1961

Submit your manuscripts athttpwwwhindawicom

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Algebra

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Stochastic AnalysisInternational Journal of


From (1) notice that the second-ordered normal is theknown multivariate normal distribution that is N119901

2(120583 Σ) =

N119901(120583 Σ) see also [13 15]In Section 2 a generalized form of the Lognormal distri-

bution is introduced which is derived from the univariatefamily of N120574(120583 120590

2) = N1

120574(120583 1205902) distributions denoted by

LN120574(120583 120590) and includes the Log-Laplace distribution as wellas the Log-UniformdistributionThe shape of theLN120574(120583 120590)members is extensively discussed while it is connected to thetailing behavior of LN120574 through the study of the cdf InSection 3 an investigation of the moments of the generalizedLognormal distribution as well as the special cases of Log-Uniform and Log-Laplace distributions is presented

The generalized error function that is briefly pro-vided here plays an important role in the development ofLN120574(120583 120590) see Section 2 The generalized error functiondenoted by Erf119886 and the generalized complementary errorfunction Erfc119886 = 1minusErf119886 119886 ge 0 [16] are defined respectivelyas

Erf119886 (119909) =Γ (119886 + 1)

radic120587int

119909

0

119890minus119905119886

119889119905 119909 isin R (3)

The generalized error function can be expressed (changing to119905119886 variable) through the lower incomplete gamma function120574(119886 119909) or the upper (complementary) incomplete gammafunction Γ(119886 119909) = Γ(119886) minus 120574(119886 119909) in the form

Erf119886 (119909) =Γ (119886)

radic120587120574 (

1

119886 119909119886) =

Γ (119886)

radic120587[Γ (

1

119886) minus Γ (

1

119886 119909119886)]

119909 isin R

(4)

see [16] Moreover adopting the series expansion form of thelower incomplete gamma function

120574 (119886 119909) = int

119909

0

119905119886minus1

119890minus119905119889119905 =

infin

sum

119896=0

(minus1)119896

119896 (119886 + 119896)119909119886+119896

119909 119886 isin R+

(5)

a series expansion form of the generalized error function isextracted

Erf119886 (119909) =Γ (119886 + 1)

radic120587

infin

sum

119896=0

(minus1)119896

119896 (119896119886 + 1)119909119896119886+1

119909 119886 isin R+ (6)

Notice that Erf2 is the known error function erf that isErf2(119909) = erf(119909) while Erf0 is the function of a straightline through the origin with slope (119890radic120587)minus1 Applying 119886 = 2the known incomplete gamma function identities such as120574(12 119909) = radic120587 erf radic119909 and Γ(12 119909) = radic120587(1 minus erf radic119909) =

radic120587 erfcradic119909 119909 ge 0 are obtained Moreover Erf119886 0 = 0 for all119886 isin R+ and

lim119909rarrplusmninfin

Erf119886119909 = plusmn1

radic120587Γ (119886) Γ (

1

119886) 119886 isin R+ (7)

as 120574(119886 119909) rarr Γ(119886) when 119909 rarr +infin

2 The 120574-Order Lognormal Distribution

Thegeneralized univariate Lognormal distribution is definedthrough the univariate generalized 120574-order normal distribu-tion as follows

Definition 1 When the logarithm of a random variable 119883follows the univariate 120574-order normal distribution that islog119883 sim N120574(120583 120590

2) then 119883 is said to follow the generalized

Lognormal distribution denoted byLN120574(120583 120590) that is 119883 sim

LN120574(120583 120590)

TheLN120574(120583 120590) is referred to as the (generalized) 120574-orderLognormal distribution (120574-GLD) Like the usual Lognormaldistribution the parameter 120583 isin R is considered to belog-scaled while the non log-scaled 120583 (ie 119890120583 when 120583 isassumed log-scaled) is referred to as the location parameter ofLN120574(120583 120590) Hence if119883 sim LN120574(120583 120590) then log119883 is a 120574-ordernormally distributed variable that is log119883 sim N120574(120583 120590

2)

Therefore the location parameter 120583 isin R of 119883 is in fact themean of119883rsquos natural logarithm that is E[log119883] = 120583 while

Var [log119883] = (120574

120574 minus 1)

2((120574minus1)120574)Γ (3 ((120574 minus 1) 120574))

Γ (((120574 minus 1) 120574))1205902 (8)

Kurt [log119883] =Γ (120574 minus 1120574) Γ (5 (120574 minus 1120574))

Γ2 (3 ((120574 minus 1)120574)) (9)

see [15] for details onN120574Let 119884 = log119883 sim N120574(120583 120590

2) with density function as in

(1) and 119883 = 119892(119884) = 119890119884 Then the density function 119891119883 of

119883 sim LN120574(120583 120590) can be written through (1) as

119891119883 (119909) = 119891119883 (119909 120583 120590 120574) = 119891119884 (119892minus1(119909))

10038161003816100381610038161003816100381610038161003816

119889

119889119909119892minus1(119909)

10038161003816100381610038161003816100381610038161003816

= 119891119884 (log119909)1

119909

=

exp minus ((120574 minus 1) 120574) 1003816100381610038161003816(log119909 minus 120583) 1205901003816100381610038161003816120574(120574minus1)

2120590 ((120574 minus 1)120574)1120574Γ ((120574 minus 1) 120574) 119909

(10)

The probability density function 119891119883 as in (10) is defined inRlowast+= R+0 that isLN120574(120583 120590) has zero thresholdTherefore

the following definition extends Definition 1

Definition 2 When the logarithm of a random variable 119883 +

120599 follows the univariate 120574-order normal distribution that islog(119883 + 120599) sim N120574(120583 120590

2) then 119883 is said to follow the genera-

lized Lognormal distribution with threshold 120599 isin R that is119883 sim LN120574(120583 120590 120599)

It is clear that when 119883 sim LN120574(120583 120590 120599 ) log(119883 minus 120599) is a120574-order normally distributed variable that is log(119883 minus 120599) sim

N120574(120583 1205902) and thus 120583 is the mean of (119883 minus 120599)rsquos natural

logarithm while Var[log119883] is the same as in (8)Let 119884 = log(119883 + 120599) sim N120574(120583 120590

2) The density function of

119883 = 119890119884minus 120599 sim LN120574(120583 120590 120599) is given by 119891119883(119909) = 119891119883(119909 minus 120599)

119909 gt 0



+ implies that

lim119909rarr120599+

119891119883 (119909)

= 120590minus11198621


expminus119911(120590 +120583


120574|119911|1(120574minus1)

)

= 120590minus11198621

120574119890(sgn 120574)(minusinfin)

(11)

and therefore

lim119909rarr120599+

119891119883 (119909) = 0 120574 isin (1 +infin)





LN120574 (120583 120590) =

D (119890120583) 120574 = 0

LU (119890120583minus120590

119890120583+120590

) 120574 = 1

LN (120583 120590) 120574 = 2

LL(1198901205831

1205901

120590) 120574 = plusmninfin

(13)






1198911198831

(119909) = lim120574rarr1+

119891119883120574

(119909)

=

1

2120590119909 119909 isin [119890

120583minus120590 119890120583+120590

]


) cup (119890120583+120590

+infin)

(14)



1

(0) = lim119909rarr0+1198911198831



2





119891119883120574

(119909) =1

2120590119909expminus

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

=

119890minus120583120590

2120590119909(1minus120590)120590

119909 isin (0 119890120583]

119890120583120590

2120590119909minus(120590+1)120590

119909 gt 119890120583

(15)


1015840 120572 120573) with 1205831015840 = 119890

120583



(0) = lim119909rarr0+


(119909) =

0 120590 lt 1

1 120590 = 1

+infin 120590 gt 1

(16)



lim120574rarr0minus

119891119883120574


119891119883120574

(119909) 119909 isin Rlowast

+ (17)


lim120574rarr0minus

119891119883120574

(119890120583) =

1

2120590119890120583( lim119896rarrinfin

119896119896

119896) sdot 1198900= +infin (18)




1198911198830


119891119883120574

(119909) =

1003816100381610038161003816log119909 minus 1205831003816100381610038161003816

2radic21205871205902119909sdot 0 sdot

1

119890= 0 (19)


as 1198911198830


0

(0) = lim119909rarr0+1198911198830

(119909) = 0)





120574


120574


+infin

(0) = (12)119890minus120583


120583)


120583120574 isin (1 +infin) is

given by

Mode119883120574 = 119890120583minus120590120574

(20)


max119891119883120574

= 119891119883120574

(Mode119883120574)

=exp (120590120574) 120574 minus 120583

2120590((120574 minus 1)120574)1120574Γ ((120574 minus 1) 120574)

(21)


(119898 120583 120590 120574) = 0 that is

(1

120590

1003816100381610038161003816log119898 minus 1205831003816100381610038161003816)

1(120574minus1)119889

119889119909

1003816100381610038161003816log119909 minus 1205831003816100381610038161003816119909=119898 = minus

120590

119898 (22)


119898 = 119890120583minus120590120574

(23)



120574

(119909) gt 0

when


gt 0 119909 gt 0

(24)

and thus 119891119883120574



119890120583) when 120574 lt 0

Similarly with (119889119889119909)119891119883120574


120583minus120590120574




lim119909rarr119890120583

119889

119889119909119891119883120574

(119909) = minus

1198621

120574

(120590119890120583)2[120590 + lim119909rarr119890120583

sgn (log119909 minus 120583)

times

10038161003816100381610038161003816100381610038161003816

120590

log119909 minus 120583

10038161003816100381610038161003816100381610038161003816

1(1minus120574)

] = +infin

(25)



119891119883120574

(119890120583) =

1

2120590((120574 minus 1)120574)1120574Γ ((120574 minus 1) 120574) 119890120583

(26)

while 119890120583minus120590120574


120574

(119890120583minus120590120574

)


120574




120574

(119890120583minus120590120574







max1198911198831

= 1198911198831

(119886) =119890120590minus120583

2120590lim120574rarr1+

((120574 minus 1) 120574)(120574minus1)120574

Γ (((120574 minus 1) 120574) + 1)

=1

119886 log (119887119886)

(27)


1

= 1198911198831



Mode119883+infin =

0 120590 lt 1

120583

119890120590 = 1

120583 120590 gt 1

(28)


120574

= 119891119883120574



eminus23 e23

0

05

1

15

x

0 1 2 3


2X sim ℒ119977(0 23)

0

05

1

15

x

0 1 2 3 4

eminus1 e1




(b3)

(b2)

(b1)

(a3)

(a2)

(a1)

x

0 1 20

05

1

2

15

eminus32


0

05

1

15

x

0 1 2 31e


x

0 1 2 30

05

1

15


x

01 2 30

05

1

15


X1112 19

X1112 19

X3410

X3410

X1112 19

X3410






120574ge1(023)

fX120574sim119977

120574lt0(023)

fX120574sim119977

120574ge1(01)

fX120574sim119977

120574ge1(032)

fX120574sim119977

120574lt0(01)

fX120574sim119977

120574lt0(032)

fX120574sim119977


(0 32)X2 sim ℒ119977




max119891119883+infin











120590minus120583 to (12120590)119890minus120583)


119889

119889120574Mode119883120574 = minus120590

120574 log (120590) 119890120583minus120590120574

(29)


Mode119883+infin =

119890120583 120590 lt 1

119890120583minus1

120590 = 1

0 120590 gt 1

(30)


120574

= 119891119883120574


+infin

= 1(2120590119890120583) provided 120590 le 1










120583minus120590120574


120583minus120590120574

|120574rarrminusinfin = 119890120583


120574

=


120574




120574

of 119883120574lt0 with 120590 gt 1


120574





120574

on 119891119883120574


120574


120574



Theorem 8 The cdf 119865119883120574


119865119883120574

(119909)

=1

2+

radic120587

2Γ ((120574 minus 1) 120574) Γ (120574 (120574 minus 1))


120574)

(120574minus1)120574 log119909 minus 120583120590

(31)

= 1 minus1

2Γ ((120574 minus 1) 120574)Γ(

120574 minus 1

120574120574 minus 1

120574(log119909 minus 120583

120590)

120574(120574minus1)

)

119909 isin Rlowast

+

(32)


119865119883120574

(119909) = 119865119883120574

(119909 120583 120590 120574) = int

119909

0

119891119883120574

(119905) 119889119905

= 120590minus11198621

120574int

119909

0


120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log 119905 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119905

(33)



119865119883120574

(119909) = 1198621

120574int

(log 119909minus120583)120590

minusinfin


120574|119908|120574(120574minus1)

119889119908

= Φ119885120574

(log119909 minus 120583

120590)

(34)

where Φ119885120574



Φ119885120574

(119911) = 1198621

120574int

119911

minusinfin


120574|119908|120574(120574minus1)

119889119908

= Φ119885120574

(0) + 1198621

120574int

119911

0


120574|119908|120574(120574minus1)

119889119908

(35)

and as 119891119885120574


Φ119885120574

(119911) =1

2+ 1198621

120574int

119911

0


120574|119908|120574(120574minus1)

119889119908

=1

2+ 1198621

120574int

119911

0

exp

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(120574 minus 1

120574)

(120574minus1)120574

119908

1003816100381610038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119908

(36)

and thus

Φ119885120574

(119911) =1

2+ 1198621

120574(

120574

120574 minus 1)

(120574minus1)120574

times int

((120574minus1)120574)(120574minus1)120574

119911

0

exp minus119906120574(120574minus1) 119889119906

(37)


Φ119885120574

(119911) =1

2+

radic120587



120574)

(120574minus1)120574

119911 119911 isin R

(38)



119865119883120574

(119909) =1

2+



120574)

(120574minus1)120574 10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

(39)


119865119883120574

(119909) =1 + sgn (log119909 minus 120583)



120574120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

)

(40)


minus1 sdot)




119865119883120574

(119909) =1

2+

((120574 minus 1)120574)(120574minus1)120574

(2120574) Γ ((120574 minus 1) 120574)(log119909 minus 120583

120590)

times

infin

sum

119896=0

(((1 minus 120574)120574)1003816100381610038161003816(log119909 minus 120583)120590

1003816100381610038161003816120574(120574minus1)

)119896

119896 [(119896 + 1) 120574 minus 1]

119909 isin Rlowast

+

(41)



119865119883120574

(119909)=1

2+1


sgn (log119909 minus 120583)

2 exp 1198991003816100381610038161003816(log119909 minus 120583)12059010038161003816100381610038161119899

times

119899minus1

sum

119896=0

119899119896

119896

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

119896119899

(42)



sum

119896=0

119909119896

119896 119909 isin R 119899 isin N

lowast= N 0

(43)



119865119883minus1

(119909) =1

2+1


times

1 + 2radic1003816100381610038161003816(log119909 minus 120583) 120590

1003816100381610038161003816


(44)




119865119883minus12

(119909) =1

2+1


times

1 + 33radic1003816100381610038161003816(log119909 minus 120583) 120590

1003816100381610038161003816 + 93

radic((log119909 minus 120583)120590)2


(45)


1198651198832

(119909) = Φ1198832

(log119909 minus 120583

120590) =

1

2+1

2Erf2 (

log119909 minus 120583radic2120590

)

=1

2+1

2erf (

log119909 minus 120583radic2120590

)

(46)





(119909)

=1

2+1

2radic120587Erf1 (

log119909 minus 120583120590

)

=1

2minus1

2sgn (log119909 minus 120583)

infin

sum

119896=0

1

(119896 + 1)(minus

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816)

119896+1

=1

2+1


2sgn (log119909 minus 120583)

times expminus10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

(47)

and hence


(119909) =

1

2119890120583120590

1199091120590 119909 isin (0 119890

120583]

1 minus119890120583120590

21199091120590 119909 isin (119890

120583 +infin)

(48)


120583 1120590 1120590) see

Theorem 3



119865119883120574

(119909)

=1

2+

((120574 minus 1)120574)(120574minus1)120574

2Γ (((120574 minus 1) 120574) + 1)(log119909 minus 120583

120590)

times[[

[

1 + (120574 minus 1)

infin

sum

119896=1

(((1 minus 120574)120574)1003816100381610038161003816(log119909 minus 120583)120590

1003816100381610038161003816120574(120574minus1)

)119896

119896 [(119896 + 1) 120574 minus 1]

]]

]

(49)


1198651198831

(119909) = lim120574rarr1+

119865119883120574

(119909) =1

2+log119909 minus 120583

2120590(1 + 0)

=log119909 minus 120583 + 120590

2120590

(50)

with 1198651198831

(119890120583minus120590

) = 0 and 1198651198831

(119890120583+120590

) = 1 Therefore

1198651198831

(119909) =

0 119909 isin (0 119890120583minus120590

)

1

2120590(log119909 minus 120583 + 120590) 119909 isin [119890

120583minus120590 119890120583+120590

]

1 119909 isin (119890120583+120590

+infin)

(51)


120583minus120590 119887 = 119890


LN1(120583 120590) = LU(119890120583minus120590

119890120583+120590

) see Theorem 3


120583|120583=0 =


120574


Q119883120574


+| 119865119883

120574

(119909) ge 119875

= exp sgn (2119875 minus 1) 120590

times [120574


120574 |2119875 minus 1|)]

(120574minus1)120574

119875 isin (0 1)

(52)






120574 11987512057412 1198751205741 1198751205742 1198751205743 1198751205744 1198751205745 1199021120574 1199023120574



120574

(14) and Q119883120574





119883120574


mean (120583119892)119883120574

= 119890E[log119883


120574

= 119890120583 as



120583sim




119891|119884| (119909) = 119891119884 (minus119909) + 119891119884 (119909) 119909 isin R+ (53)


119891|119884| (119909) = 119891119884 (minus119909) + 119891119884 (119909) 0 le 119909 le 119890

120583

119891119884 (119909) 119909 gt 119890120583

(54)


119865|119884| (119909) = int

119909

0

119891|119884| (119905) 119889119905 = int

119890120583

0

119891119884 (minus119905) 119889119905 + int

119890120583

0

119891119884 (119905) 119889119905

+ int

119909

119890120583119891119884 (119905) 119889119905 119909 isin R+

(55)



119865|119884| (119909) = 1198621

120574int

0

minusinfin


120574|119908|120574(120574minus1)

119889119908 + 1198621

120574

times int

(1120590) log 2

0


120574|119911|120574(120574minus1)

119889119911

+ 1198621

120574int

119892(119909)


120574 minus 1

120574|119911|120574(120574minus1)

119889119911

119892 (119909) =log (119909 + 119890120583) minus 120583

120590 119909 isin R+

(56)

and hence

119865|119884| (119909) = Φ119885 (0) + [Φ119885 (log 2120590

) minus Φ119885 (0)]

+ [Φ119885 (119892 (119909)) minus Φ119885 (log 2120590

)]

= Φ119885(log (119909 + 119890120583) minus 120583

120590)

(57)


x

0 1 2 3

X2simℒ119977(0 23)


sim119977

120574(023)

eminus23 e23

0

01

02

03

04

05

06

07

08

09

1

X1112 19

X3410



FX120574



(a)

120574(01)

0

01

02

03

04

05

06

07

08

09

1

X2simℒ119977(0 1)



x

0 1 2 3


X1112 19

X3410

eminus1 e1

FX120574sim119977

(b)

00

01

02

03

04

05

06

07

08

09

1

x

1 2 3

eminus32


X2simℒ119977(0 32)

X1112 19

X3410


119977120574(032)

FX120574sim119977



(c)




+ it is clear that


|119883120574minus119890120583|







120583(119905)

119883=

119890119905120583

Γ ((120574 minus 1) 120574)

infin

sum

119899=0

(119905120590)2119899

(2119899)(

120574

120574 minus 1)

2119899((120574minus1)120574)

times Γ((2119899 + 1)120574 minus 1

120574)

(58)


(119905)

119883


have

120583(119905)

119883= E [119883119905] = int

R+

119909119905119891119883 (119909) 119889119909

=1

1205901198621

120574intR+


120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119909

(59)


(1120590)


120583(119905)

119883= 1198621

120574(

120574

120574 minus 1)

(120574minus1)120574

intR

exp119905120583 + 119899(120574

120574 minus 1)

(120574minus1)120574

120590119911



exp119905(120574

120574 minus 1)

(120574minus1)120574

120590119911 =

infin

sum

119899=0

(119905120590)119899

119899(

120574

120574 minus 1)

119899((120574minus1)120574)

119911119899

(61)

it is obtained that

120583(119905)

119883= 21198621

120574(

120574

120574 minus 1)

(120574minus1)120574

119890119905120583infin

sum

119899=0

(119905120590)2119899

(2119899)(

120574

120574 minus 1)

2119899((120574minus1)120574)

times intR+

1199112119899 exp minus119911120574(120574minus1) 119889119911

(62)



intR+

119909119898119890minus119887119909119899

119889119909 =Γ ((119898 + 1) 119899)

119899119887(119898+1)119899 119899 119898 119887 isin R

lowast

+ (63)


2) we have 119872119884(119905) =

E[119890119905119884] = E[119883119905] = 120583(119905)



120583(119905)

119883=

119890119905120583

radic120587

infin

sum

119899=0

(211990521205902)119899

(2119899)Γ (119899 +

1

2) 119905 isin R+ (64)


Γ (119899 +1

2) =

(2119899)

22119899119899radic120587 119899 isin N (65)

we have

120583(119905)

119883= 119890119905120583infin

sum

119899=0

(119905120590)2119899

2119899119899= 119890119905120583infin

sum

119899=0

1

119899(1

211990521205902)

119899

= 119890119905120583+(12)(119905120590)

2

119905 isin R+

(66)


(1)

119883= E[119883] =


(119905)

119883= exp119905120583 +





LN120574(120583 120590) is given by

120583(119896)

119883=

119890119896120583

Γ ((120574 minus 1) 120574)

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899

119878119896minus119899 119896 isin N (67)

where

119878119896 =

infin

sum

119898=0

(119896120590)2119898

(2119898)(

120574

120574 minus 1)

2119898((120574minus1)120574)

Γ((2119898 + 1)120574 minus 1

120574)

119896 isin N

(68)


wehave

120583(119896)

119883= E [(119883 minus 120583119883)

119896] = int

R+

(119909 minus 120583119883)119896119891119883 (119909 120583 120590 120574) 119889119909

(69)


120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus120583119883)

119899intR+

119909119896minus119899

119891119883 (119909) 119889119909

=

119896

sum

119899=0

(119896

119899) (minus120583119883)

119899120583(119896minus119899)

119883

(70)


120583(119896)

119883=

119890119896120583

Γ ((120574 minus 1) 120574)

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898)

times (120574

120574 minus 1)

2119898((120574minus1)120574)

Γ ((2119898 + 1) ((120574 minus 1) 120574))

(71)




120583+(12)1205902


120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus1)

119899119890119896120583+(12)[119899+(119896minus119899)

2

]1205902

119896 isin N (72)

while

1205902

119883= Var [119883] = 120583

(2)

119883= 1198902120583+1205902

(1198901205902

minus 1) (73)





120583119883 =119890120583

Γ ((120574 minus 1) 120574)1198781 (74)

1205902

119883= minus 120583

2

119883+

1198902120583

Γ ((120574 minus 1) 120574)1198782 (75)

1198621198812

119883= Γ(

120574 minus 1

120574)1198782

11987821

minus 1 (76)

120582119883 = minus 119862119881minus3

119883minus 119862119881minus1

119883+

1198903120583

1205903119883Γ ((120574 minus 1) 120574)

1198783 (77)

120581119883 = minus 119862119881minus4

119883minus 6119862119881

minus2

119883minus 4

120582119883

119862119881119883

+1198904120583

1205904119883Γ ((120574 minus 1) 120574)

1198784

(78)



120583(1)


1205902

119883= 120583(2)

119883= 1205832

119883+ [Γ(

120574 minus 1

120574)]

minus1

(11989021205831198782 minus 2119890

1205831205831198831198781) (79)


(3)

1198831205903

119883


120582119883 = minus119862119881minus3

119883+ [1205903

119883Γ(

120574 minus 1

120574)]

minus1

times (11989031205831198783 minus 3119890

21205831205831198831198782 + 3119890

1205831205831198831198781)

(80)


(4)

1198831205904

119883 which


120581119883 = 119862119881minus4

119883+ [1205904

119883Γ(

120574 minus 1

120574)]

minus1

times (11989041205831198784 minus 4119890

31205831205831198831198783 + 6119890

21205831205832

1198831198782 minus 4119890

1205831205833

1198831198781)

(81)



radic120587119890(1198992

1205902


120583119883 = 119890120583+(12)120590

2

1205902

119883= 1198902120583+1205902

(1198901205902

minus 1)

119862119881119883 =radic1198901205902

minus 1

120582119883 = (1198901205902

+ 2)radic1198901205902

minus 1 120581119883 = 11989041205902

+ 211989031205902

+ 311989021205902

minus 3

(82)




120574

le

120583119883120574



Med119883120574 = (120583119892)119883120574

= 119890120583lt 120583119883

120574

(83)



Mode119883120574 = 119890120583minus120590120574

lt 119890120583= Med119883120574 (84)


Mode119883120574 = 119890120583= Med119883120574 (85)



Mode119883120574 = 0 lt 119890120583= Med119883120574 (86)





120583(119905)

119883=

119887119905minus 119886119905

119905 log (119887119886) 119905 isin R (87)

120583(119896)

119883=

(119886 minus 119887)119896

log119896 (119887119886)+

1

log (119887119886)

times

119896minus1

sum

119899=0

(119896

119899)(119886 minus 119887)

119899(119887119896minus119899

minus 119886119896minus119899

)

(119896 minus 119899) log119899 (119887119886) 119896 isin N

(88)


120583119883 =119887 minus 119886

log (119887119886) (89)

1205902

119883=

(119887 minus 119886)2

log2 (119887119886)+(119887 minus 119886) (119887 + 119886)

2 log (119887119886) (90)

119862119881119883 = radic1 +119887 + 119886

2 (119887 minus 119886)log 119887

119886 (91)

120582119883 =1

1205903119883

[1198873minus 1198863

3 log (119887119886)minus 3

(119887 minus 119886)2(119887 + 119886)

2 log2 (119887119886)+ 2

(119887 minus 119886)3

log3 (119887119886)]

(92)

120581119883 =1

1205904119883

[1198874minus 1198864

4 log (119887119886)minus 4

(119887 minus 119886) (1198873minus 1198863)

3 log2 (119887119886)

+3(119887 minus 119886)

3(119887 + 119886)

log3 (119887119886)minus 3

(119887 minus 119886)4

log4 (119887119886)]

(93)


120583(119905)

119883120574

=119890119905120583

119905120590Γ ((120574 minus 1) 120574 + 1)

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)(

120574

120574 minus 1)

2119898((120574minus1)120574)

times Γ((2119898 + 1)120574 minus 1

120574+ 1)

(94)


120583(119905)

119883= lim120574rarr1+

120583(119905)

119883120574

=119890119905120583

119905120590

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)=119890119905(120583+120590)

minus 119890119905(120583minus120590)

2119905120590

119905 isin R

(95)


119883= E[119883] = (12120590)

(119890120583+120590

minus 119890120583minus120590


120583(119896)

119883= lim120574rarr1+

120583(119896)

119883120574

= 119890119896120583119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898 + 1)

119896 isin N

(96)


120583(119896)

119883=119890119896120583

2120590

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 119890(119896minus119899)120590


(119896 minus 119899) 119896 isin N

(97)


Var[119883] = 120583(2)

119883minus 1205832


For 119896 = 3 and 119896 = 4 through 120583(3)119883

and 120583(4)119883 we obtain (92) and

(93) respectively


120583(119905)

119883=

1205831199051205902

1205902 minus 1199052gt 120583119905 120590 gt 119905 119905 isin R (98)

120583(119896)

119883= 120583119896119896

sum

119899=0

(119896

119899)

1205902(119899+1)


2] 120590 gt 119896 119896 isin N

(99)


120583119883 =1205831205902

1205902 minus 1gt 120583 120590 gt 1 (100)

1205902

119883=

12058321205902(21205902+ 1)

(1205902 minus 4) (1205902 minus 1)2 120590 gt 2 (101)

119862119881119883 =1

120590

radic21205902+ 1

1205902 minus 4 120590 gt 2 (102)

120582119883 =2 (15120590

4+ 71205902+ 2)

120590 (1205902 minus 9)radic

1205902minus 4

(21205902 + 1)3 120590 gt 3 (103)


120581119883 =3 (81205908+ 212120590

6+ 95120590

4+ 33120590

2+ 12) (120590

2minus 4)

(1205902 minus 16) (1205902 minus 9) (21205902 + 1)2

120590 gt 4

(104)


120583(119905)

119883= 120583(119905)

119883plusmninfin

= 120583119905infin

sum

119896=0

(119905

120590)

2119896

119905 isin R (105)


120583(119905)

119883plusmninfin

=1

2120583119905[

infin

sum

119896=0

(119905

120590)

119896

+

infin

sum

119896=0

(minus119905

120590)

119896

]

=1

2120583119905(

120590

120590 minus 119905+

120590

120590 + 119905)

(106)



120583(119896)

119883= 1205831198961205902119896

sum

119899=0

(119896

119899)

(minus120583119883120583)119899




(2)

119883minus 1205832

119883


119883and 120583(4)




119880sim LU(119890

120583minus120590 119890120583+120590


119886 119890119887)


(119905)

LU = (119890119905119887minus 119890119905119886)(1119905(119887 minus 119886))



120583 1120590 1120590) due to


(119905)

LL = 119890119905120590(1 minus 11990521205902)minus1

4 Conclusion



Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











+ implies that

lim119909rarr120599+

119891119883 (119909)

= 120590minus11198621


expminus119911(120590 +120583


120574|119911|1(120574minus1)

)

= 120590minus11198621

120574119890(sgn 120574)(minusinfin)

(11)

and therefore

lim119909rarr120599+

119891119883 (119909) = 0 120574 isin (1 +infin)





LN120574 (120583 120590) =

D (119890120583) 120574 = 0

LU (119890120583minus120590

119890120583+120590

) 120574 = 1

LN (120583 120590) 120574 = 2

LL(1198901205831

1205901

120590) 120574 = plusmninfin

(13)






1198911198831

(119909) = lim120574rarr1+

119891119883120574

(119909)

=

1

2120590119909 119909 isin [119890

120583minus120590 119890120583+120590

]


) cup (119890120583+120590

+infin)

(14)



1

(0) = lim119909rarr0+1198911198831



2





119891119883120574

(119909) =1

2120590119909expminus

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

=

119890minus120583120590

2120590119909(1minus120590)120590

119909 isin (0 119890120583]

119890120583120590

2120590119909minus(120590+1)120590

119909 gt 119890120583

(15)


1015840 120572 120573) with 1205831015840 = 119890

120583



(0) = lim119909rarr0+


(119909) =

0 120590 lt 1

1 120590 = 1

+infin 120590 gt 1

(16)



lim120574rarr0minus

119891119883120574


119891119883120574

(119909) 119909 isin Rlowast

+ (17)


lim120574rarr0minus

119891119883120574

(119890120583) =

1

2120590119890120583( lim119896rarrinfin

119896119896

119896) sdot 1198900= +infin (18)




1198911198830


119891119883120574

(119909) =

1003816100381610038161003816log119909 minus 1205831003816100381610038161003816

2radic21205871205902119909sdot 0 sdot

1

119890= 0 (19)


as 1198911198830


0

(0) = lim119909rarr0+1198911198830

(119909) = 0)





120574


120574


+infin

(0) = (12)119890minus120583


120583)


120583120574 isin (1 +infin) is

given by

Mode119883120574 = 119890120583minus120590120574

(20)


max119891119883120574

= 119891119883120574

(Mode119883120574)

=exp (120590120574) 120574 minus 120583

2120590((120574 minus 1)120574)1120574Γ ((120574 minus 1) 120574)

(21)


(119898 120583 120590 120574) = 0 that is

(1

120590

1003816100381610038161003816log119898 minus 1205831003816100381610038161003816)

1(120574minus1)119889

119889119909

1003816100381610038161003816log119909 minus 1205831003816100381610038161003816119909=119898 = minus

120590

119898 (22)


119898 = 119890120583minus120590120574

(23)



120574

(119909) gt 0

when


gt 0 119909 gt 0

(24)

and thus 119891119883120574



119890120583) when 120574 lt 0

Similarly with (119889119889119909)119891119883120574


120583minus120590120574




lim119909rarr119890120583

119889

119889119909119891119883120574

(119909) = minus

1198621

120574

(120590119890120583)2[120590 + lim119909rarr119890120583

sgn (log119909 minus 120583)

times

10038161003816100381610038161003816100381610038161003816

120590

log119909 minus 120583

10038161003816100381610038161003816100381610038161003816

1(1minus120574)

] = +infin

(25)



119891119883120574

(119890120583) =

1

2120590((120574 minus 1)120574)1120574Γ ((120574 minus 1) 120574) 119890120583

(26)

while 119890120583minus120590120574


120574

(119890120583minus120590120574

)


120574




120574

(119890120583minus120590120574







max1198911198831

= 1198911198831

(119886) =119890120590minus120583

2120590lim120574rarr1+

((120574 minus 1) 120574)(120574minus1)120574

Γ (((120574 minus 1) 120574) + 1)

=1

119886 log (119887119886)

(27)


1

= 1198911198831



Mode119883+infin =

0 120590 lt 1

120583

119890120590 = 1

120583 120590 gt 1

(28)


120574

= 119891119883120574



eminus23 e23

0

05

1

15

x

0 1 2 3


2X sim ℒ119977(0 23)

0

05

1

15

x

0 1 2 3 4

eminus1 e1




(b3)

(b2)

(b1)

(a3)

(a2)

(a1)

x

0 1 20

05

1

2

15

eminus32


0

05

1

15

x

0 1 2 31e


x

0 1 2 30

05

1

15


x

01 2 30

05

1

15


X1112 19

X1112 19

X3410

X3410

X1112 19

X3410






120574ge1(023)

fX120574sim119977

120574lt0(023)

fX120574sim119977

120574ge1(01)

fX120574sim119977

120574ge1(032)

fX120574sim119977

120574lt0(01)

fX120574sim119977

120574lt0(032)

fX120574sim119977


(0 32)X2 sim ℒ119977




max119891119883+infin











120590minus120583 to (12120590)119890minus120583)


119889

119889120574Mode119883120574 = minus120590

120574 log (120590) 119890120583minus120590120574

(29)


Mode119883+infin =

119890120583 120590 lt 1

119890120583minus1

120590 = 1

0 120590 gt 1

(30)


120574

= 119891119883120574


+infin

= 1(2120590119890120583) provided 120590 le 1










120583minus120590120574


120583minus120590120574

|120574rarrminusinfin = 119890120583


120574

=


120574




120574

of 119883120574lt0 with 120590 gt 1


120574





120574

on 119891119883120574


120574


120574



Theorem 8 The cdf 119865119883120574


119865119883120574

(119909)

=1

2+

radic120587

2Γ ((120574 minus 1) 120574) Γ (120574 (120574 minus 1))


120574)

(120574minus1)120574 log119909 minus 120583120590

(31)

= 1 minus1

2Γ ((120574 minus 1) 120574)Γ(

120574 minus 1

120574120574 minus 1

120574(log119909 minus 120583

120590)

120574(120574minus1)

)

119909 isin Rlowast

+

(32)


119865119883120574

(119909) = 119865119883120574

(119909 120583 120590 120574) = int

119909

0

119891119883120574

(119905) 119889119905

= 120590minus11198621

120574int

119909

0


120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log 119905 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119905

(33)



119865119883120574

(119909) = 1198621

120574int

(log 119909minus120583)120590

minusinfin


120574|119908|120574(120574minus1)

119889119908

= Φ119885120574

(log119909 minus 120583

120590)

(34)

where Φ119885120574



Φ119885120574

(119911) = 1198621

120574int

119911

minusinfin


120574|119908|120574(120574minus1)

119889119908

= Φ119885120574

(0) + 1198621

120574int

119911

0


120574|119908|120574(120574minus1)

119889119908

(35)

and as 119891119885120574


Φ119885120574

(119911) =1

2+ 1198621

120574int

119911

0


120574|119908|120574(120574minus1)

119889119908

=1

2+ 1198621

120574int

119911

0

exp

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(120574 minus 1

120574)

(120574minus1)120574

119908

1003816100381610038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119908

(36)

and thus

Φ119885120574

(119911) =1

2+ 1198621

120574(

120574

120574 minus 1)

(120574minus1)120574

times int

((120574minus1)120574)(120574minus1)120574

119911

0

exp minus119906120574(120574minus1) 119889119906

(37)


Φ119885120574

(119911) =1

2+

radic120587



120574)

(120574minus1)120574

119911 119911 isin R

(38)



119865119883120574

(119909) =1

2+



120574)

(120574minus1)120574 10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

(39)


119865119883120574

(119909) =1 + sgn (log119909 minus 120583)



120574120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

)

(40)


minus1 sdot)




119865119883120574

(119909) =1

2+

((120574 minus 1)120574)(120574minus1)120574

(2120574) Γ ((120574 minus 1) 120574)(log119909 minus 120583

120590)

times

infin

sum

119896=0

(((1 minus 120574)120574)1003816100381610038161003816(log119909 minus 120583)120590

1003816100381610038161003816120574(120574minus1)

)119896

119896 [(119896 + 1) 120574 minus 1]

119909 isin Rlowast

+

(41)



119865119883120574

(119909)=1

2+1


sgn (log119909 minus 120583)

2 exp 1198991003816100381610038161003816(log119909 minus 120583)12059010038161003816100381610038161119899

times

119899minus1

sum

119896=0

119899119896

119896

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

119896119899

(42)



sum

119896=0

119909119896

119896 119909 isin R 119899 isin N

lowast= N 0

(43)



119865119883minus1

(119909) =1

2+1


times

1 + 2radic1003816100381610038161003816(log119909 minus 120583) 120590

1003816100381610038161003816


(44)




119865119883minus12

(119909) =1

2+1


times

1 + 33radic1003816100381610038161003816(log119909 minus 120583) 120590

1003816100381610038161003816 + 93

radic((log119909 minus 120583)120590)2


(45)


1198651198832

(119909) = Φ1198832

(log119909 minus 120583

120590) =

1

2+1

2Erf2 (

log119909 minus 120583radic2120590

)

=1

2+1

2erf (

log119909 minus 120583radic2120590

)

(46)





(119909)

=1

2+1

2radic120587Erf1 (

log119909 minus 120583120590

)

=1

2minus1

2sgn (log119909 minus 120583)

infin

sum

119896=0

1

(119896 + 1)(minus

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816)

119896+1

=1

2+1


2sgn (log119909 minus 120583)

times expminus10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

(47)

and hence


(119909) =

1

2119890120583120590

1199091120590 119909 isin (0 119890

120583]

1 minus119890120583120590

21199091120590 119909 isin (119890

120583 +infin)

(48)


120583 1120590 1120590) see

Theorem 3



119865119883120574

(119909)

=1

2+

((120574 minus 1)120574)(120574minus1)120574

2Γ (((120574 minus 1) 120574) + 1)(log119909 minus 120583

120590)

times[[

[

1 + (120574 minus 1)

infin

sum

119896=1

(((1 minus 120574)120574)1003816100381610038161003816(log119909 minus 120583)120590

1003816100381610038161003816120574(120574minus1)

)119896

119896 [(119896 + 1) 120574 minus 1]

]]

]

(49)


1198651198831

(119909) = lim120574rarr1+

119865119883120574

(119909) =1

2+log119909 minus 120583

2120590(1 + 0)

=log119909 minus 120583 + 120590

2120590

(50)

with 1198651198831

(119890120583minus120590

) = 0 and 1198651198831

(119890120583+120590

) = 1 Therefore

1198651198831

(119909) =

0 119909 isin (0 119890120583minus120590

)

1

2120590(log119909 minus 120583 + 120590) 119909 isin [119890

120583minus120590 119890120583+120590

]

1 119909 isin (119890120583+120590

+infin)

(51)


120583minus120590 119887 = 119890


LN1(120583 120590) = LU(119890120583minus120590

119890120583+120590

) see Theorem 3


120583|120583=0 =


120574


Q119883120574


+| 119865119883

120574

(119909) ge 119875

= exp sgn (2119875 minus 1) 120590

times [120574


120574 |2119875 minus 1|)]

(120574minus1)120574

119875 isin (0 1)

(52)






120574 11987512057412 1198751205741 1198751205742 1198751205743 1198751205744 1198751205745 1199021120574 1199023120574



120574

(14) and Q119883120574





119883120574


mean (120583119892)119883120574

= 119890E[log119883


120574

= 119890120583 as



120583sim




119891|119884| (119909) = 119891119884 (minus119909) + 119891119884 (119909) 119909 isin R+ (53)


119891|119884| (119909) = 119891119884 (minus119909) + 119891119884 (119909) 0 le 119909 le 119890

120583

119891119884 (119909) 119909 gt 119890120583

(54)


119865|119884| (119909) = int

119909

0

119891|119884| (119905) 119889119905 = int

119890120583

0

119891119884 (minus119905) 119889119905 + int

119890120583

0

119891119884 (119905) 119889119905

+ int

119909

119890120583119891119884 (119905) 119889119905 119909 isin R+

(55)



119865|119884| (119909) = 1198621

120574int

0

minusinfin


120574|119908|120574(120574minus1)

119889119908 + 1198621

120574

times int

(1120590) log 2

0


120574|119911|120574(120574minus1)

119889119911

+ 1198621

120574int

119892(119909)


120574 minus 1

120574|119911|120574(120574minus1)

119889119911

119892 (119909) =log (119909 + 119890120583) minus 120583

120590 119909 isin R+

(56)

and hence

119865|119884| (119909) = Φ119885 (0) + [Φ119885 (log 2120590

) minus Φ119885 (0)]

+ [Φ119885 (119892 (119909)) minus Φ119885 (log 2120590

)]

= Φ119885(log (119909 + 119890120583) minus 120583

120590)

(57)


x

0 1 2 3

X2simℒ119977(0 23)


sim119977

120574(023)

eminus23 e23

0

01

02

03

04

05

06

07

08

09

1

X1112 19

X3410



FX120574



(a)

120574(01)

0

01

02

03

04

05

06

07

08

09

1

X2simℒ119977(0 1)



x

0 1 2 3


X1112 19

X3410

eminus1 e1

FX120574sim119977

(b)

00

01

02

03

04

05

06

07

08

09

1

x

1 2 3

eminus32


X2simℒ119977(0 32)

X1112 19

X3410


119977120574(032)

FX120574sim119977



(c)




+ it is clear that


|119883120574minus119890120583|







120583(119905)

119883=

119890119905120583

Γ ((120574 minus 1) 120574)

infin

sum

119899=0

(119905120590)2119899

(2119899)(

120574

120574 minus 1)

2119899((120574minus1)120574)

times Γ((2119899 + 1)120574 minus 1

120574)

(58)


(119905)

119883


have

120583(119905)

119883= E [119883119905] = int

R+

119909119905119891119883 (119909) 119889119909

=1

1205901198621

120574intR+


120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119909

(59)


(1120590)


120583(119905)

119883= 1198621

120574(

120574

120574 minus 1)

(120574minus1)120574

intR

exp119905120583 + 119899(120574

120574 minus 1)

(120574minus1)120574

120590119911



exp119905(120574

120574 minus 1)

(120574minus1)120574

120590119911 =

infin

sum

119899=0

(119905120590)119899

119899(

120574

120574 minus 1)

119899((120574minus1)120574)

119911119899

(61)

it is obtained that

120583(119905)

119883= 21198621

120574(

120574

120574 minus 1)

(120574minus1)120574

119890119905120583infin

sum

119899=0

(119905120590)2119899

(2119899)(

120574

120574 minus 1)

2119899((120574minus1)120574)

times intR+

1199112119899 exp minus119911120574(120574minus1) 119889119911

(62)



intR+

119909119898119890minus119887119909119899

119889119909 =Γ ((119898 + 1) 119899)

119899119887(119898+1)119899 119899 119898 119887 isin R

lowast

+ (63)


2) we have 119872119884(119905) =

E[119890119905119884] = E[119883119905] = 120583(119905)



120583(119905)

119883=

119890119905120583

radic120587

infin

sum

119899=0

(211990521205902)119899

(2119899)Γ (119899 +

1

2) 119905 isin R+ (64)


Γ (119899 +1

2) =

(2119899)

22119899119899radic120587 119899 isin N (65)

we have

120583(119905)

119883= 119890119905120583infin

sum

119899=0

(119905120590)2119899

2119899119899= 119890119905120583infin

sum

119899=0

1

119899(1

211990521205902)

119899

= 119890119905120583+(12)(119905120590)

2

119905 isin R+

(66)


(1)

119883= E[119883] =


(119905)

119883= exp119905120583 +





LN120574(120583 120590) is given by

120583(119896)

119883=

119890119896120583

Γ ((120574 minus 1) 120574)

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899

119878119896minus119899 119896 isin N (67)

where

119878119896 =

infin

sum

119898=0

(119896120590)2119898

(2119898)(

120574

120574 minus 1)

2119898((120574minus1)120574)

Γ((2119898 + 1)120574 minus 1

120574)

119896 isin N

(68)


wehave

120583(119896)

119883= E [(119883 minus 120583119883)

119896] = int

R+

(119909 minus 120583119883)119896119891119883 (119909 120583 120590 120574) 119889119909

(69)


120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus120583119883)

119899intR+

119909119896minus119899

119891119883 (119909) 119889119909

=

119896

sum

119899=0

(119896

119899) (minus120583119883)

119899120583(119896minus119899)

119883

(70)


120583(119896)

119883=

119890119896120583

Γ ((120574 minus 1) 120574)

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898)

times (120574

120574 minus 1)

2119898((120574minus1)120574)

Γ ((2119898 + 1) ((120574 minus 1) 120574))

(71)




120583+(12)1205902


120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus1)

119899119890119896120583+(12)[119899+(119896minus119899)

2

]1205902

119896 isin N (72)

while

1205902

119883= Var [119883] = 120583

(2)

119883= 1198902120583+1205902

(1198901205902

minus 1) (73)





120583119883 =119890120583

Γ ((120574 minus 1) 120574)1198781 (74)

1205902

119883= minus 120583

2

119883+

1198902120583

Γ ((120574 minus 1) 120574)1198782 (75)

1198621198812

119883= Γ(

120574 minus 1

120574)1198782

11987821

minus 1 (76)

120582119883 = minus 119862119881minus3

119883minus 119862119881minus1

119883+

1198903120583

1205903119883Γ ((120574 minus 1) 120574)

1198783 (77)

120581119883 = minus 119862119881minus4

119883minus 6119862119881

minus2

119883minus 4

120582119883

119862119881119883

+1198904120583

1205904119883Γ ((120574 minus 1) 120574)

1198784

(78)



120583(1)


1205902

119883= 120583(2)

119883= 1205832

119883+ [Γ(

120574 minus 1

120574)]

minus1

(11989021205831198782 minus 2119890

1205831205831198831198781) (79)


(3)

1198831205903

119883


120582119883 = minus119862119881minus3

119883+ [1205903

119883Γ(

120574 minus 1

120574)]

minus1

times (11989031205831198783 minus 3119890

21205831205831198831198782 + 3119890

1205831205831198831198781)

(80)


(4)

1198831205904

119883 which


120581119883 = 119862119881minus4

119883+ [1205904

119883Γ(

120574 minus 1

120574)]

minus1

times (11989041205831198784 minus 4119890

31205831205831198831198783 + 6119890

21205831205832

1198831198782 minus 4119890

1205831205833

1198831198781)

(81)



radic120587119890(1198992

1205902


120583119883 = 119890120583+(12)120590

2

1205902

119883= 1198902120583+1205902

(1198901205902

minus 1)

119862119881119883 =radic1198901205902

minus 1

120582119883 = (1198901205902

+ 2)radic1198901205902

minus 1 120581119883 = 11989041205902

+ 211989031205902

+ 311989021205902

minus 3

(82)




120574

le

120583119883120574



Med119883120574 = (120583119892)119883120574

= 119890120583lt 120583119883

120574

(83)



Mode119883120574 = 119890120583minus120590120574

lt 119890120583= Med119883120574 (84)


Mode119883120574 = 119890120583= Med119883120574 (85)



Mode119883120574 = 0 lt 119890120583= Med119883120574 (86)





120583(119905)

119883=

119887119905minus 119886119905

119905 log (119887119886) 119905 isin R (87)

120583(119896)

119883=

(119886 minus 119887)119896

log119896 (119887119886)+

1

log (119887119886)

times

119896minus1

sum

119899=0

(119896

119899)(119886 minus 119887)

119899(119887119896minus119899

minus 119886119896minus119899

)

(119896 minus 119899) log119899 (119887119886) 119896 isin N

(88)


120583119883 =119887 minus 119886

log (119887119886) (89)

1205902

119883=

(119887 minus 119886)2

log2 (119887119886)+(119887 minus 119886) (119887 + 119886)

2 log (119887119886) (90)

119862119881119883 = radic1 +119887 + 119886

2 (119887 minus 119886)log 119887

119886 (91)

120582119883 =1

1205903119883

[1198873minus 1198863

3 log (119887119886)minus 3

(119887 minus 119886)2(119887 + 119886)

2 log2 (119887119886)+ 2

(119887 minus 119886)3

log3 (119887119886)]

(92)

120581119883 =1

1205904119883

[1198874minus 1198864

4 log (119887119886)minus 4

(119887 minus 119886) (1198873minus 1198863)

3 log2 (119887119886)

+3(119887 minus 119886)

3(119887 + 119886)

log3 (119887119886)minus 3

(119887 minus 119886)4

log4 (119887119886)]

(93)


120583(119905)

119883120574

=119890119905120583

119905120590Γ ((120574 minus 1) 120574 + 1)

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)(

120574

120574 minus 1)

2119898((120574minus1)120574)

times Γ((2119898 + 1)120574 minus 1

120574+ 1)

(94)


120583(119905)

119883= lim120574rarr1+

120583(119905)

119883120574

=119890119905120583

119905120590

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)=119890119905(120583+120590)

minus 119890119905(120583minus120590)

2119905120590

119905 isin R

(95)


119883= E[119883] = (12120590)

(119890120583+120590

minus 119890120583minus120590


120583(119896)

119883= lim120574rarr1+

120583(119896)

119883120574

= 119890119896120583119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898 + 1)

119896 isin N

(96)


120583(119896)

119883=119890119896120583

2120590

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 119890(119896minus119899)120590


(119896 minus 119899) 119896 isin N

(97)


Var[119883] = 120583(2)

119883minus 1205832


For 119896 = 3 and 119896 = 4 through 120583(3)119883

and 120583(4)119883 we obtain (92) and

(93) respectively


120583(119905)

119883=

1205831199051205902

1205902 minus 1199052gt 120583119905 120590 gt 119905 119905 isin R (98)

120583(119896)

119883= 120583119896119896

sum

119899=0

(119896

119899)

1205902(119899+1)


2] 120590 gt 119896 119896 isin N

(99)


120583119883 =1205831205902

1205902 minus 1gt 120583 120590 gt 1 (100)

1205902

119883=

12058321205902(21205902+ 1)

(1205902 minus 4) (1205902 minus 1)2 120590 gt 2 (101)

119862119881119883 =1

120590

radic21205902+ 1

1205902 minus 4 120590 gt 2 (102)

120582119883 =2 (15120590

4+ 71205902+ 2)

120590 (1205902 minus 9)radic

1205902minus 4

(21205902 + 1)3 120590 gt 3 (103)


120581119883 =3 (81205908+ 212120590

6+ 95120590

4+ 33120590

2+ 12) (120590

2minus 4)

(1205902 minus 16) (1205902 minus 9) (21205902 + 1)2

120590 gt 4

(104)


120583(119905)

119883= 120583(119905)

119883plusmninfin

= 120583119905infin

sum

119896=0

(119905

120590)

2119896

119905 isin R (105)


120583(119905)

119883plusmninfin

=1

2120583119905[

infin

sum

119896=0

(119905

120590)

119896

+

infin

sum

119896=0

(minus119905

120590)

119896

]

=1

2120583119905(

120590

120590 minus 119905+

120590

120590 + 119905)

(106)



120583(119896)

119883= 1205831198961205902119896

sum

119899=0

(119896

119899)

(minus120583119883120583)119899




(2)

119883minus 1205832

119883


119883and 120583(4)




119880sim LU(119890

120583minus120590 119890120583+120590


119886 119890119887)


(119905)

LU = (119890119905119887minus 119890119905119886)(1119905(119887 minus 119886))



120583 1120590 1120590) due to


(119905)

LL = 119890119905120590(1 minus 11990521205902)minus1

4 Conclusion



Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120574


120574


+infin

(0) = (12)119890minus120583


120583)


120583120574 isin (1 +infin) is

given by

Mode119883120574 = 119890120583minus120590120574

(20)


max119891119883120574

= 119891119883120574

(Mode119883120574)

=exp (120590120574) 120574 minus 120583

2120590((120574 minus 1)120574)1120574Γ ((120574 minus 1) 120574)

(21)


(119898 120583 120590 120574) = 0 that is

(1

120590

1003816100381610038161003816log119898 minus 1205831003816100381610038161003816)

1(120574minus1)119889

119889119909

1003816100381610038161003816log119909 minus 1205831003816100381610038161003816119909=119898 = minus

120590

119898 (22)


119898 = 119890120583minus120590120574

(23)



120574

(119909) gt 0

when


gt 0 119909 gt 0

(24)

and thus 119891119883120574



119890120583) when 120574 lt 0

Similarly with (119889119889119909)119891119883120574


120583minus120590120574




lim119909rarr119890120583

119889

119889119909119891119883120574

(119909) = minus

1198621

120574

(120590119890120583)2[120590 + lim119909rarr119890120583

sgn (log119909 minus 120583)

times

10038161003816100381610038161003816100381610038161003816

120590

log119909 minus 120583

10038161003816100381610038161003816100381610038161003816

1(1minus120574)

] = +infin

(25)



119891119883120574

(119890120583) =

1

2120590((120574 minus 1)120574)1120574Γ ((120574 minus 1) 120574) 119890120583

(26)

while 119890120583minus120590120574


120574

(119890120583minus120590120574

)


120574




120574

(119890120583minus120590120574







max1198911198831

= 1198911198831

(119886) =119890120590minus120583

2120590lim120574rarr1+

((120574 minus 1) 120574)(120574minus1)120574

Γ (((120574 minus 1) 120574) + 1)

=1

119886 log (119887119886)

(27)


1

= 1198911198831



Mode119883+infin =

0 120590 lt 1

120583

119890120590 = 1

120583 120590 gt 1

(28)


120574

= 119891119883120574



eminus23 e23

0

05

1

15

x

0 1 2 3


2X sim ℒ119977(0 23)

0

05

1

15

x

0 1 2 3 4

eminus1 e1




(b3)

(b2)

(b1)

(a3)

(a2)

(a1)

x

0 1 20

05

1

2

15

eminus32


0

05

1

15

x

0 1 2 31e


x

0 1 2 30

05

1

15


x

01 2 30

05

1

15


X1112 19

X1112 19

X3410

X3410

X1112 19

X3410






120574ge1(023)

fX120574sim119977

120574lt0(023)

fX120574sim119977

120574ge1(01)

fX120574sim119977

120574ge1(032)

fX120574sim119977

120574lt0(01)

fX120574sim119977

120574lt0(032)

fX120574sim119977


(0 32)X2 sim ℒ119977




max119891119883+infin











120590minus120583 to (12120590)119890minus120583)


119889

119889120574Mode119883120574 = minus120590

120574 log (120590) 119890120583minus120590120574

(29)


Mode119883+infin =

119890120583 120590 lt 1

119890120583minus1

120590 = 1

0 120590 gt 1

(30)


120574

= 119891119883120574


+infin

= 1(2120590119890120583) provided 120590 le 1










120583minus120590120574


120583minus120590120574

|120574rarrminusinfin = 119890120583


120574

=


120574




120574

of 119883120574lt0 with 120590 gt 1


120574





120574

on 119891119883120574


120574


120574



Theorem 8 The cdf 119865119883120574


119865119883120574

(119909)

=1

2+

radic120587

2Γ ((120574 minus 1) 120574) Γ (120574 (120574 minus 1))


120574)

(120574minus1)120574 log119909 minus 120583120590

(31)

= 1 minus1

2Γ ((120574 minus 1) 120574)Γ(

120574 minus 1

120574120574 minus 1

120574(log119909 minus 120583

120590)

120574(120574minus1)

)

119909 isin Rlowast

+

(32)


119865119883120574

(119909) = 119865119883120574

(119909 120583 120590 120574) = int

119909

0

119891119883120574

(119905) 119889119905

= 120590minus11198621

120574int

119909

0


120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log 119905 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119905

(33)



119865119883120574

(119909) = 1198621

120574int

(log 119909minus120583)120590

minusinfin


120574|119908|120574(120574minus1)

119889119908

= Φ119885120574

(log119909 minus 120583

120590)

(34)

where Φ119885120574



Φ119885120574

(119911) = 1198621

120574int

119911

minusinfin


120574|119908|120574(120574minus1)

119889119908

= Φ119885120574

(0) + 1198621

120574int

119911

0


120574|119908|120574(120574minus1)

119889119908

(35)

and as 119891119885120574


Φ119885120574

(119911) =1

2+ 1198621

120574int

119911

0


120574|119908|120574(120574minus1)

119889119908

=1

2+ 1198621

120574int

119911

0

exp

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(120574 minus 1

120574)

(120574minus1)120574

119908

1003816100381610038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119908

(36)

and thus

Φ119885120574

(119911) =1

2+ 1198621

120574(

120574

120574 minus 1)

(120574minus1)120574

times int

((120574minus1)120574)(120574minus1)120574

119911

0

exp minus119906120574(120574minus1) 119889119906

(37)


Φ119885120574

(119911) =1

2+

radic120587



120574)

(120574minus1)120574

119911 119911 isin R

(38)



119865119883120574

(119909) =1

2+



120574)

(120574minus1)120574 10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

(39)


119865119883120574

(119909) =1 + sgn (log119909 minus 120583)



120574120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

)

(40)


minus1 sdot)




119865119883120574

(119909) =1

2+

((120574 minus 1)120574)(120574minus1)120574

(2120574) Γ ((120574 minus 1) 120574)(log119909 minus 120583

120590)

times

infin

sum

119896=0

(((1 minus 120574)120574)1003816100381610038161003816(log119909 minus 120583)120590

1003816100381610038161003816120574(120574minus1)

)119896

119896 [(119896 + 1) 120574 minus 1]

119909 isin Rlowast

+

(41)



119865119883120574

(119909)=1

2+1


sgn (log119909 minus 120583)

2 exp 1198991003816100381610038161003816(log119909 minus 120583)12059010038161003816100381610038161119899

times

119899minus1

sum

119896=0

119899119896

119896

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

119896119899

(42)



sum

119896=0

119909119896

119896 119909 isin R 119899 isin N

lowast= N 0

(43)



119865119883minus1

(119909) =1

2+1


times

1 + 2radic1003816100381610038161003816(log119909 minus 120583) 120590

1003816100381610038161003816


(44)




119865119883minus12

(119909) =1

2+1


times

1 + 33radic1003816100381610038161003816(log119909 minus 120583) 120590

1003816100381610038161003816 + 93

radic((log119909 minus 120583)120590)2


(45)


1198651198832

(119909) = Φ1198832

(log119909 minus 120583

120590) =

1

2+1

2Erf2 (

log119909 minus 120583radic2120590

)

=1

2+1

2erf (

log119909 minus 120583radic2120590

)

(46)





(119909)

=1

2+1

2radic120587Erf1 (

log119909 minus 120583120590

)

=1

2minus1

2sgn (log119909 minus 120583)

infin

sum

119896=0

1

(119896 + 1)(minus

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816)

119896+1

=1

2+1


2sgn (log119909 minus 120583)

times expminus10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

(47)

and hence


(119909) =

1

2119890120583120590

1199091120590 119909 isin (0 119890

120583]

1 minus119890120583120590

21199091120590 119909 isin (119890

120583 +infin)

(48)


120583 1120590 1120590) see

Theorem 3



119865119883120574

(119909)

=1

2+

((120574 minus 1)120574)(120574minus1)120574

2Γ (((120574 minus 1) 120574) + 1)(log119909 minus 120583

120590)

times[[

[

1 + (120574 minus 1)

infin

sum

119896=1

(((1 minus 120574)120574)1003816100381610038161003816(log119909 minus 120583)120590

1003816100381610038161003816120574(120574minus1)

)119896

119896 [(119896 + 1) 120574 minus 1]

]]

]

(49)


1198651198831

(119909) = lim120574rarr1+

119865119883120574

(119909) =1

2+log119909 minus 120583

2120590(1 + 0)

=log119909 minus 120583 + 120590

2120590

(50)

with 1198651198831

(119890120583minus120590

) = 0 and 1198651198831

(119890120583+120590

) = 1 Therefore

1198651198831

(119909) =

0 119909 isin (0 119890120583minus120590

)

1

2120590(log119909 minus 120583 + 120590) 119909 isin [119890

120583minus120590 119890120583+120590

]

1 119909 isin (119890120583+120590

+infin)

(51)


120583minus120590 119887 = 119890


LN1(120583 120590) = LU(119890120583minus120590

119890120583+120590

) see Theorem 3


120583|120583=0 =


120574


Q119883120574


+| 119865119883

120574

(119909) ge 119875

= exp sgn (2119875 minus 1) 120590

times [120574


120574 |2119875 minus 1|)]

(120574minus1)120574

119875 isin (0 1)

(52)






120574 11987512057412 1198751205741 1198751205742 1198751205743 1198751205744 1198751205745 1199021120574 1199023120574



120574

(14) and Q119883120574





119883120574


mean (120583119892)119883120574

= 119890E[log119883


120574

= 119890120583 as



120583sim




119891|119884| (119909) = 119891119884 (minus119909) + 119891119884 (119909) 119909 isin R+ (53)


119891|119884| (119909) = 119891119884 (minus119909) + 119891119884 (119909) 0 le 119909 le 119890

120583

119891119884 (119909) 119909 gt 119890120583

(54)


119865|119884| (119909) = int

119909

0

119891|119884| (119905) 119889119905 = int

119890120583

0

119891119884 (minus119905) 119889119905 + int

119890120583

0

119891119884 (119905) 119889119905

+ int

119909

119890120583119891119884 (119905) 119889119905 119909 isin R+

(55)



119865|119884| (119909) = 1198621

120574int

0

minusinfin


120574|119908|120574(120574minus1)

119889119908 + 1198621

120574

times int

(1120590) log 2

0


120574|119911|120574(120574minus1)

119889119911

+ 1198621

120574int

119892(119909)


120574 minus 1

120574|119911|120574(120574minus1)

119889119911

119892 (119909) =log (119909 + 119890120583) minus 120583

120590 119909 isin R+

(56)

and hence

119865|119884| (119909) = Φ119885 (0) + [Φ119885 (log 2120590

) minus Φ119885 (0)]

+ [Φ119885 (119892 (119909)) minus Φ119885 (log 2120590

)]

= Φ119885(log (119909 + 119890120583) minus 120583

120590)

(57)


x

0 1 2 3

X2simℒ119977(0 23)


sim119977

120574(023)

eminus23 e23

0

01

02

03

04

05

06

07

08

09

1

X1112 19

X3410



FX120574



(a)

120574(01)

0

01

02

03

04

05

06

07

08

09

1

X2simℒ119977(0 1)



x

0 1 2 3


X1112 19

X3410

eminus1 e1

FX120574sim119977

(b)

00

01

02

03

04

05

06

07

08

09

1

x

1 2 3

eminus32


X2simℒ119977(0 32)

X1112 19

X3410


119977120574(032)

FX120574sim119977



(c)




+ it is clear that


|119883120574minus119890120583|







120583(119905)

119883=

119890119905120583

Γ ((120574 minus 1) 120574)

infin

sum

119899=0

(119905120590)2119899

(2119899)(

120574

120574 minus 1)

2119899((120574minus1)120574)

times Γ((2119899 + 1)120574 minus 1

120574)

(58)


(119905)

119883


have

120583(119905)

119883= E [119883119905] = int

R+

119909119905119891119883 (119909) 119889119909

=1

1205901198621

120574intR+


120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119909

(59)


(1120590)


120583(119905)

119883= 1198621

120574(

120574

120574 minus 1)

(120574minus1)120574

intR

exp119905120583 + 119899(120574

120574 minus 1)

(120574minus1)120574

120590119911



exp119905(120574

120574 minus 1)

(120574minus1)120574

120590119911 =

infin

sum

119899=0

(119905120590)119899

119899(

120574

120574 minus 1)

119899((120574minus1)120574)

119911119899

(61)

it is obtained that

120583(119905)

119883= 21198621

120574(

120574

120574 minus 1)

(120574minus1)120574

119890119905120583infin

sum

119899=0

(119905120590)2119899

(2119899)(

120574

120574 minus 1)

2119899((120574minus1)120574)

times intR+

1199112119899 exp minus119911120574(120574minus1) 119889119911

(62)



intR+

119909119898119890minus119887119909119899

119889119909 =Γ ((119898 + 1) 119899)

119899119887(119898+1)119899 119899 119898 119887 isin R

lowast

+ (63)


2) we have 119872119884(119905) =

E[119890119905119884] = E[119883119905] = 120583(119905)



120583(119905)

119883=

119890119905120583

radic120587

infin

sum

119899=0

(211990521205902)119899

(2119899)Γ (119899 +

1

2) 119905 isin R+ (64)


Γ (119899 +1

2) =

(2119899)

22119899119899radic120587 119899 isin N (65)

we have

120583(119905)

119883= 119890119905120583infin

sum

119899=0

(119905120590)2119899

2119899119899= 119890119905120583infin

sum

119899=0

1

119899(1

211990521205902)

119899

= 119890119905120583+(12)(119905120590)

2

119905 isin R+

(66)


(1)

119883= E[119883] =


(119905)

119883= exp119905120583 +





LN120574(120583 120590) is given by

120583(119896)

119883=

119890119896120583

Γ ((120574 minus 1) 120574)

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899

119878119896minus119899 119896 isin N (67)

where

119878119896 =

infin

sum

119898=0

(119896120590)2119898

(2119898)(

120574

120574 minus 1)

2119898((120574minus1)120574)

Γ((2119898 + 1)120574 minus 1

120574)

119896 isin N

(68)


wehave

120583(119896)

119883= E [(119883 minus 120583119883)

119896] = int

R+

(119909 minus 120583119883)119896119891119883 (119909 120583 120590 120574) 119889119909

(69)


120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus120583119883)

119899intR+

119909119896minus119899

119891119883 (119909) 119889119909

=

119896

sum

119899=0

(119896

119899) (minus120583119883)

119899120583(119896minus119899)

119883

(70)


120583(119896)

119883=

119890119896120583

Γ ((120574 minus 1) 120574)

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898)

times (120574

120574 minus 1)

2119898((120574minus1)120574)

Γ ((2119898 + 1) ((120574 minus 1) 120574))

(71)




120583+(12)1205902


120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus1)

119899119890119896120583+(12)[119899+(119896minus119899)

2

]1205902

119896 isin N (72)

while

1205902

119883= Var [119883] = 120583

(2)

119883= 1198902120583+1205902

(1198901205902

minus 1) (73)





120583119883 =119890120583

Γ ((120574 minus 1) 120574)1198781 (74)

1205902

119883= minus 120583

2

119883+

1198902120583

Γ ((120574 minus 1) 120574)1198782 (75)

1198621198812

119883= Γ(

120574 minus 1

120574)1198782

11987821

minus 1 (76)

120582119883 = minus 119862119881minus3

119883minus 119862119881minus1

119883+

1198903120583

1205903119883Γ ((120574 minus 1) 120574)

1198783 (77)

120581119883 = minus 119862119881minus4

119883minus 6119862119881

minus2

119883minus 4

120582119883

119862119881119883

+1198904120583

1205904119883Γ ((120574 minus 1) 120574)

1198784

(78)



120583(1)


1205902

119883= 120583(2)

119883= 1205832

119883+ [Γ(

120574 minus 1

120574)]

minus1

(11989021205831198782 minus 2119890

1205831205831198831198781) (79)


(3)

1198831205903

119883


120582119883 = minus119862119881minus3

119883+ [1205903

119883Γ(

120574 minus 1

120574)]

minus1

times (11989031205831198783 minus 3119890

21205831205831198831198782 + 3119890

1205831205831198831198781)

(80)


(4)

1198831205904

119883 which


120581119883 = 119862119881minus4

119883+ [1205904

119883Γ(

120574 minus 1

120574)]

minus1

times (11989041205831198784 minus 4119890

31205831205831198831198783 + 6119890

21205831205832

1198831198782 minus 4119890

1205831205833

1198831198781)

(81)



radic120587119890(1198992

1205902


120583119883 = 119890120583+(12)120590

2

1205902

119883= 1198902120583+1205902

(1198901205902

minus 1)

119862119881119883 =radic1198901205902

minus 1

120582119883 = (1198901205902

+ 2)radic1198901205902

minus 1 120581119883 = 11989041205902

+ 211989031205902

+ 311989021205902

minus 3

(82)




120574

le

120583119883120574



Med119883120574 = (120583119892)119883120574

= 119890120583lt 120583119883

120574

(83)



Mode119883120574 = 119890120583minus120590120574

lt 119890120583= Med119883120574 (84)


Mode119883120574 = 119890120583= Med119883120574 (85)



Mode119883120574 = 0 lt 119890120583= Med119883120574 (86)





120583(119905)

119883=

119887119905minus 119886119905

119905 log (119887119886) 119905 isin R (87)

120583(119896)

119883=

(119886 minus 119887)119896

log119896 (119887119886)+

1

log (119887119886)

times

119896minus1

sum

119899=0

(119896

119899)(119886 minus 119887)

119899(119887119896minus119899

minus 119886119896minus119899

)

(119896 minus 119899) log119899 (119887119886) 119896 isin N

(88)


120583119883 =119887 minus 119886

log (119887119886) (89)

1205902

119883=

(119887 minus 119886)2

log2 (119887119886)+(119887 minus 119886) (119887 + 119886)

2 log (119887119886) (90)

119862119881119883 = radic1 +119887 + 119886

2 (119887 minus 119886)log 119887

119886 (91)

120582119883 =1

1205903119883

[1198873minus 1198863

3 log (119887119886)minus 3

(119887 minus 119886)2(119887 + 119886)

2 log2 (119887119886)+ 2

(119887 minus 119886)3

log3 (119887119886)]

(92)

120581119883 =1

1205904119883

[1198874minus 1198864

4 log (119887119886)minus 4

(119887 minus 119886) (1198873minus 1198863)

3 log2 (119887119886)

+3(119887 minus 119886)

3(119887 + 119886)

log3 (119887119886)minus 3

(119887 minus 119886)4

log4 (119887119886)]

(93)


120583(119905)

119883120574

=119890119905120583

119905120590Γ ((120574 minus 1) 120574 + 1)

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)(

120574

120574 minus 1)

2119898((120574minus1)120574)

times Γ((2119898 + 1)120574 minus 1

120574+ 1)

(94)


120583(119905)

119883= lim120574rarr1+

120583(119905)

119883120574

=119890119905120583

119905120590

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)=119890119905(120583+120590)

minus 119890119905(120583minus120590)

2119905120590

119905 isin R

(95)


119883= E[119883] = (12120590)

(119890120583+120590

minus 119890120583minus120590


120583(119896)

119883= lim120574rarr1+

120583(119896)

119883120574

= 119890119896120583119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898 + 1)

119896 isin N

(96)


120583(119896)

119883=119890119896120583

2120590

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 119890(119896minus119899)120590


(119896 minus 119899) 119896 isin N

(97)


Var[119883] = 120583(2)

119883minus 1205832


For 119896 = 3 and 119896 = 4 through 120583(3)119883

and 120583(4)119883 we obtain (92) and

(93) respectively


120583(119905)

119883=

1205831199051205902

1205902 minus 1199052gt 120583119905 120590 gt 119905 119905 isin R (98)

120583(119896)

119883= 120583119896119896

sum

119899=0

(119896

119899)

1205902(119899+1)


2] 120590 gt 119896 119896 isin N

(99)


120583119883 =1205831205902

1205902 minus 1gt 120583 120590 gt 1 (100)

1205902

119883=

12058321205902(21205902+ 1)

(1205902 minus 4) (1205902 minus 1)2 120590 gt 2 (101)

119862119881119883 =1

120590

radic21205902+ 1

1205902 minus 4 120590 gt 2 (102)

120582119883 =2 (15120590

4+ 71205902+ 2)

120590 (1205902 minus 9)radic

1205902minus 4

(21205902 + 1)3 120590 gt 3 (103)


120581119883 =3 (81205908+ 212120590

6+ 95120590

4+ 33120590

2+ 12) (120590

2minus 4)

(1205902 minus 16) (1205902 minus 9) (21205902 + 1)2

120590 gt 4

(104)


120583(119905)

119883= 120583(119905)

119883plusmninfin

= 120583119905infin

sum

119896=0

(119905

120590)

2119896

119905 isin R (105)


120583(119905)

119883plusmninfin

=1

2120583119905[

infin

sum

119896=0

(119905

120590)

119896

+

infin

sum

119896=0

(minus119905

120590)

119896

]

=1

2120583119905(

120590

120590 minus 119905+

120590

120590 + 119905)

(106)



120583(119896)

119883= 1205831198961205902119896

sum

119899=0

(119896

119899)

(minus120583119883120583)119899




(2)

119883minus 1205832

119883


119883and 120583(4)




119880sim LU(119890

120583minus120590 119890120583+120590


119886 119890119887)


(119905)

LU = (119890119905119887minus 119890119905119886)(1119905(119887 minus 119886))



120583 1120590 1120590) due to


(119905)

LL = 119890119905120590(1 minus 11990521205902)minus1

4 Conclusion



Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










eminus23 e23

0

05

1

15

x

0 1 2 3


2X sim ℒ119977(0 23)

0

05

1

15

x

0 1 2 3 4

eminus1 e1




(b3)

(b2)

(b1)

(a3)

(a2)

(a1)

x

0 1 20

05

1

2

15

eminus32


0

05

1

15

x

0 1 2 31e


x

0 1 2 30

05

1

15


x

01 2 30

05

1

15


X1112 19

X1112 19

X3410

X3410

X1112 19

X3410






120574ge1(023)

fX120574sim119977

120574lt0(023)

fX120574sim119977

120574ge1(01)

fX120574sim119977

120574ge1(032)

fX120574sim119977

120574lt0(01)

fX120574sim119977

120574lt0(032)

fX120574sim119977


(0 32)X2 sim ℒ119977




max119891119883+infin











120590minus120583 to (12120590)119890minus120583)


119889

119889120574Mode119883120574 = minus120590

120574 log (120590) 119890120583minus120590120574

(29)


Mode119883+infin =

119890120583 120590 lt 1

119890120583minus1

120590 = 1

0 120590 gt 1

(30)


120574

= 119891119883120574


+infin

= 1(2120590119890120583) provided 120590 le 1










120583minus120590120574


120583minus120590120574

|120574rarrminusinfin = 119890120583


120574

=


120574




120574

of 119883120574lt0 with 120590 gt 1


120574





120574

on 119891119883120574


120574


120574



Theorem 8 The cdf 119865119883120574


119865119883120574

(119909)

=1

2+

radic120587

2Γ ((120574 minus 1) 120574) Γ (120574 (120574 minus 1))


120574)

(120574minus1)120574 log119909 minus 120583120590

(31)

= 1 minus1

2Γ ((120574 minus 1) 120574)Γ(

120574 minus 1

120574120574 minus 1

120574(log119909 minus 120583

120590)

120574(120574minus1)

)

119909 isin Rlowast

+

(32)


119865119883120574

(119909) = 119865119883120574

(119909 120583 120590 120574) = int

119909

0

119891119883120574

(119905) 119889119905

= 120590minus11198621

120574int

119909

0


120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log 119905 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119905

(33)



119865119883120574

(119909) = 1198621

120574int

(log 119909minus120583)120590

minusinfin


120574|119908|120574(120574minus1)

119889119908

= Φ119885120574

(log119909 minus 120583

120590)

(34)

where Φ119885120574



Φ119885120574

(119911) = 1198621

120574int

119911

minusinfin


120574|119908|120574(120574minus1)

119889119908

= Φ119885120574

(0) + 1198621

120574int

119911

0


120574|119908|120574(120574minus1)

119889119908

(35)

and as 119891119885120574


Φ119885120574

(119911) =1

2+ 1198621

120574int

119911

0


120574|119908|120574(120574minus1)

119889119908

=1

2+ 1198621

120574int

119911

0

exp

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(120574 minus 1

120574)

(120574minus1)120574

119908

1003816100381610038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119908

(36)

and thus

Φ119885120574

(119911) =1

2+ 1198621

120574(

120574

120574 minus 1)

(120574minus1)120574

times int

((120574minus1)120574)(120574minus1)120574

119911

0

exp minus119906120574(120574minus1) 119889119906

(37)


Φ119885120574

(119911) =1

2+

radic120587



120574)

(120574minus1)120574

119911 119911 isin R

(38)



119865119883120574

(119909) =1

2+



120574)

(120574minus1)120574 10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

(39)


119865119883120574

(119909) =1 + sgn (log119909 minus 120583)



120574120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

)

(40)


minus1 sdot)




119865119883120574

(119909) =1

2+

((120574 minus 1)120574)(120574minus1)120574

(2120574) Γ ((120574 minus 1) 120574)(log119909 minus 120583

120590)

times

infin

sum

119896=0

(((1 minus 120574)120574)1003816100381610038161003816(log119909 minus 120583)120590

1003816100381610038161003816120574(120574minus1)

)119896

119896 [(119896 + 1) 120574 minus 1]

119909 isin Rlowast

+

(41)



119865119883120574

(119909)=1

2+1


sgn (log119909 minus 120583)

2 exp 1198991003816100381610038161003816(log119909 minus 120583)12059010038161003816100381610038161119899

times

119899minus1

sum

119896=0

119899119896

119896

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

119896119899

(42)



sum

119896=0

119909119896

119896 119909 isin R 119899 isin N

lowast= N 0

(43)



119865119883minus1

(119909) =1

2+1


times

1 + 2radic1003816100381610038161003816(log119909 minus 120583) 120590

1003816100381610038161003816


(44)




119865119883minus12

(119909) =1

2+1


times

1 + 33radic1003816100381610038161003816(log119909 minus 120583) 120590

1003816100381610038161003816 + 93

radic((log119909 minus 120583)120590)2


(45)


1198651198832

(119909) = Φ1198832

(log119909 minus 120583

120590) =

1

2+1

2Erf2 (

log119909 minus 120583radic2120590

)

=1

2+1

2erf (

log119909 minus 120583radic2120590

)

(46)





(119909)

=1

2+1

2radic120587Erf1 (

log119909 minus 120583120590

)

=1

2minus1

2sgn (log119909 minus 120583)

infin

sum

119896=0

1

(119896 + 1)(minus

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816)

119896+1

=1

2+1


2sgn (log119909 minus 120583)

times expminus10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

(47)

and hence


(119909) =

1

2119890120583120590

1199091120590 119909 isin (0 119890

120583]

1 minus119890120583120590

21199091120590 119909 isin (119890

120583 +infin)

(48)


120583 1120590 1120590) see

Theorem 3



119865119883120574

(119909)

=1

2+

((120574 minus 1)120574)(120574minus1)120574

2Γ (((120574 minus 1) 120574) + 1)(log119909 minus 120583

120590)

times[[

[

1 + (120574 minus 1)

infin

sum

119896=1

(((1 minus 120574)120574)1003816100381610038161003816(log119909 minus 120583)120590

1003816100381610038161003816120574(120574minus1)

)119896

119896 [(119896 + 1) 120574 minus 1]

]]

]

(49)


1198651198831

(119909) = lim120574rarr1+

119865119883120574

(119909) =1

2+log119909 minus 120583

2120590(1 + 0)

=log119909 minus 120583 + 120590

2120590

(50)

with 1198651198831

(119890120583minus120590

) = 0 and 1198651198831

(119890120583+120590

) = 1 Therefore

1198651198831

(119909) =

0 119909 isin (0 119890120583minus120590

)

1

2120590(log119909 minus 120583 + 120590) 119909 isin [119890

120583minus120590 119890120583+120590

]

1 119909 isin (119890120583+120590

+infin)

(51)


120583minus120590 119887 = 119890


LN1(120583 120590) = LU(119890120583minus120590

119890120583+120590

) see Theorem 3


120583|120583=0 =


120574


Q119883120574


+| 119865119883

120574

(119909) ge 119875

= exp sgn (2119875 minus 1) 120590

times [120574


120574 |2119875 minus 1|)]

(120574minus1)120574

119875 isin (0 1)

(52)






120574 11987512057412 1198751205741 1198751205742 1198751205743 1198751205744 1198751205745 1199021120574 1199023120574



120574

(14) and Q119883120574





119883120574


mean (120583119892)119883120574

= 119890E[log119883


120574

= 119890120583 as



120583sim




119891|119884| (119909) = 119891119884 (minus119909) + 119891119884 (119909) 119909 isin R+ (53)


119891|119884| (119909) = 119891119884 (minus119909) + 119891119884 (119909) 0 le 119909 le 119890

120583

119891119884 (119909) 119909 gt 119890120583

(54)


119865|119884| (119909) = int

119909

0

119891|119884| (119905) 119889119905 = int

119890120583

0

119891119884 (minus119905) 119889119905 + int

119890120583

0

119891119884 (119905) 119889119905

+ int

119909

119890120583119891119884 (119905) 119889119905 119909 isin R+

(55)



119865|119884| (119909) = 1198621

120574int

0

minusinfin


120574|119908|120574(120574minus1)

119889119908 + 1198621

120574

times int

(1120590) log 2

0


120574|119911|120574(120574minus1)

119889119911

+ 1198621

120574int

119892(119909)


120574 minus 1

120574|119911|120574(120574minus1)

119889119911

119892 (119909) =log (119909 + 119890120583) minus 120583

120590 119909 isin R+

(56)

and hence

119865|119884| (119909) = Φ119885 (0) + [Φ119885 (log 2120590

) minus Φ119885 (0)]

+ [Φ119885 (119892 (119909)) minus Φ119885 (log 2120590

)]

= Φ119885(log (119909 + 119890120583) minus 120583

120590)

(57)


x

0 1 2 3

X2simℒ119977(0 23)


sim119977

120574(023)

eminus23 e23

0

01

02

03

04

05

06

07

08

09

1

X1112 19

X3410



FX120574



(a)

120574(01)

0

01

02

03

04

05

06

07

08

09

1

X2simℒ119977(0 1)



x

0 1 2 3


X1112 19

X3410

eminus1 e1

FX120574sim119977

(b)

00

01

02

03

04

05

06

07

08

09

1

x

1 2 3

eminus32


X2simℒ119977(0 32)

X1112 19

X3410


119977120574(032)

FX120574sim119977



(c)




+ it is clear that


|119883120574minus119890120583|







120583(119905)

119883=

119890119905120583

Γ ((120574 minus 1) 120574)

infin

sum

119899=0

(119905120590)2119899

(2119899)(

120574

120574 minus 1)

2119899((120574minus1)120574)

times Γ((2119899 + 1)120574 minus 1

120574)

(58)


(119905)

119883


have

120583(119905)

119883= E [119883119905] = int

R+

119909119905119891119883 (119909) 119889119909

=1

1205901198621

120574intR+


120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119909

(59)


(1120590)


120583(119905)

119883= 1198621

120574(

120574

120574 minus 1)

(120574minus1)120574

intR

exp119905120583 + 119899(120574

120574 minus 1)

(120574minus1)120574

120590119911



exp119905(120574

120574 minus 1)

(120574minus1)120574

120590119911 =

infin

sum

119899=0

(119905120590)119899

119899(

120574

120574 minus 1)

119899((120574minus1)120574)

119911119899

(61)

it is obtained that

120583(119905)

119883= 21198621

120574(

120574

120574 minus 1)

(120574minus1)120574

119890119905120583infin

sum

119899=0

(119905120590)2119899

(2119899)(

120574

120574 minus 1)

2119899((120574minus1)120574)

times intR+

1199112119899 exp minus119911120574(120574minus1) 119889119911

(62)



intR+

119909119898119890minus119887119909119899

119889119909 =Γ ((119898 + 1) 119899)

119899119887(119898+1)119899 119899 119898 119887 isin R

lowast

+ (63)


2) we have 119872119884(119905) =

E[119890119905119884] = E[119883119905] = 120583(119905)



120583(119905)

119883=

119890119905120583

radic120587

infin

sum

119899=0

(211990521205902)119899

(2119899)Γ (119899 +

1

2) 119905 isin R+ (64)


Γ (119899 +1

2) =

(2119899)

22119899119899radic120587 119899 isin N (65)

we have

120583(119905)

119883= 119890119905120583infin

sum

119899=0

(119905120590)2119899

2119899119899= 119890119905120583infin

sum

119899=0

1

119899(1

211990521205902)

119899

= 119890119905120583+(12)(119905120590)

2

119905 isin R+

(66)


(1)

119883= E[119883] =


(119905)

119883= exp119905120583 +





LN120574(120583 120590) is given by

120583(119896)

119883=

119890119896120583

Γ ((120574 minus 1) 120574)

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899

119878119896minus119899 119896 isin N (67)

where

119878119896 =

infin

sum

119898=0

(119896120590)2119898

(2119898)(

120574

120574 minus 1)

2119898((120574minus1)120574)

Γ((2119898 + 1)120574 minus 1

120574)

119896 isin N

(68)


wehave

120583(119896)

119883= E [(119883 minus 120583119883)

119896] = int

R+

(119909 minus 120583119883)119896119891119883 (119909 120583 120590 120574) 119889119909

(69)


120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus120583119883)

119899intR+

119909119896minus119899

119891119883 (119909) 119889119909

=

119896

sum

119899=0

(119896

119899) (minus120583119883)

119899120583(119896minus119899)

119883

(70)


120583(119896)

119883=

119890119896120583

Γ ((120574 minus 1) 120574)

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898)

times (120574

120574 minus 1)

2119898((120574minus1)120574)

Γ ((2119898 + 1) ((120574 minus 1) 120574))

(71)




120583+(12)1205902


120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus1)

119899119890119896120583+(12)[119899+(119896minus119899)

2

]1205902

119896 isin N (72)

while

1205902

119883= Var [119883] = 120583

(2)

119883= 1198902120583+1205902

(1198901205902

minus 1) (73)





120583119883 =119890120583

Γ ((120574 minus 1) 120574)1198781 (74)

1205902

119883= minus 120583

2

119883+

1198902120583

Γ ((120574 minus 1) 120574)1198782 (75)

1198621198812

119883= Γ(

120574 minus 1

120574)1198782

11987821

minus 1 (76)

120582119883 = minus 119862119881minus3

119883minus 119862119881minus1

119883+

1198903120583

1205903119883Γ ((120574 minus 1) 120574)

1198783 (77)

120581119883 = minus 119862119881minus4

119883minus 6119862119881

minus2

119883minus 4

120582119883

119862119881119883

+1198904120583

1205904119883Γ ((120574 minus 1) 120574)

1198784

(78)



120583(1)


1205902

119883= 120583(2)

119883= 1205832

119883+ [Γ(

120574 minus 1

120574)]

minus1

(11989021205831198782 minus 2119890

1205831205831198831198781) (79)


(3)

1198831205903

119883


120582119883 = minus119862119881minus3

119883+ [1205903

119883Γ(

120574 minus 1

120574)]

minus1

times (11989031205831198783 minus 3119890

21205831205831198831198782 + 3119890

1205831205831198831198781)

(80)


(4)

1198831205904

119883 which


120581119883 = 119862119881minus4

119883+ [1205904

119883Γ(

120574 minus 1

120574)]

minus1

times (11989041205831198784 minus 4119890

31205831205831198831198783 + 6119890

21205831205832

1198831198782 minus 4119890

1205831205833

1198831198781)

(81)



radic120587119890(1198992

1205902


120583119883 = 119890120583+(12)120590

2

1205902

119883= 1198902120583+1205902

(1198901205902

minus 1)

119862119881119883 =radic1198901205902

minus 1

120582119883 = (1198901205902

+ 2)radic1198901205902

minus 1 120581119883 = 11989041205902

+ 211989031205902

+ 311989021205902

minus 3

(82)




120574

le

120583119883120574



Med119883120574 = (120583119892)119883120574

= 119890120583lt 120583119883

120574

(83)



Mode119883120574 = 119890120583minus120590120574

lt 119890120583= Med119883120574 (84)


Mode119883120574 = 119890120583= Med119883120574 (85)



Mode119883120574 = 0 lt 119890120583= Med119883120574 (86)





120583(119905)

119883=

119887119905minus 119886119905

119905 log (119887119886) 119905 isin R (87)

120583(119896)

119883=

(119886 minus 119887)119896

log119896 (119887119886)+

1

log (119887119886)

times

119896minus1

sum

119899=0

(119896

119899)(119886 minus 119887)

119899(119887119896minus119899

minus 119886119896minus119899

)

(119896 minus 119899) log119899 (119887119886) 119896 isin N

(88)


120583119883 =119887 minus 119886

log (119887119886) (89)

1205902

119883=

(119887 minus 119886)2

log2 (119887119886)+(119887 minus 119886) (119887 + 119886)

2 log (119887119886) (90)

119862119881119883 = radic1 +119887 + 119886

2 (119887 minus 119886)log 119887

119886 (91)

120582119883 =1

1205903119883

[1198873minus 1198863

3 log (119887119886)minus 3

(119887 minus 119886)2(119887 + 119886)

2 log2 (119887119886)+ 2

(119887 minus 119886)3

log3 (119887119886)]

(92)

120581119883 =1

1205904119883

[1198874minus 1198864

4 log (119887119886)minus 4

(119887 minus 119886) (1198873minus 1198863)

3 log2 (119887119886)

+3(119887 minus 119886)

3(119887 + 119886)

log3 (119887119886)minus 3

(119887 minus 119886)4

log4 (119887119886)]

(93)


120583(119905)

119883120574

=119890119905120583

119905120590Γ ((120574 minus 1) 120574 + 1)

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)(

120574

120574 minus 1)

2119898((120574minus1)120574)

times Γ((2119898 + 1)120574 minus 1

120574+ 1)

(94)


120583(119905)

119883= lim120574rarr1+

120583(119905)

119883120574

=119890119905120583

119905120590

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)=119890119905(120583+120590)

minus 119890119905(120583minus120590)

2119905120590

119905 isin R

(95)


119883= E[119883] = (12120590)

(119890120583+120590

minus 119890120583minus120590


120583(119896)

119883= lim120574rarr1+

120583(119896)

119883120574

= 119890119896120583119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898 + 1)

119896 isin N

(96)


120583(119896)

119883=119890119896120583

2120590

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 119890(119896minus119899)120590


(119896 minus 119899) 119896 isin N

(97)


Var[119883] = 120583(2)

119883minus 1205832


For 119896 = 3 and 119896 = 4 through 120583(3)119883

and 120583(4)119883 we obtain (92) and

(93) respectively


120583(119905)

119883=

1205831199051205902

1205902 minus 1199052gt 120583119905 120590 gt 119905 119905 isin R (98)

120583(119896)

119883= 120583119896119896

sum

119899=0

(119896

119899)

1205902(119899+1)


2] 120590 gt 119896 119896 isin N

(99)


120583119883 =1205831205902

1205902 minus 1gt 120583 120590 gt 1 (100)

1205902

119883=

12058321205902(21205902+ 1)

(1205902 minus 4) (1205902 minus 1)2 120590 gt 2 (101)

119862119881119883 =1

120590

radic21205902+ 1

1205902 minus 4 120590 gt 2 (102)

120582119883 =2 (15120590

4+ 71205902+ 2)

120590 (1205902 minus 9)radic

1205902minus 4

(21205902 + 1)3 120590 gt 3 (103)


120581119883 =3 (81205908+ 212120590

6+ 95120590

4+ 33120590

2+ 12) (120590

2minus 4)

(1205902 minus 16) (1205902 minus 9) (21205902 + 1)2

120590 gt 4

(104)


120583(119905)

119883= 120583(119905)

119883plusmninfin

= 120583119905infin

sum

119896=0

(119905

120590)

2119896

119905 isin R (105)


120583(119905)

119883plusmninfin

=1

2120583119905[

infin

sum

119896=0

(119905

120590)

119896

+

infin

sum

119896=0

(minus119905

120590)

119896

]

=1

2120583119905(

120590

120590 minus 119905+

120590

120590 + 119905)

(106)



120583(119896)

119883= 1205831198961205902119896

sum

119899=0

(119896

119899)

(minus120583119883120583)119899




(2)

119883minus 1205832

119883


119883and 120583(4)




119880sim LU(119890

120583minus120590 119890120583+120590


119886 119890119887)


(119905)

LU = (119890119905119887minus 119890119905119886)(1119905(119887 minus 119886))



120583 1120590 1120590) due to


(119905)

LL = 119890119905120590(1 minus 11990521205902)minus1

4 Conclusion



Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










max119891119883+infin











120590minus120583 to (12120590)119890minus120583)


119889

119889120574Mode119883120574 = minus120590

120574 log (120590) 119890120583minus120590120574

(29)


Mode119883+infin =

119890120583 120590 lt 1

119890120583minus1

120590 = 1

0 120590 gt 1

(30)


120574

= 119891119883120574


+infin

= 1(2120590119890120583) provided 120590 le 1










120583minus120590120574


120583minus120590120574

|120574rarrminusinfin = 119890120583


120574

=


120574




120574

of 119883120574lt0 with 120590 gt 1


120574





120574

on 119891119883120574


120574


120574



Theorem 8 The cdf 119865119883120574


119865119883120574

(119909)

=1

2+

radic120587

2Γ ((120574 minus 1) 120574) Γ (120574 (120574 minus 1))


120574)

(120574minus1)120574 log119909 minus 120583120590

(31)

= 1 minus1

2Γ ((120574 minus 1) 120574)Γ(

120574 minus 1

120574120574 minus 1

120574(log119909 minus 120583

120590)

120574(120574minus1)

)

119909 isin Rlowast

+

(32)


119865119883120574

(119909) = 119865119883120574

(119909 120583 120590 120574) = int

119909

0

119891119883120574

(119905) 119889119905

= 120590minus11198621

120574int

119909

0


120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log 119905 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119905

(33)



119865119883120574

(119909) = 1198621

120574int

(log 119909minus120583)120590

minusinfin


120574|119908|120574(120574minus1)

119889119908

= Φ119885120574

(log119909 minus 120583

120590)

(34)

where Φ119885120574



Φ119885120574

(119911) = 1198621

120574int

119911

minusinfin


120574|119908|120574(120574minus1)

119889119908

= Φ119885120574

(0) + 1198621

120574int

119911

0


120574|119908|120574(120574minus1)

119889119908

(35)

and as 119891119885120574


Φ119885120574

(119911) =1

2+ 1198621

120574int

119911

0


120574|119908|120574(120574minus1)

119889119908

=1

2+ 1198621

120574int

119911

0

exp

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(120574 minus 1

120574)

(120574minus1)120574

119908

1003816100381610038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119908

(36)

and thus

Φ119885120574

(119911) =1

2+ 1198621

120574(

120574

120574 minus 1)

(120574minus1)120574

times int

((120574minus1)120574)(120574minus1)120574

119911

0

exp minus119906120574(120574minus1) 119889119906

(37)


Φ119885120574

(119911) =1

2+

radic120587



120574)

(120574minus1)120574

119911 119911 isin R

(38)



119865119883120574

(119909) =1

2+



120574)

(120574minus1)120574 10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

(39)


119865119883120574

(119909) =1 + sgn (log119909 minus 120583)



120574120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

)

(40)


minus1 sdot)




119865119883120574

(119909) =1

2+

((120574 minus 1)120574)(120574minus1)120574

(2120574) Γ ((120574 minus 1) 120574)(log119909 minus 120583

120590)

times

infin

sum

119896=0

(((1 minus 120574)120574)1003816100381610038161003816(log119909 minus 120583)120590

1003816100381610038161003816120574(120574minus1)

)119896

119896 [(119896 + 1) 120574 minus 1]

119909 isin Rlowast

+

(41)



119865119883120574

(119909)=1

2+1


sgn (log119909 minus 120583)

2 exp 1198991003816100381610038161003816(log119909 minus 120583)12059010038161003816100381610038161119899

times

119899minus1

sum

119896=0

119899119896

119896

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

119896119899

(42)



sum

119896=0

119909119896

119896 119909 isin R 119899 isin N

lowast= N 0

(43)



119865119883minus1

(119909) =1

2+1


times

1 + 2radic1003816100381610038161003816(log119909 minus 120583) 120590

1003816100381610038161003816


(44)




119865119883minus12

(119909) =1

2+1


times

1 + 33radic1003816100381610038161003816(log119909 minus 120583) 120590

1003816100381610038161003816 + 93

radic((log119909 minus 120583)120590)2


(45)


1198651198832

(119909) = Φ1198832

(log119909 minus 120583

120590) =

1

2+1

2Erf2 (

log119909 minus 120583radic2120590

)

=1

2+1

2erf (

log119909 minus 120583radic2120590

)

(46)





(119909)

=1

2+1

2radic120587Erf1 (

log119909 minus 120583120590

)

=1

2minus1

2sgn (log119909 minus 120583)

infin

sum

119896=0

1

(119896 + 1)(minus

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816)

119896+1

=1

2+1


2sgn (log119909 minus 120583)

times expminus10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

(47)

and hence


(119909) =

1

2119890120583120590

1199091120590 119909 isin (0 119890

120583]

1 minus119890120583120590

21199091120590 119909 isin (119890

120583 +infin)

(48)


120583 1120590 1120590) see

Theorem 3



119865119883120574

(119909)

=1

2+

((120574 minus 1)120574)(120574minus1)120574

2Γ (((120574 minus 1) 120574) + 1)(log119909 minus 120583

120590)

times[[

[

1 + (120574 minus 1)

infin

sum

119896=1

(((1 minus 120574)120574)1003816100381610038161003816(log119909 minus 120583)120590

1003816100381610038161003816120574(120574minus1)

)119896

119896 [(119896 + 1) 120574 minus 1]

]]

]

(49)


1198651198831

(119909) = lim120574rarr1+

119865119883120574

(119909) =1

2+log119909 minus 120583

2120590(1 + 0)

=log119909 minus 120583 + 120590

2120590

(50)

with 1198651198831

(119890120583minus120590

) = 0 and 1198651198831

(119890120583+120590

) = 1 Therefore

1198651198831

(119909) =

0 119909 isin (0 119890120583minus120590

)

1

2120590(log119909 minus 120583 + 120590) 119909 isin [119890

120583minus120590 119890120583+120590

]

1 119909 isin (119890120583+120590

+infin)

(51)


120583minus120590 119887 = 119890


LN1(120583 120590) = LU(119890120583minus120590

119890120583+120590

) see Theorem 3


120583|120583=0 =


120574


Q119883120574


+| 119865119883

120574

(119909) ge 119875

= exp sgn (2119875 minus 1) 120590

times [120574


120574 |2119875 minus 1|)]

(120574minus1)120574

119875 isin (0 1)

(52)






120574 11987512057412 1198751205741 1198751205742 1198751205743 1198751205744 1198751205745 1199021120574 1199023120574



120574

(14) and Q119883120574





119883120574


mean (120583119892)119883120574

= 119890E[log119883


120574

= 119890120583 as



120583sim




119891|119884| (119909) = 119891119884 (minus119909) + 119891119884 (119909) 119909 isin R+ (53)


119891|119884| (119909) = 119891119884 (minus119909) + 119891119884 (119909) 0 le 119909 le 119890

120583

119891119884 (119909) 119909 gt 119890120583

(54)


119865|119884| (119909) = int

119909

0

119891|119884| (119905) 119889119905 = int

119890120583

0

119891119884 (minus119905) 119889119905 + int

119890120583

0

119891119884 (119905) 119889119905

+ int

119909

119890120583119891119884 (119905) 119889119905 119909 isin R+

(55)



119865|119884| (119909) = 1198621

120574int

0

minusinfin


120574|119908|120574(120574minus1)

119889119908 + 1198621

120574

times int

(1120590) log 2

0


120574|119911|120574(120574minus1)

119889119911

+ 1198621

120574int

119892(119909)


120574 minus 1

120574|119911|120574(120574minus1)

119889119911

119892 (119909) =log (119909 + 119890120583) minus 120583

120590 119909 isin R+

(56)

and hence

119865|119884| (119909) = Φ119885 (0) + [Φ119885 (log 2120590

) minus Φ119885 (0)]

+ [Φ119885 (119892 (119909)) minus Φ119885 (log 2120590

)]

= Φ119885(log (119909 + 119890120583) minus 120583

120590)

(57)


x

0 1 2 3

X2simℒ119977(0 23)


sim119977

120574(023)

eminus23 e23

0

01

02

03

04

05

06

07

08

09

1

X1112 19

X3410



FX120574



(a)

120574(01)

0

01

02

03

04

05

06

07

08

09

1

X2simℒ119977(0 1)



x

0 1 2 3


X1112 19

X3410

eminus1 e1

FX120574sim119977

(b)

00

01

02

03

04

05

06

07

08

09

1

x

1 2 3

eminus32


X2simℒ119977(0 32)

X1112 19

X3410


119977120574(032)

FX120574sim119977



(c)




+ it is clear that


|119883120574minus119890120583|







120583(119905)

119883=

119890119905120583

Γ ((120574 minus 1) 120574)

infin

sum

119899=0

(119905120590)2119899

(2119899)(

120574

120574 minus 1)

2119899((120574minus1)120574)

times Γ((2119899 + 1)120574 minus 1

120574)

(58)


(119905)

119883


have

120583(119905)

119883= E [119883119905] = int

R+

119909119905119891119883 (119909) 119889119909

=1

1205901198621

120574intR+


120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119909

(59)


(1120590)


120583(119905)

119883= 1198621

120574(

120574

120574 minus 1)

(120574minus1)120574

intR

exp119905120583 + 119899(120574

120574 minus 1)

(120574minus1)120574

120590119911



exp119905(120574

120574 minus 1)

(120574minus1)120574

120590119911 =

infin

sum

119899=0

(119905120590)119899

119899(

120574

120574 minus 1)

119899((120574minus1)120574)

119911119899

(61)

it is obtained that

120583(119905)

119883= 21198621

120574(

120574

120574 minus 1)

(120574minus1)120574

119890119905120583infin

sum

119899=0

(119905120590)2119899

(2119899)(

120574

120574 minus 1)

2119899((120574minus1)120574)

times intR+

1199112119899 exp minus119911120574(120574minus1) 119889119911

(62)



intR+

119909119898119890minus119887119909119899

119889119909 =Γ ((119898 + 1) 119899)

119899119887(119898+1)119899 119899 119898 119887 isin R

lowast

+ (63)


2) we have 119872119884(119905) =

E[119890119905119884] = E[119883119905] = 120583(119905)



120583(119905)

119883=

119890119905120583

radic120587

infin

sum

119899=0

(211990521205902)119899

(2119899)Γ (119899 +

1

2) 119905 isin R+ (64)


Γ (119899 +1

2) =

(2119899)

22119899119899radic120587 119899 isin N (65)

we have

120583(119905)

119883= 119890119905120583infin

sum

119899=0

(119905120590)2119899

2119899119899= 119890119905120583infin

sum

119899=0

1

119899(1

211990521205902)

119899

= 119890119905120583+(12)(119905120590)

2

119905 isin R+

(66)


(1)

119883= E[119883] =


(119905)

119883= exp119905120583 +





LN120574(120583 120590) is given by

120583(119896)

119883=

119890119896120583

Γ ((120574 minus 1) 120574)

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899

119878119896minus119899 119896 isin N (67)

where

119878119896 =

infin

sum

119898=0

(119896120590)2119898

(2119898)(

120574

120574 minus 1)

2119898((120574minus1)120574)

Γ((2119898 + 1)120574 minus 1

120574)

119896 isin N

(68)


wehave

120583(119896)

119883= E [(119883 minus 120583119883)

119896] = int

R+

(119909 minus 120583119883)119896119891119883 (119909 120583 120590 120574) 119889119909

(69)


120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus120583119883)

119899intR+

119909119896minus119899

119891119883 (119909) 119889119909

=

119896

sum

119899=0

(119896

119899) (minus120583119883)

119899120583(119896minus119899)

119883

(70)


120583(119896)

119883=

119890119896120583

Γ ((120574 minus 1) 120574)

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898)

times (120574

120574 minus 1)

2119898((120574minus1)120574)

Γ ((2119898 + 1) ((120574 minus 1) 120574))

(71)




120583+(12)1205902


120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus1)

119899119890119896120583+(12)[119899+(119896minus119899)

2

]1205902

119896 isin N (72)

while

1205902

119883= Var [119883] = 120583

(2)

119883= 1198902120583+1205902

(1198901205902

minus 1) (73)





120583119883 =119890120583

Γ ((120574 minus 1) 120574)1198781 (74)

1205902

119883= minus 120583

2

119883+

1198902120583

Γ ((120574 minus 1) 120574)1198782 (75)

1198621198812

119883= Γ(

120574 minus 1

120574)1198782

11987821

minus 1 (76)

120582119883 = minus 119862119881minus3

119883minus 119862119881minus1

119883+

1198903120583

1205903119883Γ ((120574 minus 1) 120574)

1198783 (77)

120581119883 = minus 119862119881minus4

119883minus 6119862119881

minus2

119883minus 4

120582119883

119862119881119883

+1198904120583

1205904119883Γ ((120574 minus 1) 120574)

1198784

(78)



120583(1)


1205902

119883= 120583(2)

119883= 1205832

119883+ [Γ(

120574 minus 1

120574)]

minus1

(11989021205831198782 minus 2119890

1205831205831198831198781) (79)


(3)

1198831205903

119883


120582119883 = minus119862119881minus3

119883+ [1205903

119883Γ(

120574 minus 1

120574)]

minus1

times (11989031205831198783 minus 3119890

21205831205831198831198782 + 3119890

1205831205831198831198781)

(80)


(4)

1198831205904

119883 which


120581119883 = 119862119881minus4

119883+ [1205904

119883Γ(

120574 minus 1

120574)]

minus1

times (11989041205831198784 minus 4119890

31205831205831198831198783 + 6119890

21205831205832

1198831198782 minus 4119890

1205831205833

1198831198781)

(81)



radic120587119890(1198992

1205902


120583119883 = 119890120583+(12)120590

2

1205902

119883= 1198902120583+1205902

(1198901205902

minus 1)

119862119881119883 =radic1198901205902

minus 1

120582119883 = (1198901205902

+ 2)radic1198901205902

minus 1 120581119883 = 11989041205902

+ 211989031205902

+ 311989021205902

minus 3

(82)




120574

le

120583119883120574



Med119883120574 = (120583119892)119883120574

= 119890120583lt 120583119883

120574

(83)



Mode119883120574 = 119890120583minus120590120574

lt 119890120583= Med119883120574 (84)


Mode119883120574 = 119890120583= Med119883120574 (85)



Mode119883120574 = 0 lt 119890120583= Med119883120574 (86)





120583(119905)

119883=

119887119905minus 119886119905

119905 log (119887119886) 119905 isin R (87)

120583(119896)

119883=

(119886 minus 119887)119896

log119896 (119887119886)+

1

log (119887119886)

times

119896minus1

sum

119899=0

(119896

119899)(119886 minus 119887)

119899(119887119896minus119899

minus 119886119896minus119899

)

(119896 minus 119899) log119899 (119887119886) 119896 isin N

(88)


120583119883 =119887 minus 119886

log (119887119886) (89)

1205902

119883=

(119887 minus 119886)2

log2 (119887119886)+(119887 minus 119886) (119887 + 119886)

2 log (119887119886) (90)

119862119881119883 = radic1 +119887 + 119886

2 (119887 minus 119886)log 119887

119886 (91)

120582119883 =1

1205903119883

[1198873minus 1198863

3 log (119887119886)minus 3

(119887 minus 119886)2(119887 + 119886)

2 log2 (119887119886)+ 2

(119887 minus 119886)3

log3 (119887119886)]

(92)

120581119883 =1

1205904119883

[1198874minus 1198864

4 log (119887119886)minus 4

(119887 minus 119886) (1198873minus 1198863)

3 log2 (119887119886)

+3(119887 minus 119886)

3(119887 + 119886)

log3 (119887119886)minus 3

(119887 minus 119886)4

log4 (119887119886)]

(93)


120583(119905)

119883120574

=119890119905120583

119905120590Γ ((120574 minus 1) 120574 + 1)

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)(

120574

120574 minus 1)

2119898((120574minus1)120574)

times Γ((2119898 + 1)120574 minus 1

120574+ 1)

(94)


120583(119905)

119883= lim120574rarr1+

120583(119905)

119883120574

=119890119905120583

119905120590

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)=119890119905(120583+120590)

minus 119890119905(120583minus120590)

2119905120590

119905 isin R

(95)


119883= E[119883] = (12120590)

(119890120583+120590

minus 119890120583minus120590


120583(119896)

119883= lim120574rarr1+

120583(119896)

119883120574

= 119890119896120583119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898 + 1)

119896 isin N

(96)


120583(119896)

119883=119890119896120583

2120590

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 119890(119896minus119899)120590


(119896 minus 119899) 119896 isin N

(97)


Var[119883] = 120583(2)

119883minus 1205832


For 119896 = 3 and 119896 = 4 through 120583(3)119883

and 120583(4)119883 we obtain (92) and

(93) respectively


120583(119905)

119883=

1205831199051205902

1205902 minus 1199052gt 120583119905 120590 gt 119905 119905 isin R (98)

120583(119896)

119883= 120583119896119896

sum

119899=0

(119896

119899)

1205902(119899+1)


2] 120590 gt 119896 119896 isin N

(99)


120583119883 =1205831205902

1205902 minus 1gt 120583 120590 gt 1 (100)

1205902

119883=

12058321205902(21205902+ 1)

(1205902 minus 4) (1205902 minus 1)2 120590 gt 2 (101)

119862119881119883 =1

120590

radic21205902+ 1

1205902 minus 4 120590 gt 2 (102)

120582119883 =2 (15120590

4+ 71205902+ 2)

120590 (1205902 minus 9)radic

1205902minus 4

(21205902 + 1)3 120590 gt 3 (103)


120581119883 =3 (81205908+ 212120590

6+ 95120590

4+ 33120590

2+ 12) (120590

2minus 4)

(1205902 minus 16) (1205902 minus 9) (21205902 + 1)2

120590 gt 4

(104)


120583(119905)

119883= 120583(119905)

119883plusmninfin

= 120583119905infin

sum

119896=0

(119905

120590)

2119896

119905 isin R (105)


120583(119905)

119883plusmninfin

=1

2120583119905[

infin

sum

119896=0

(119905

120590)

119896

+

infin

sum

119896=0

(minus119905

120590)

119896

]

=1

2120583119905(

120590

120590 minus 119905+

120590

120590 + 119905)

(106)



120583(119896)

119883= 1205831198961205902119896

sum

119899=0

(119896

119899)

(minus120583119883120583)119899




(2)

119883minus 1205832

119883


119883and 120583(4)




119880sim LU(119890

120583minus120590 119890120583+120590


119886 119890119887)


(119905)

LU = (119890119905119887minus 119890119905119886)(1119905(119887 minus 119886))



120583 1120590 1120590) due to


(119905)

LL = 119890119905120590(1 minus 11990521205902)minus1

4 Conclusion



Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119865119883120574

(119909) = 1198621

120574int

(log 119909minus120583)120590

minusinfin


120574|119908|120574(120574minus1)

119889119908

= Φ119885120574

(log119909 minus 120583

120590)

(34)

where Φ119885120574



Φ119885120574

(119911) = 1198621

120574int

119911

minusinfin


120574|119908|120574(120574minus1)

119889119908

= Φ119885120574

(0) + 1198621

120574int

119911

0


120574|119908|120574(120574minus1)

119889119908

(35)

and as 119891119885120574


Φ119885120574

(119911) =1

2+ 1198621

120574int

119911

0


120574|119908|120574(120574minus1)

119889119908

=1

2+ 1198621

120574int

119911

0

exp

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(120574 minus 1

120574)

(120574minus1)120574

119908

1003816100381610038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119908

(36)

and thus

Φ119885120574

(119911) =1

2+ 1198621

120574(

120574

120574 minus 1)

(120574minus1)120574

times int

((120574minus1)120574)(120574minus1)120574

119911

0

exp minus119906120574(120574minus1) 119889119906

(37)


Φ119885120574

(119911) =1

2+

radic120587



120574)

(120574minus1)120574

119911 119911 isin R

(38)



119865119883120574

(119909) =1

2+



120574)

(120574minus1)120574 10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

(39)


119865119883120574

(119909) =1 + sgn (log119909 minus 120583)



120574120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

)

(40)


minus1 sdot)




119865119883120574

(119909) =1

2+

((120574 minus 1)120574)(120574minus1)120574

(2120574) Γ ((120574 minus 1) 120574)(log119909 minus 120583

120590)

times

infin

sum

119896=0

(((1 minus 120574)120574)1003816100381610038161003816(log119909 minus 120583)120590

1003816100381610038161003816120574(120574minus1)

)119896

119896 [(119896 + 1) 120574 minus 1]

119909 isin Rlowast

+

(41)



119865119883120574

(119909)=1

2+1


sgn (log119909 minus 120583)

2 exp 1198991003816100381610038161003816(log119909 minus 120583)12059010038161003816100381610038161119899

times

119899minus1

sum

119896=0

119899119896

119896

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

119896119899

(42)



sum

119896=0

119909119896

119896 119909 isin R 119899 isin N

lowast= N 0

(43)



119865119883minus1

(119909) =1

2+1


times

1 + 2radic1003816100381610038161003816(log119909 minus 120583) 120590

1003816100381610038161003816


(44)




119865119883minus12

(119909) =1

2+1


times

1 + 33radic1003816100381610038161003816(log119909 minus 120583) 120590

1003816100381610038161003816 + 93

radic((log119909 minus 120583)120590)2


(45)


1198651198832

(119909) = Φ1198832

(log119909 minus 120583

120590) =

1

2+1

2Erf2 (

log119909 minus 120583radic2120590

)

=1

2+1

2erf (

log119909 minus 120583radic2120590

)

(46)





(119909)

=1

2+1

2radic120587Erf1 (

log119909 minus 120583120590

)

=1

2minus1

2sgn (log119909 minus 120583)

infin

sum

119896=0

1

(119896 + 1)(minus

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816)

119896+1

=1

2+1


2sgn (log119909 minus 120583)

times expminus10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

(47)

and hence


(119909) =

1

2119890120583120590

1199091120590 119909 isin (0 119890

120583]

1 minus119890120583120590

21199091120590 119909 isin (119890

120583 +infin)

(48)


120583 1120590 1120590) see

Theorem 3



119865119883120574

(119909)

=1

2+

((120574 minus 1)120574)(120574minus1)120574

2Γ (((120574 minus 1) 120574) + 1)(log119909 minus 120583

120590)

times[[

[

1 + (120574 minus 1)

infin

sum

119896=1

(((1 minus 120574)120574)1003816100381610038161003816(log119909 minus 120583)120590

1003816100381610038161003816120574(120574minus1)

)119896

119896 [(119896 + 1) 120574 minus 1]

]]

]

(49)


1198651198831

(119909) = lim120574rarr1+

119865119883120574

(119909) =1

2+log119909 minus 120583

2120590(1 + 0)

=log119909 minus 120583 + 120590

2120590

(50)

with 1198651198831

(119890120583minus120590

) = 0 and 1198651198831

(119890120583+120590

) = 1 Therefore

1198651198831

(119909) =

0 119909 isin (0 119890120583minus120590

)

1

2120590(log119909 minus 120583 + 120590) 119909 isin [119890

120583minus120590 119890120583+120590

]

1 119909 isin (119890120583+120590

+infin)

(51)


120583minus120590 119887 = 119890


LN1(120583 120590) = LU(119890120583minus120590

119890120583+120590

) see Theorem 3


120583|120583=0 =


120574


Q119883120574


+| 119865119883

120574

(119909) ge 119875

= exp sgn (2119875 minus 1) 120590

times [120574


120574 |2119875 minus 1|)]

(120574minus1)120574

119875 isin (0 1)

(52)






120574 11987512057412 1198751205741 1198751205742 1198751205743 1198751205744 1198751205745 1199021120574 1199023120574



120574

(14) and Q119883120574





119883120574


mean (120583119892)119883120574

= 119890E[log119883


120574

= 119890120583 as



120583sim




119891|119884| (119909) = 119891119884 (minus119909) + 119891119884 (119909) 119909 isin R+ (53)


119891|119884| (119909) = 119891119884 (minus119909) + 119891119884 (119909) 0 le 119909 le 119890

120583

119891119884 (119909) 119909 gt 119890120583

(54)


119865|119884| (119909) = int

119909

0

119891|119884| (119905) 119889119905 = int

119890120583

0

119891119884 (minus119905) 119889119905 + int

119890120583

0

119891119884 (119905) 119889119905

+ int

119909

119890120583119891119884 (119905) 119889119905 119909 isin R+

(55)



119865|119884| (119909) = 1198621

120574int

0

minusinfin


120574|119908|120574(120574minus1)

119889119908 + 1198621

120574

times int

(1120590) log 2

0


120574|119911|120574(120574minus1)

119889119911

+ 1198621

120574int

119892(119909)


120574 minus 1

120574|119911|120574(120574minus1)

119889119911

119892 (119909) =log (119909 + 119890120583) minus 120583

120590 119909 isin R+

(56)

and hence

119865|119884| (119909) = Φ119885 (0) + [Φ119885 (log 2120590

) minus Φ119885 (0)]

+ [Φ119885 (119892 (119909)) minus Φ119885 (log 2120590

)]

= Φ119885(log (119909 + 119890120583) minus 120583

120590)

(57)


x

0 1 2 3

X2simℒ119977(0 23)


sim119977

120574(023)

eminus23 e23

0

01

02

03

04

05

06

07

08

09

1

X1112 19

X3410



FX120574



(a)

120574(01)

0

01

02

03

04

05

06

07

08

09

1

X2simℒ119977(0 1)



x

0 1 2 3


X1112 19

X3410

eminus1 e1

FX120574sim119977

(b)

00

01

02

03

04

05

06

07

08

09

1

x

1 2 3

eminus32


X2simℒ119977(0 32)

X1112 19

X3410


119977120574(032)

FX120574sim119977



(c)




+ it is clear that


|119883120574minus119890120583|







120583(119905)

119883=

119890119905120583

Γ ((120574 minus 1) 120574)

infin

sum

119899=0

(119905120590)2119899

(2119899)(

120574

120574 minus 1)

2119899((120574minus1)120574)

times Γ((2119899 + 1)120574 minus 1

120574)

(58)


(119905)

119883


have

120583(119905)

119883= E [119883119905] = int

R+

119909119905119891119883 (119909) 119889119909

=1

1205901198621

120574intR+


120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119909

(59)


(1120590)


120583(119905)

119883= 1198621

120574(

120574

120574 minus 1)

(120574minus1)120574

intR

exp119905120583 + 119899(120574

120574 minus 1)

(120574minus1)120574

120590119911



exp119905(120574

120574 minus 1)

(120574minus1)120574

120590119911 =

infin

sum

119899=0

(119905120590)119899

119899(

120574

120574 minus 1)

119899((120574minus1)120574)

119911119899

(61)

it is obtained that

120583(119905)

119883= 21198621

120574(

120574

120574 minus 1)

(120574minus1)120574

119890119905120583infin

sum

119899=0

(119905120590)2119899

(2119899)(

120574

120574 minus 1)

2119899((120574minus1)120574)

times intR+

1199112119899 exp minus119911120574(120574minus1) 119889119911

(62)



intR+

119909119898119890minus119887119909119899

119889119909 =Γ ((119898 + 1) 119899)

119899119887(119898+1)119899 119899 119898 119887 isin R

lowast

+ (63)


2) we have 119872119884(119905) =

E[119890119905119884] = E[119883119905] = 120583(119905)



120583(119905)

119883=

119890119905120583

radic120587

infin

sum

119899=0

(211990521205902)119899

(2119899)Γ (119899 +

1

2) 119905 isin R+ (64)


Γ (119899 +1

2) =

(2119899)

22119899119899radic120587 119899 isin N (65)

we have

120583(119905)

119883= 119890119905120583infin

sum

119899=0

(119905120590)2119899

2119899119899= 119890119905120583infin

sum

119899=0

1

119899(1

211990521205902)

119899

= 119890119905120583+(12)(119905120590)

2

119905 isin R+

(66)


(1)

119883= E[119883] =


(119905)

119883= exp119905120583 +





LN120574(120583 120590) is given by

120583(119896)

119883=

119890119896120583

Γ ((120574 minus 1) 120574)

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899

119878119896minus119899 119896 isin N (67)

where

119878119896 =

infin

sum

119898=0

(119896120590)2119898

(2119898)(

120574

120574 minus 1)

2119898((120574minus1)120574)

Γ((2119898 + 1)120574 minus 1

120574)

119896 isin N

(68)


wehave

120583(119896)

119883= E [(119883 minus 120583119883)

119896] = int

R+

(119909 minus 120583119883)119896119891119883 (119909 120583 120590 120574) 119889119909

(69)


120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus120583119883)

119899intR+

119909119896minus119899

119891119883 (119909) 119889119909

=

119896

sum

119899=0

(119896

119899) (minus120583119883)

119899120583(119896minus119899)

119883

(70)


120583(119896)

119883=

119890119896120583

Γ ((120574 minus 1) 120574)

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898)

times (120574

120574 minus 1)

2119898((120574minus1)120574)

Γ ((2119898 + 1) ((120574 minus 1) 120574))

(71)




120583+(12)1205902


120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus1)

119899119890119896120583+(12)[119899+(119896minus119899)

2

]1205902

119896 isin N (72)

while

1205902

119883= Var [119883] = 120583

(2)

119883= 1198902120583+1205902

(1198901205902

minus 1) (73)





120583119883 =119890120583

Γ ((120574 minus 1) 120574)1198781 (74)

1205902

119883= minus 120583

2

119883+

1198902120583

Γ ((120574 minus 1) 120574)1198782 (75)

1198621198812

119883= Γ(

120574 minus 1

120574)1198782

11987821

minus 1 (76)

120582119883 = minus 119862119881minus3

119883minus 119862119881minus1

119883+

1198903120583

1205903119883Γ ((120574 minus 1) 120574)

1198783 (77)

120581119883 = minus 119862119881minus4

119883minus 6119862119881

minus2

119883minus 4

120582119883

119862119881119883

+1198904120583

1205904119883Γ ((120574 minus 1) 120574)

1198784

(78)



120583(1)


1205902

119883= 120583(2)

119883= 1205832

119883+ [Γ(

120574 minus 1

120574)]

minus1

(11989021205831198782 minus 2119890

1205831205831198831198781) (79)


(3)

1198831205903

119883


120582119883 = minus119862119881minus3

119883+ [1205903

119883Γ(

120574 minus 1

120574)]

minus1

times (11989031205831198783 minus 3119890

21205831205831198831198782 + 3119890

1205831205831198831198781)

(80)


(4)

1198831205904

119883 which


120581119883 = 119862119881minus4

119883+ [1205904

119883Γ(

120574 minus 1

120574)]

minus1

times (11989041205831198784 minus 4119890

31205831205831198831198783 + 6119890

21205831205832

1198831198782 minus 4119890

1205831205833

1198831198781)

(81)



radic120587119890(1198992

1205902


120583119883 = 119890120583+(12)120590

2

1205902

119883= 1198902120583+1205902

(1198901205902

minus 1)

119862119881119883 =radic1198901205902

minus 1

120582119883 = (1198901205902

+ 2)radic1198901205902

minus 1 120581119883 = 11989041205902

+ 211989031205902

+ 311989021205902

minus 3

(82)




120574

le

120583119883120574



Med119883120574 = (120583119892)119883120574

= 119890120583lt 120583119883

120574

(83)



Mode119883120574 = 119890120583minus120590120574

lt 119890120583= Med119883120574 (84)


Mode119883120574 = 119890120583= Med119883120574 (85)



Mode119883120574 = 0 lt 119890120583= Med119883120574 (86)





120583(119905)

119883=

119887119905minus 119886119905

119905 log (119887119886) 119905 isin R (87)

120583(119896)

119883=

(119886 minus 119887)119896

log119896 (119887119886)+

1

log (119887119886)

times

119896minus1

sum

119899=0

(119896

119899)(119886 minus 119887)

119899(119887119896minus119899

minus 119886119896minus119899

)

(119896 minus 119899) log119899 (119887119886) 119896 isin N

(88)


120583119883 =119887 minus 119886

log (119887119886) (89)

1205902

119883=

(119887 minus 119886)2

log2 (119887119886)+(119887 minus 119886) (119887 + 119886)

2 log (119887119886) (90)

119862119881119883 = radic1 +119887 + 119886

2 (119887 minus 119886)log 119887

119886 (91)

120582119883 =1

1205903119883

[1198873minus 1198863

3 log (119887119886)minus 3

(119887 minus 119886)2(119887 + 119886)

2 log2 (119887119886)+ 2

(119887 minus 119886)3

log3 (119887119886)]

(92)

120581119883 =1

1205904119883

[1198874minus 1198864

4 log (119887119886)minus 4

(119887 minus 119886) (1198873minus 1198863)

3 log2 (119887119886)

+3(119887 minus 119886)

3(119887 + 119886)

log3 (119887119886)minus 3

(119887 minus 119886)4

log4 (119887119886)]

(93)


120583(119905)

119883120574

=119890119905120583

119905120590Γ ((120574 minus 1) 120574 + 1)

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)(

120574

120574 minus 1)

2119898((120574minus1)120574)

times Γ((2119898 + 1)120574 minus 1

120574+ 1)

(94)


120583(119905)

119883= lim120574rarr1+

120583(119905)

119883120574

=119890119905120583

119905120590

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)=119890119905(120583+120590)

minus 119890119905(120583minus120590)

2119905120590

119905 isin R

(95)


119883= E[119883] = (12120590)

(119890120583+120590

minus 119890120583minus120590


120583(119896)

119883= lim120574rarr1+

120583(119896)

119883120574

= 119890119896120583119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898 + 1)

119896 isin N

(96)


120583(119896)

119883=119890119896120583

2120590

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 119890(119896minus119899)120590


(119896 minus 119899) 119896 isin N

(97)


Var[119883] = 120583(2)

119883minus 1205832


For 119896 = 3 and 119896 = 4 through 120583(3)119883

and 120583(4)119883 we obtain (92) and

(93) respectively


120583(119905)

119883=

1205831199051205902

1205902 minus 1199052gt 120583119905 120590 gt 119905 119905 isin R (98)

120583(119896)

119883= 120583119896119896

sum

119899=0

(119896

119899)

1205902(119899+1)


2] 120590 gt 119896 119896 isin N

(99)


120583119883 =1205831205902

1205902 minus 1gt 120583 120590 gt 1 (100)

1205902

119883=

12058321205902(21205902+ 1)

(1205902 minus 4) (1205902 minus 1)2 120590 gt 2 (101)

119862119881119883 =1

120590

radic21205902+ 1

1205902 minus 4 120590 gt 2 (102)

120582119883 =2 (15120590

4+ 71205902+ 2)

120590 (1205902 minus 9)radic

1205902minus 4

(21205902 + 1)3 120590 gt 3 (103)


120581119883 =3 (81205908+ 212120590

6+ 95120590

4+ 33120590

2+ 12) (120590

2minus 4)

(1205902 minus 16) (1205902 minus 9) (21205902 + 1)2

120590 gt 4

(104)


120583(119905)

119883= 120583(119905)

119883plusmninfin

= 120583119905infin

sum

119896=0

(119905

120590)

2119896

119905 isin R (105)


120583(119905)

119883plusmninfin

=1

2120583119905[

infin

sum

119896=0

(119905

120590)

119896

+

infin

sum

119896=0

(minus119905

120590)

119896

]

=1

2120583119905(

120590

120590 minus 119905+

120590

120590 + 119905)

(106)



120583(119896)

119883= 1205831198961205902119896

sum

119899=0

(119896

119899)

(minus120583119883120583)119899




(2)

119883minus 1205832

119883


119883and 120583(4)




119880sim LU(119890

120583minus120590 119890120583+120590


119886 119890119887)


(119905)

LU = (119890119905119887minus 119890119905119886)(1119905(119887 minus 119886))



120583 1120590 1120590) due to


(119905)

LL = 119890119905120590(1 minus 11990521205902)minus1

4 Conclusion



Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119865119883minus12

(119909) =1

2+1


times

1 + 33radic1003816100381610038161003816(log119909 minus 120583) 120590

1003816100381610038161003816 + 93

radic((log119909 minus 120583)120590)2


(45)


1198651198832

(119909) = Φ1198832

(log119909 minus 120583

120590) =

1

2+1

2Erf2 (

log119909 minus 120583radic2120590

)

=1

2+1

2erf (

log119909 minus 120583radic2120590

)

(46)





(119909)

=1

2+1

2radic120587Erf1 (

log119909 minus 120583120590

)

=1

2minus1

2sgn (log119909 minus 120583)

infin

sum

119896=0

1

(119896 + 1)(minus

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816)

119896+1

=1

2+1


2sgn (log119909 minus 120583)

times expminus10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

(47)

and hence


(119909) =

1

2119890120583120590

1199091120590 119909 isin (0 119890

120583]

1 minus119890120583120590

21199091120590 119909 isin (119890

120583 +infin)

(48)


120583 1120590 1120590) see

Theorem 3



119865119883120574

(119909)

=1

2+

((120574 minus 1)120574)(120574minus1)120574

2Γ (((120574 minus 1) 120574) + 1)(log119909 minus 120583

120590)

times[[

[

1 + (120574 minus 1)

infin

sum

119896=1

(((1 minus 120574)120574)1003816100381610038161003816(log119909 minus 120583)120590

1003816100381610038161003816120574(120574minus1)

)119896

119896 [(119896 + 1) 120574 minus 1]

]]

]

(49)


1198651198831

(119909) = lim120574rarr1+

119865119883120574

(119909) =1

2+log119909 minus 120583

2120590(1 + 0)

=log119909 minus 120583 + 120590

2120590

(50)

with 1198651198831

(119890120583minus120590

) = 0 and 1198651198831

(119890120583+120590

) = 1 Therefore

1198651198831

(119909) =

0 119909 isin (0 119890120583minus120590

)

1

2120590(log119909 minus 120583 + 120590) 119909 isin [119890

120583minus120590 119890120583+120590

]

1 119909 isin (119890120583+120590

+infin)

(51)


120583minus120590 119887 = 119890


LN1(120583 120590) = LU(119890120583minus120590

119890120583+120590

) see Theorem 3


120583|120583=0 =


120574


Q119883120574


+| 119865119883

120574

(119909) ge 119875

= exp sgn (2119875 minus 1) 120590

times [120574


120574 |2119875 minus 1|)]

(120574minus1)120574

119875 isin (0 1)

(52)






120574 11987512057412 1198751205741 1198751205742 1198751205743 1198751205744 1198751205745 1199021120574 1199023120574



120574

(14) and Q119883120574





119883120574


mean (120583119892)119883120574

= 119890E[log119883


120574

= 119890120583 as



120583sim




119891|119884| (119909) = 119891119884 (minus119909) + 119891119884 (119909) 119909 isin R+ (53)


119891|119884| (119909) = 119891119884 (minus119909) + 119891119884 (119909) 0 le 119909 le 119890

120583

119891119884 (119909) 119909 gt 119890120583

(54)


119865|119884| (119909) = int

119909

0

119891|119884| (119905) 119889119905 = int

119890120583

0

119891119884 (minus119905) 119889119905 + int

119890120583

0

119891119884 (119905) 119889119905

+ int

119909

119890120583119891119884 (119905) 119889119905 119909 isin R+

(55)



119865|119884| (119909) = 1198621

120574int

0

minusinfin


120574|119908|120574(120574minus1)

119889119908 + 1198621

120574

times int

(1120590) log 2

0


120574|119911|120574(120574minus1)

119889119911

+ 1198621

120574int

119892(119909)


120574 minus 1

120574|119911|120574(120574minus1)

119889119911

119892 (119909) =log (119909 + 119890120583) minus 120583

120590 119909 isin R+

(56)

and hence

119865|119884| (119909) = Φ119885 (0) + [Φ119885 (log 2120590

) minus Φ119885 (0)]

+ [Φ119885 (119892 (119909)) minus Φ119885 (log 2120590

)]

= Φ119885(log (119909 + 119890120583) minus 120583

120590)

(57)


x

0 1 2 3

X2simℒ119977(0 23)


sim119977

120574(023)

eminus23 e23

0

01

02

03

04

05

06

07

08

09

1

X1112 19

X3410



FX120574



(a)

120574(01)

0

01

02

03

04

05

06

07

08

09

1

X2simℒ119977(0 1)



x

0 1 2 3


X1112 19

X3410

eminus1 e1

FX120574sim119977

(b)

00

01

02

03

04

05

06

07

08

09

1

x

1 2 3

eminus32


X2simℒ119977(0 32)

X1112 19

X3410


119977120574(032)

FX120574sim119977



(c)




+ it is clear that


|119883120574minus119890120583|







120583(119905)

119883=

119890119905120583

Γ ((120574 minus 1) 120574)

infin

sum

119899=0

(119905120590)2119899

(2119899)(

120574

120574 minus 1)

2119899((120574minus1)120574)

times Γ((2119899 + 1)120574 minus 1

120574)

(58)


(119905)

119883


have

120583(119905)

119883= E [119883119905] = int

R+

119909119905119891119883 (119909) 119889119909

=1

1205901198621

120574intR+


120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119909

(59)


(1120590)


120583(119905)

119883= 1198621

120574(

120574

120574 minus 1)

(120574minus1)120574

intR

exp119905120583 + 119899(120574

120574 minus 1)

(120574minus1)120574

120590119911



exp119905(120574

120574 minus 1)

(120574minus1)120574

120590119911 =

infin

sum

119899=0

(119905120590)119899

119899(

120574

120574 minus 1)

119899((120574minus1)120574)

119911119899

(61)

it is obtained that

120583(119905)

119883= 21198621

120574(

120574

120574 minus 1)

(120574minus1)120574

119890119905120583infin

sum

119899=0

(119905120590)2119899

(2119899)(

120574

120574 minus 1)

2119899((120574minus1)120574)

times intR+

1199112119899 exp minus119911120574(120574minus1) 119889119911

(62)



intR+

119909119898119890minus119887119909119899

119889119909 =Γ ((119898 + 1) 119899)

119899119887(119898+1)119899 119899 119898 119887 isin R

lowast

+ (63)


2) we have 119872119884(119905) =

E[119890119905119884] = E[119883119905] = 120583(119905)



120583(119905)

119883=

119890119905120583

radic120587

infin

sum

119899=0

(211990521205902)119899

(2119899)Γ (119899 +

1

2) 119905 isin R+ (64)


Γ (119899 +1

2) =

(2119899)

22119899119899radic120587 119899 isin N (65)

we have

120583(119905)

119883= 119890119905120583infin

sum

119899=0

(119905120590)2119899

2119899119899= 119890119905120583infin

sum

119899=0

1

119899(1

211990521205902)

119899

= 119890119905120583+(12)(119905120590)

2

119905 isin R+

(66)


(1)

119883= E[119883] =


(119905)

119883= exp119905120583 +





LN120574(120583 120590) is given by

120583(119896)

119883=

119890119896120583

Γ ((120574 minus 1) 120574)

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899

119878119896minus119899 119896 isin N (67)

where

119878119896 =

infin

sum

119898=0

(119896120590)2119898

(2119898)(

120574

120574 minus 1)

2119898((120574minus1)120574)

Γ((2119898 + 1)120574 minus 1

120574)

119896 isin N

(68)


wehave

120583(119896)

119883= E [(119883 minus 120583119883)

119896] = int

R+

(119909 minus 120583119883)119896119891119883 (119909 120583 120590 120574) 119889119909

(69)


120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus120583119883)

119899intR+

119909119896minus119899

119891119883 (119909) 119889119909

=

119896

sum

119899=0

(119896

119899) (minus120583119883)

119899120583(119896minus119899)

119883

(70)


120583(119896)

119883=

119890119896120583

Γ ((120574 minus 1) 120574)

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898)

times (120574

120574 minus 1)

2119898((120574minus1)120574)

Γ ((2119898 + 1) ((120574 minus 1) 120574))

(71)




120583+(12)1205902


120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus1)

119899119890119896120583+(12)[119899+(119896minus119899)

2

]1205902

119896 isin N (72)

while

1205902

119883= Var [119883] = 120583

(2)

119883= 1198902120583+1205902

(1198901205902

minus 1) (73)





120583119883 =119890120583

Γ ((120574 minus 1) 120574)1198781 (74)

1205902

119883= minus 120583

2

119883+

1198902120583

Γ ((120574 minus 1) 120574)1198782 (75)

1198621198812

119883= Γ(

120574 minus 1

120574)1198782

11987821

minus 1 (76)

120582119883 = minus 119862119881minus3

119883minus 119862119881minus1

119883+

1198903120583

1205903119883Γ ((120574 minus 1) 120574)

1198783 (77)

120581119883 = minus 119862119881minus4

119883minus 6119862119881

minus2

119883minus 4

120582119883

119862119881119883

+1198904120583

1205904119883Γ ((120574 minus 1) 120574)

1198784

(78)



120583(1)


1205902

119883= 120583(2)

119883= 1205832

119883+ [Γ(

120574 minus 1

120574)]

minus1

(11989021205831198782 minus 2119890

1205831205831198831198781) (79)


(3)

1198831205903

119883


120582119883 = minus119862119881minus3

119883+ [1205903

119883Γ(

120574 minus 1

120574)]

minus1

times (11989031205831198783 minus 3119890

21205831205831198831198782 + 3119890

1205831205831198831198781)

(80)


(4)

1198831205904

119883 which


120581119883 = 119862119881minus4

119883+ [1205904

119883Γ(

120574 minus 1

120574)]

minus1

times (11989041205831198784 minus 4119890

31205831205831198831198783 + 6119890

21205831205832

1198831198782 minus 4119890

1205831205833

1198831198781)

(81)



radic120587119890(1198992

1205902


120583119883 = 119890120583+(12)120590

2

1205902

119883= 1198902120583+1205902

(1198901205902

minus 1)

119862119881119883 =radic1198901205902

minus 1

120582119883 = (1198901205902

+ 2)radic1198901205902

minus 1 120581119883 = 11989041205902

+ 211989031205902

+ 311989021205902

minus 3

(82)




120574

le

120583119883120574



Med119883120574 = (120583119892)119883120574

= 119890120583lt 120583119883

120574

(83)



Mode119883120574 = 119890120583minus120590120574

lt 119890120583= Med119883120574 (84)


Mode119883120574 = 119890120583= Med119883120574 (85)



Mode119883120574 = 0 lt 119890120583= Med119883120574 (86)





120583(119905)

119883=

119887119905minus 119886119905

119905 log (119887119886) 119905 isin R (87)

120583(119896)

119883=

(119886 minus 119887)119896

log119896 (119887119886)+

1

log (119887119886)

times

119896minus1

sum

119899=0

(119896

119899)(119886 minus 119887)

119899(119887119896minus119899

minus 119886119896minus119899

)

(119896 minus 119899) log119899 (119887119886) 119896 isin N

(88)


120583119883 =119887 minus 119886

log (119887119886) (89)

1205902

119883=

(119887 minus 119886)2

log2 (119887119886)+(119887 minus 119886) (119887 + 119886)

2 log (119887119886) (90)

119862119881119883 = radic1 +119887 + 119886

2 (119887 minus 119886)log 119887

119886 (91)

120582119883 =1

1205903119883

[1198873minus 1198863

3 log (119887119886)minus 3

(119887 minus 119886)2(119887 + 119886)

2 log2 (119887119886)+ 2

(119887 minus 119886)3

log3 (119887119886)]

(92)

120581119883 =1

1205904119883

[1198874minus 1198864

4 log (119887119886)minus 4

(119887 minus 119886) (1198873minus 1198863)

3 log2 (119887119886)

+3(119887 minus 119886)

3(119887 + 119886)

log3 (119887119886)minus 3

(119887 minus 119886)4

log4 (119887119886)]

(93)


120583(119905)

119883120574

=119890119905120583

119905120590Γ ((120574 minus 1) 120574 + 1)

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)(

120574

120574 minus 1)

2119898((120574minus1)120574)

times Γ((2119898 + 1)120574 minus 1

120574+ 1)

(94)


120583(119905)

119883= lim120574rarr1+

120583(119905)

119883120574

=119890119905120583

119905120590

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)=119890119905(120583+120590)

minus 119890119905(120583minus120590)

2119905120590

119905 isin R

(95)


119883= E[119883] = (12120590)

(119890120583+120590

minus 119890120583minus120590


120583(119896)

119883= lim120574rarr1+

120583(119896)

119883120574

= 119890119896120583119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898 + 1)

119896 isin N

(96)


120583(119896)

119883=119890119896120583

2120590

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 119890(119896minus119899)120590


(119896 minus 119899) 119896 isin N

(97)


Var[119883] = 120583(2)

119883minus 1205832


For 119896 = 3 and 119896 = 4 through 120583(3)119883

and 120583(4)119883 we obtain (92) and

(93) respectively


120583(119905)

119883=

1205831199051205902

1205902 minus 1199052gt 120583119905 120590 gt 119905 119905 isin R (98)

120583(119896)

119883= 120583119896119896

sum

119899=0

(119896

119899)

1205902(119899+1)


2] 120590 gt 119896 119896 isin N

(99)


120583119883 =1205831205902

1205902 minus 1gt 120583 120590 gt 1 (100)

1205902

119883=

12058321205902(21205902+ 1)

(1205902 minus 4) (1205902 minus 1)2 120590 gt 2 (101)

119862119881119883 =1

120590

radic21205902+ 1

1205902 minus 4 120590 gt 2 (102)

120582119883 =2 (15120590

4+ 71205902+ 2)

120590 (1205902 minus 9)radic

1205902minus 4

(21205902 + 1)3 120590 gt 3 (103)


120581119883 =3 (81205908+ 212120590

6+ 95120590

4+ 33120590

2+ 12) (120590

2minus 4)

(1205902 minus 16) (1205902 minus 9) (21205902 + 1)2

120590 gt 4

(104)


120583(119905)

119883= 120583(119905)

119883plusmninfin

= 120583119905infin

sum

119896=0

(119905

120590)

2119896

119905 isin R (105)


120583(119905)

119883plusmninfin

=1

2120583119905[

infin

sum

119896=0

(119905

120590)

119896

+

infin

sum

119896=0

(minus119905

120590)

119896

]

=1

2120583119905(

120590

120590 minus 119905+

120590

120590 + 119905)

(106)



120583(119896)

119883= 1205831198961205902119896

sum

119899=0

(119896

119899)

(minus120583119883120583)119899




(2)

119883minus 1205832

119883


119883and 120583(4)




119880sim LU(119890

120583minus120590 119890120583+120590


119886 119890119887)


(119905)

LU = (119890119905119887minus 119890119905119886)(1119905(119887 minus 119886))



120583 1120590 1120590) due to


(119905)

LL = 119890119905120590(1 minus 11990521205902)minus1

4 Conclusion



Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120574 11987512057412 1198751205741 1198751205742 1198751205743 1198751205744 1198751205745 1199021120574 1199023120574



120574

(14) and Q119883120574





119883120574


mean (120583119892)119883120574

= 119890E[log119883


120574

= 119890120583 as



120583sim




119891|119884| (119909) = 119891119884 (minus119909) + 119891119884 (119909) 119909 isin R+ (53)


119891|119884| (119909) = 119891119884 (minus119909) + 119891119884 (119909) 0 le 119909 le 119890

120583

119891119884 (119909) 119909 gt 119890120583

(54)


119865|119884| (119909) = int

119909

0

119891|119884| (119905) 119889119905 = int

119890120583

0

119891119884 (minus119905) 119889119905 + int

119890120583

0

119891119884 (119905) 119889119905

+ int

119909

119890120583119891119884 (119905) 119889119905 119909 isin R+

(55)



119865|119884| (119909) = 1198621

120574int

0

minusinfin


120574|119908|120574(120574minus1)

119889119908 + 1198621

120574

times int

(1120590) log 2

0


120574|119911|120574(120574minus1)

119889119911

+ 1198621

120574int

119892(119909)


120574 minus 1

120574|119911|120574(120574minus1)

119889119911

119892 (119909) =log (119909 + 119890120583) minus 120583

120590 119909 isin R+

(56)

and hence

119865|119884| (119909) = Φ119885 (0) + [Φ119885 (log 2120590

) minus Φ119885 (0)]

+ [Φ119885 (119892 (119909)) minus Φ119885 (log 2120590

)]

= Φ119885(log (119909 + 119890120583) minus 120583

120590)

(57)


x

0 1 2 3

X2simℒ119977(0 23)


sim119977

120574(023)

eminus23 e23

0

01

02

03

04

05

06

07

08

09

1

X1112 19

X3410



FX120574



(a)

120574(01)

0

01

02

03

04

05

06

07

08

09

1

X2simℒ119977(0 1)



x

0 1 2 3


X1112 19

X3410

eminus1 e1

FX120574sim119977

(b)

00

01

02

03

04

05

06

07

08

09

1

x

1 2 3

eminus32


X2simℒ119977(0 32)

X1112 19

X3410


119977120574(032)

FX120574sim119977



(c)




+ it is clear that


|119883120574minus119890120583|







120583(119905)

119883=

119890119905120583

Γ ((120574 minus 1) 120574)

infin

sum

119899=0

(119905120590)2119899

(2119899)(

120574

120574 minus 1)

2119899((120574minus1)120574)

times Γ((2119899 + 1)120574 minus 1

120574)

(58)


(119905)

119883


have

120583(119905)

119883= E [119883119905] = int

R+

119909119905119891119883 (119909) 119889119909

=1

1205901198621

120574intR+


120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119909

(59)


(1120590)


120583(119905)

119883= 1198621

120574(

120574

120574 minus 1)

(120574minus1)120574

intR

exp119905120583 + 119899(120574

120574 minus 1)

(120574minus1)120574

120590119911



exp119905(120574

120574 minus 1)

(120574minus1)120574

120590119911 =

infin

sum

119899=0

(119905120590)119899

119899(

120574

120574 minus 1)

119899((120574minus1)120574)

119911119899

(61)

it is obtained that

120583(119905)

119883= 21198621

120574(

120574

120574 minus 1)

(120574minus1)120574

119890119905120583infin

sum

119899=0

(119905120590)2119899

(2119899)(

120574

120574 minus 1)

2119899((120574minus1)120574)

times intR+

1199112119899 exp minus119911120574(120574minus1) 119889119911

(62)



intR+

119909119898119890minus119887119909119899

119889119909 =Γ ((119898 + 1) 119899)

119899119887(119898+1)119899 119899 119898 119887 isin R

lowast

+ (63)


2) we have 119872119884(119905) =

E[119890119905119884] = E[119883119905] = 120583(119905)



120583(119905)

119883=

119890119905120583

radic120587

infin

sum

119899=0

(211990521205902)119899

(2119899)Γ (119899 +

1

2) 119905 isin R+ (64)


Γ (119899 +1

2) =

(2119899)

22119899119899radic120587 119899 isin N (65)

we have

120583(119905)

119883= 119890119905120583infin

sum

119899=0

(119905120590)2119899

2119899119899= 119890119905120583infin

sum

119899=0

1

119899(1

211990521205902)

119899

= 119890119905120583+(12)(119905120590)

2

119905 isin R+

(66)


(1)

119883= E[119883] =


(119905)

119883= exp119905120583 +





LN120574(120583 120590) is given by

120583(119896)

119883=

119890119896120583

Γ ((120574 minus 1) 120574)

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899

119878119896minus119899 119896 isin N (67)

where

119878119896 =

infin

sum

119898=0

(119896120590)2119898

(2119898)(

120574

120574 minus 1)

2119898((120574minus1)120574)

Γ((2119898 + 1)120574 minus 1

120574)

119896 isin N

(68)


wehave

120583(119896)

119883= E [(119883 minus 120583119883)

119896] = int

R+

(119909 minus 120583119883)119896119891119883 (119909 120583 120590 120574) 119889119909

(69)


120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus120583119883)

119899intR+

119909119896minus119899

119891119883 (119909) 119889119909

=

119896

sum

119899=0

(119896

119899) (minus120583119883)

119899120583(119896minus119899)

119883

(70)


120583(119896)

119883=

119890119896120583

Γ ((120574 minus 1) 120574)

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898)

times (120574

120574 minus 1)

2119898((120574minus1)120574)

Γ ((2119898 + 1) ((120574 minus 1) 120574))

(71)




120583+(12)1205902


120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus1)

119899119890119896120583+(12)[119899+(119896minus119899)

2

]1205902

119896 isin N (72)

while

1205902

119883= Var [119883] = 120583

(2)

119883= 1198902120583+1205902

(1198901205902

minus 1) (73)





120583119883 =119890120583

Γ ((120574 minus 1) 120574)1198781 (74)

1205902

119883= minus 120583

2

119883+

1198902120583

Γ ((120574 minus 1) 120574)1198782 (75)

1198621198812

119883= Γ(

120574 minus 1

120574)1198782

11987821

minus 1 (76)

120582119883 = minus 119862119881minus3

119883minus 119862119881minus1

119883+

1198903120583

1205903119883Γ ((120574 minus 1) 120574)

1198783 (77)

120581119883 = minus 119862119881minus4

119883minus 6119862119881

minus2

119883minus 4

120582119883

119862119881119883

+1198904120583

1205904119883Γ ((120574 minus 1) 120574)

1198784

(78)



120583(1)


1205902

119883= 120583(2)

119883= 1205832

119883+ [Γ(

120574 minus 1

120574)]

minus1

(11989021205831198782 minus 2119890

1205831205831198831198781) (79)


(3)

1198831205903

119883


120582119883 = minus119862119881minus3

119883+ [1205903

119883Γ(

120574 minus 1

120574)]

minus1

times (11989031205831198783 minus 3119890

21205831205831198831198782 + 3119890

1205831205831198831198781)

(80)


(4)

1198831205904

119883 which


120581119883 = 119862119881minus4

119883+ [1205904

119883Γ(

120574 minus 1

120574)]

minus1

times (11989041205831198784 minus 4119890

31205831205831198831198783 + 6119890

21205831205832

1198831198782 minus 4119890

1205831205833

1198831198781)

(81)



radic120587119890(1198992

1205902


120583119883 = 119890120583+(12)120590

2

1205902

119883= 1198902120583+1205902

(1198901205902

minus 1)

119862119881119883 =radic1198901205902

minus 1

120582119883 = (1198901205902

+ 2)radic1198901205902

minus 1 120581119883 = 11989041205902

+ 211989031205902

+ 311989021205902

minus 3

(82)




120574

le

120583119883120574



Med119883120574 = (120583119892)119883120574

= 119890120583lt 120583119883

120574

(83)



Mode119883120574 = 119890120583minus120590120574

lt 119890120583= Med119883120574 (84)


Mode119883120574 = 119890120583= Med119883120574 (85)



Mode119883120574 = 0 lt 119890120583= Med119883120574 (86)





120583(119905)

119883=

119887119905minus 119886119905

119905 log (119887119886) 119905 isin R (87)

120583(119896)

119883=

(119886 minus 119887)119896

log119896 (119887119886)+

1

log (119887119886)

times

119896minus1

sum

119899=0

(119896

119899)(119886 minus 119887)

119899(119887119896minus119899

minus 119886119896minus119899

)

(119896 minus 119899) log119899 (119887119886) 119896 isin N

(88)


120583119883 =119887 minus 119886

log (119887119886) (89)

1205902

119883=

(119887 minus 119886)2

log2 (119887119886)+(119887 minus 119886) (119887 + 119886)

2 log (119887119886) (90)

119862119881119883 = radic1 +119887 + 119886

2 (119887 minus 119886)log 119887

119886 (91)

120582119883 =1

1205903119883

[1198873minus 1198863

3 log (119887119886)minus 3

(119887 minus 119886)2(119887 + 119886)

2 log2 (119887119886)+ 2

(119887 minus 119886)3

log3 (119887119886)]

(92)

120581119883 =1

1205904119883

[1198874minus 1198864

4 log (119887119886)minus 4

(119887 minus 119886) (1198873minus 1198863)

3 log2 (119887119886)

+3(119887 minus 119886)

3(119887 + 119886)

log3 (119887119886)minus 3

(119887 minus 119886)4

log4 (119887119886)]

(93)


120583(119905)

119883120574

=119890119905120583

119905120590Γ ((120574 minus 1) 120574 + 1)

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)(

120574

120574 minus 1)

2119898((120574minus1)120574)

times Γ((2119898 + 1)120574 minus 1

120574+ 1)

(94)


120583(119905)

119883= lim120574rarr1+

120583(119905)

119883120574

=119890119905120583

119905120590

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)=119890119905(120583+120590)

minus 119890119905(120583minus120590)

2119905120590

119905 isin R

(95)


119883= E[119883] = (12120590)

(119890120583+120590

minus 119890120583minus120590


120583(119896)

119883= lim120574rarr1+

120583(119896)

119883120574

= 119890119896120583119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898 + 1)

119896 isin N

(96)


120583(119896)

119883=119890119896120583

2120590

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 119890(119896minus119899)120590


(119896 minus 119899) 119896 isin N

(97)


Var[119883] = 120583(2)

119883minus 1205832


For 119896 = 3 and 119896 = 4 through 120583(3)119883

and 120583(4)119883 we obtain (92) and

(93) respectively


120583(119905)

119883=

1205831199051205902

1205902 minus 1199052gt 120583119905 120590 gt 119905 119905 isin R (98)

120583(119896)

119883= 120583119896119896

sum

119899=0

(119896

119899)

1205902(119899+1)


2] 120590 gt 119896 119896 isin N

(99)


120583119883 =1205831205902

1205902 minus 1gt 120583 120590 gt 1 (100)

1205902

119883=

12058321205902(21205902+ 1)

(1205902 minus 4) (1205902 minus 1)2 120590 gt 2 (101)

119862119881119883 =1

120590

radic21205902+ 1

1205902 minus 4 120590 gt 2 (102)

120582119883 =2 (15120590

4+ 71205902+ 2)

120590 (1205902 minus 9)radic

1205902minus 4

(21205902 + 1)3 120590 gt 3 (103)


120581119883 =3 (81205908+ 212120590

6+ 95120590

4+ 33120590

2+ 12) (120590

2minus 4)

(1205902 minus 16) (1205902 minus 9) (21205902 + 1)2

120590 gt 4

(104)


120583(119905)

119883= 120583(119905)

119883plusmninfin

= 120583119905infin

sum

119896=0

(119905

120590)

2119896

119905 isin R (105)


120583(119905)

119883plusmninfin

=1

2120583119905[

infin

sum

119896=0

(119905

120590)

119896

+

infin

sum

119896=0

(minus119905

120590)

119896

]

=1

2120583119905(

120590

120590 minus 119905+

120590

120590 + 119905)

(106)



120583(119896)

119883= 1205831198961205902119896

sum

119899=0

(119896

119899)

(minus120583119883120583)119899




(2)

119883minus 1205832

119883


119883and 120583(4)




119880sim LU(119890

120583minus120590 119890120583+120590


119886 119890119887)


(119905)

LU = (119890119905119887minus 119890119905119886)(1119905(119887 minus 119886))



120583 1120590 1120590) due to


(119905)

LL = 119890119905120590(1 minus 11990521205902)minus1

4 Conclusion



Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x

0 1 2 3

X2simℒ119977(0 23)


sim119977

120574(023)

eminus23 e23

0

01

02

03

04

05

06

07

08

09

1

X1112 19

X3410



FX120574



(a)

120574(01)

0

01

02

03

04

05

06

07

08

09

1

X2simℒ119977(0 1)



x

0 1 2 3


X1112 19

X3410

eminus1 e1

FX120574sim119977

(b)

00

01

02

03

04

05

06

07

08

09

1

x

1 2 3

eminus32


X2simℒ119977(0 32)

X1112 19

X3410


119977120574(032)

FX120574sim119977



(c)




+ it is clear that


|119883120574minus119890120583|







120583(119905)

119883=

119890119905120583

Γ ((120574 minus 1) 120574)

infin

sum

119899=0

(119905120590)2119899

(2119899)(

120574

120574 minus 1)

2119899((120574minus1)120574)

times Γ((2119899 + 1)120574 minus 1

120574)

(58)


(119905)

119883


have

120583(119905)

119883= E [119883119905] = int

R+

119909119905119891119883 (119909) 119889119909

=1

1205901198621

120574intR+


120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119909

(59)


(1120590)


120583(119905)

119883= 1198621

120574(

120574

120574 minus 1)

(120574minus1)120574

intR

exp119905120583 + 119899(120574

120574 minus 1)

(120574minus1)120574

120590119911



exp119905(120574

120574 minus 1)

(120574minus1)120574

120590119911 =

infin

sum

119899=0

(119905120590)119899

119899(

120574

120574 minus 1)

119899((120574minus1)120574)

119911119899

(61)

it is obtained that

120583(119905)

119883= 21198621

120574(

120574

120574 minus 1)

(120574minus1)120574

119890119905120583infin

sum

119899=0

(119905120590)2119899

(2119899)(

120574

120574 minus 1)

2119899((120574minus1)120574)

times intR+

1199112119899 exp minus119911120574(120574minus1) 119889119911

(62)



intR+

119909119898119890minus119887119909119899

119889119909 =Γ ((119898 + 1) 119899)

119899119887(119898+1)119899 119899 119898 119887 isin R

lowast

+ (63)


2) we have 119872119884(119905) =

E[119890119905119884] = E[119883119905] = 120583(119905)



120583(119905)

119883=

119890119905120583

radic120587

infin

sum

119899=0

(211990521205902)119899

(2119899)Γ (119899 +

1

2) 119905 isin R+ (64)


Γ (119899 +1

2) =

(2119899)

22119899119899radic120587 119899 isin N (65)

we have

120583(119905)

119883= 119890119905120583infin

sum

119899=0

(119905120590)2119899

2119899119899= 119890119905120583infin

sum

119899=0

1

119899(1

211990521205902)

119899

= 119890119905120583+(12)(119905120590)

2

119905 isin R+

(66)


(1)

119883= E[119883] =


(119905)

119883= exp119905120583 +





LN120574(120583 120590) is given by

120583(119896)

119883=

119890119896120583

Γ ((120574 minus 1) 120574)

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899

119878119896minus119899 119896 isin N (67)

where

119878119896 =

infin

sum

119898=0

(119896120590)2119898

(2119898)(

120574

120574 minus 1)

2119898((120574minus1)120574)

Γ((2119898 + 1)120574 minus 1

120574)

119896 isin N

(68)


wehave

120583(119896)

119883= E [(119883 minus 120583119883)

119896] = int

R+

(119909 minus 120583119883)119896119891119883 (119909 120583 120590 120574) 119889119909

(69)


120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus120583119883)

119899intR+

119909119896minus119899

119891119883 (119909) 119889119909

=

119896

sum

119899=0

(119896

119899) (minus120583119883)

119899120583(119896minus119899)

119883

(70)


120583(119896)

119883=

119890119896120583

Γ ((120574 minus 1) 120574)

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898)

times (120574

120574 minus 1)

2119898((120574minus1)120574)

Γ ((2119898 + 1) ((120574 minus 1) 120574))

(71)




120583+(12)1205902


120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus1)

119899119890119896120583+(12)[119899+(119896minus119899)

2

]1205902

119896 isin N (72)

while

1205902

119883= Var [119883] = 120583

(2)

119883= 1198902120583+1205902

(1198901205902

minus 1) (73)





120583119883 =119890120583

Γ ((120574 minus 1) 120574)1198781 (74)

1205902

119883= minus 120583

2

119883+

1198902120583

Γ ((120574 minus 1) 120574)1198782 (75)

1198621198812

119883= Γ(

120574 minus 1

120574)1198782

11987821

minus 1 (76)

120582119883 = minus 119862119881minus3

119883minus 119862119881minus1

119883+

1198903120583

1205903119883Γ ((120574 minus 1) 120574)

1198783 (77)

120581119883 = minus 119862119881minus4

119883minus 6119862119881

minus2

119883minus 4

120582119883

119862119881119883

+1198904120583

1205904119883Γ ((120574 minus 1) 120574)

1198784

(78)



120583(1)


1205902

119883= 120583(2)

119883= 1205832

119883+ [Γ(

120574 minus 1

120574)]

minus1

(11989021205831198782 minus 2119890

1205831205831198831198781) (79)


(3)

1198831205903

119883


120582119883 = minus119862119881minus3

119883+ [1205903

119883Γ(

120574 minus 1

120574)]

minus1

times (11989031205831198783 minus 3119890

21205831205831198831198782 + 3119890

1205831205831198831198781)

(80)


(4)

1198831205904

119883 which


120581119883 = 119862119881minus4

119883+ [1205904

119883Γ(

120574 minus 1

120574)]

minus1

times (11989041205831198784 minus 4119890

31205831205831198831198783 + 6119890

21205831205832

1198831198782 minus 4119890

1205831205833

1198831198781)

(81)



radic120587119890(1198992

1205902


120583119883 = 119890120583+(12)120590

2

1205902

119883= 1198902120583+1205902

(1198901205902

minus 1)

119862119881119883 =radic1198901205902

minus 1

120582119883 = (1198901205902

+ 2)radic1198901205902

minus 1 120581119883 = 11989041205902

+ 211989031205902

+ 311989021205902

minus 3

(82)




120574

le

120583119883120574



Med119883120574 = (120583119892)119883120574

= 119890120583lt 120583119883

120574

(83)



Mode119883120574 = 119890120583minus120590120574

lt 119890120583= Med119883120574 (84)


Mode119883120574 = 119890120583= Med119883120574 (85)



Mode119883120574 = 0 lt 119890120583= Med119883120574 (86)





120583(119905)

119883=

119887119905minus 119886119905

119905 log (119887119886) 119905 isin R (87)

120583(119896)

119883=

(119886 minus 119887)119896

log119896 (119887119886)+

1

log (119887119886)

times

119896minus1

sum

119899=0

(119896

119899)(119886 minus 119887)

119899(119887119896minus119899

minus 119886119896minus119899

)

(119896 minus 119899) log119899 (119887119886) 119896 isin N

(88)


120583119883 =119887 minus 119886

log (119887119886) (89)

1205902

119883=

(119887 minus 119886)2

log2 (119887119886)+(119887 minus 119886) (119887 + 119886)

2 log (119887119886) (90)

119862119881119883 = radic1 +119887 + 119886

2 (119887 minus 119886)log 119887

119886 (91)

120582119883 =1

1205903119883

[1198873minus 1198863

3 log (119887119886)minus 3

(119887 minus 119886)2(119887 + 119886)

2 log2 (119887119886)+ 2

(119887 minus 119886)3

log3 (119887119886)]

(92)

120581119883 =1

1205904119883

[1198874minus 1198864

4 log (119887119886)minus 4

(119887 minus 119886) (1198873minus 1198863)

3 log2 (119887119886)

+3(119887 minus 119886)

3(119887 + 119886)

log3 (119887119886)minus 3

(119887 minus 119886)4

log4 (119887119886)]

(93)


120583(119905)

119883120574

=119890119905120583

119905120590Γ ((120574 minus 1) 120574 + 1)

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)(

120574

120574 minus 1)

2119898((120574minus1)120574)

times Γ((2119898 + 1)120574 minus 1

120574+ 1)

(94)


120583(119905)

119883= lim120574rarr1+

120583(119905)

119883120574

=119890119905120583

119905120590

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)=119890119905(120583+120590)

minus 119890119905(120583minus120590)

2119905120590

119905 isin R

(95)


119883= E[119883] = (12120590)

(119890120583+120590

minus 119890120583minus120590


120583(119896)

119883= lim120574rarr1+

120583(119896)

119883120574

= 119890119896120583119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898 + 1)

119896 isin N

(96)


120583(119896)

119883=119890119896120583

2120590

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 119890(119896minus119899)120590


(119896 minus 119899) 119896 isin N

(97)


Var[119883] = 120583(2)

119883minus 1205832


For 119896 = 3 and 119896 = 4 through 120583(3)119883

and 120583(4)119883 we obtain (92) and

(93) respectively


120583(119905)

119883=

1205831199051205902

1205902 minus 1199052gt 120583119905 120590 gt 119905 119905 isin R (98)

120583(119896)

119883= 120583119896119896

sum

119899=0

(119896

119899)

1205902(119899+1)


2] 120590 gt 119896 119896 isin N

(99)


120583119883 =1205831205902

1205902 minus 1gt 120583 120590 gt 1 (100)

1205902

119883=

12058321205902(21205902+ 1)

(1205902 minus 4) (1205902 minus 1)2 120590 gt 2 (101)

119862119881119883 =1

120590

radic21205902+ 1

1205902 minus 4 120590 gt 2 (102)

120582119883 =2 (15120590

4+ 71205902+ 2)

120590 (1205902 minus 9)radic

1205902minus 4

(21205902 + 1)3 120590 gt 3 (103)


120581119883 =3 (81205908+ 212120590

6+ 95120590

4+ 33120590

2+ 12) (120590

2minus 4)

(1205902 minus 16) (1205902 minus 9) (21205902 + 1)2

120590 gt 4

(104)


120583(119905)

119883= 120583(119905)

119883plusmninfin

= 120583119905infin

sum

119896=0

(119905

120590)

2119896

119905 isin R (105)


120583(119905)

119883plusmninfin

=1

2120583119905[

infin

sum

119896=0

(119905

120590)

119896

+

infin

sum

119896=0

(minus119905

120590)

119896

]

=1

2120583119905(

120590

120590 minus 119905+

120590

120590 + 119905)

(106)



120583(119896)

119883= 1205831198961205902119896

sum

119899=0

(119896

119899)

(minus120583119883120583)119899




(2)

119883minus 1205832

119883


119883and 120583(4)




119880sim LU(119890

120583minus120590 119890120583+120590


119886 119890119887)


(119905)

LU = (119890119905119887minus 119890119905119886)(1119905(119887 minus 119886))



120583 1120590 1120590) due to


(119905)

LL = 119890119905120590(1 minus 11990521205902)minus1

4 Conclusion



Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120583(119905)

119883=

119890119905120583

Γ ((120574 minus 1) 120574)

infin

sum

119899=0

(119905120590)2119899

(2119899)(

120574

120574 minus 1)

2119899((120574minus1)120574)

times Γ((2119899 + 1)120574 minus 1

120574)

(58)


(119905)

119883


have

120583(119905)

119883= E [119883119905] = int

R+

119909119905119891119883 (119909) 119889119909

=1

1205901198621

120574intR+


120574 minus 1

120574

10038161003816100381610038161003816100381610038161003816

log119909 minus 120583120590

10038161003816100381610038161003816100381610038161003816

120574(120574minus1)

119889119909

(59)


(1120590)


120583(119905)

119883= 1198621

120574(

120574

120574 minus 1)

(120574minus1)120574

intR

exp119905120583 + 119899(120574

120574 minus 1)

(120574minus1)120574

120590119911



exp119905(120574

120574 minus 1)

(120574minus1)120574

120590119911 =

infin

sum

119899=0

(119905120590)119899

119899(

120574

120574 minus 1)

119899((120574minus1)120574)

119911119899

(61)

it is obtained that

120583(119905)

119883= 21198621

120574(

120574

120574 minus 1)

(120574minus1)120574

119890119905120583infin

sum

119899=0

(119905120590)2119899

(2119899)(

120574

120574 minus 1)

2119899((120574minus1)120574)

times intR+

1199112119899 exp minus119911120574(120574minus1) 119889119911

(62)



intR+

119909119898119890minus119887119909119899

119889119909 =Γ ((119898 + 1) 119899)

119899119887(119898+1)119899 119899 119898 119887 isin R

lowast

+ (63)


2) we have 119872119884(119905) =

E[119890119905119884] = E[119883119905] = 120583(119905)



120583(119905)

119883=

119890119905120583

radic120587

infin

sum

119899=0

(211990521205902)119899

(2119899)Γ (119899 +

1

2) 119905 isin R+ (64)


Γ (119899 +1

2) =

(2119899)

22119899119899radic120587 119899 isin N (65)

we have

120583(119905)

119883= 119890119905120583infin

sum

119899=0

(119905120590)2119899

2119899119899= 119890119905120583infin

sum

119899=0

1

119899(1

211990521205902)

119899

= 119890119905120583+(12)(119905120590)

2

119905 isin R+

(66)


(1)

119883= E[119883] =


(119905)

119883= exp119905120583 +





LN120574(120583 120590) is given by

120583(119896)

119883=

119890119896120583

Γ ((120574 minus 1) 120574)

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899

119878119896minus119899 119896 isin N (67)

where

119878119896 =

infin

sum

119898=0

(119896120590)2119898

(2119898)(

120574

120574 minus 1)

2119898((120574minus1)120574)

Γ((2119898 + 1)120574 minus 1

120574)

119896 isin N

(68)


wehave

120583(119896)

119883= E [(119883 minus 120583119883)

119896] = int

R+

(119909 minus 120583119883)119896119891119883 (119909 120583 120590 120574) 119889119909

(69)


120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus120583119883)

119899intR+

119909119896minus119899

119891119883 (119909) 119889119909

=

119896

sum

119899=0

(119896

119899) (minus120583119883)

119899120583(119896minus119899)

119883

(70)


120583(119896)

119883=

119890119896120583

Γ ((120574 minus 1) 120574)

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898)

times (120574

120574 minus 1)

2119898((120574minus1)120574)

Γ ((2119898 + 1) ((120574 minus 1) 120574))

(71)




120583+(12)1205902


120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus1)

119899119890119896120583+(12)[119899+(119896minus119899)

2

]1205902

119896 isin N (72)

while

1205902

119883= Var [119883] = 120583

(2)

119883= 1198902120583+1205902

(1198901205902

minus 1) (73)





120583119883 =119890120583

Γ ((120574 minus 1) 120574)1198781 (74)

1205902

119883= minus 120583

2

119883+

1198902120583

Γ ((120574 minus 1) 120574)1198782 (75)

1198621198812

119883= Γ(

120574 minus 1

120574)1198782

11987821

minus 1 (76)

120582119883 = minus 119862119881minus3

119883minus 119862119881minus1

119883+

1198903120583

1205903119883Γ ((120574 minus 1) 120574)

1198783 (77)

120581119883 = minus 119862119881minus4

119883minus 6119862119881

minus2

119883minus 4

120582119883

119862119881119883

+1198904120583

1205904119883Γ ((120574 minus 1) 120574)

1198784

(78)



120583(1)


1205902

119883= 120583(2)

119883= 1205832

119883+ [Γ(

120574 minus 1

120574)]

minus1

(11989021205831198782 minus 2119890

1205831205831198831198781) (79)


(3)

1198831205903

119883


120582119883 = minus119862119881minus3

119883+ [1205903

119883Γ(

120574 minus 1

120574)]

minus1

times (11989031205831198783 minus 3119890

21205831205831198831198782 + 3119890

1205831205831198831198781)

(80)


(4)

1198831205904

119883 which


120581119883 = 119862119881minus4

119883+ [1205904

119883Γ(

120574 minus 1

120574)]

minus1

times (11989041205831198784 minus 4119890

31205831205831198831198783 + 6119890

21205831205832

1198831198782 minus 4119890

1205831205833

1198831198781)

(81)



radic120587119890(1198992

1205902


120583119883 = 119890120583+(12)120590

2

1205902

119883= 1198902120583+1205902

(1198901205902

minus 1)

119862119881119883 =radic1198901205902

minus 1

120582119883 = (1198901205902

+ 2)radic1198901205902

minus 1 120581119883 = 11989041205902

+ 211989031205902

+ 311989021205902

minus 3

(82)




120574

le

120583119883120574



Med119883120574 = (120583119892)119883120574

= 119890120583lt 120583119883

120574

(83)



Mode119883120574 = 119890120583minus120590120574

lt 119890120583= Med119883120574 (84)


Mode119883120574 = 119890120583= Med119883120574 (85)



Mode119883120574 = 0 lt 119890120583= Med119883120574 (86)





120583(119905)

119883=

119887119905minus 119886119905

119905 log (119887119886) 119905 isin R (87)

120583(119896)

119883=

(119886 minus 119887)119896

log119896 (119887119886)+

1

log (119887119886)

times

119896minus1

sum

119899=0

(119896

119899)(119886 minus 119887)

119899(119887119896minus119899

minus 119886119896minus119899

)

(119896 minus 119899) log119899 (119887119886) 119896 isin N

(88)


120583119883 =119887 minus 119886

log (119887119886) (89)

1205902

119883=

(119887 minus 119886)2

log2 (119887119886)+(119887 minus 119886) (119887 + 119886)

2 log (119887119886) (90)

119862119881119883 = radic1 +119887 + 119886

2 (119887 minus 119886)log 119887

119886 (91)

120582119883 =1

1205903119883

[1198873minus 1198863

3 log (119887119886)minus 3

(119887 minus 119886)2(119887 + 119886)

2 log2 (119887119886)+ 2

(119887 minus 119886)3

log3 (119887119886)]

(92)

120581119883 =1

1205904119883

[1198874minus 1198864

4 log (119887119886)minus 4

(119887 minus 119886) (1198873minus 1198863)

3 log2 (119887119886)

+3(119887 minus 119886)

3(119887 + 119886)

log3 (119887119886)minus 3

(119887 minus 119886)4

log4 (119887119886)]

(93)


120583(119905)

119883120574

=119890119905120583

119905120590Γ ((120574 minus 1) 120574 + 1)

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)(

120574

120574 minus 1)

2119898((120574minus1)120574)

times Γ((2119898 + 1)120574 minus 1

120574+ 1)

(94)


120583(119905)

119883= lim120574rarr1+

120583(119905)

119883120574

=119890119905120583

119905120590

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)=119890119905(120583+120590)

minus 119890119905(120583minus120590)

2119905120590

119905 isin R

(95)


119883= E[119883] = (12120590)

(119890120583+120590

minus 119890120583minus120590


120583(119896)

119883= lim120574rarr1+

120583(119896)

119883120574

= 119890119896120583119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898 + 1)

119896 isin N

(96)


120583(119896)

119883=119890119896120583

2120590

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 119890(119896minus119899)120590


(119896 minus 119899) 119896 isin N

(97)


Var[119883] = 120583(2)

119883minus 1205832


For 119896 = 3 and 119896 = 4 through 120583(3)119883

and 120583(4)119883 we obtain (92) and

(93) respectively


120583(119905)

119883=

1205831199051205902

1205902 minus 1199052gt 120583119905 120590 gt 119905 119905 isin R (98)

120583(119896)

119883= 120583119896119896

sum

119899=0

(119896

119899)

1205902(119899+1)


2] 120590 gt 119896 119896 isin N

(99)


120583119883 =1205831205902

1205902 minus 1gt 120583 120590 gt 1 (100)

1205902

119883=

12058321205902(21205902+ 1)

(1205902 minus 4) (1205902 minus 1)2 120590 gt 2 (101)

119862119881119883 =1

120590

radic21205902+ 1

1205902 minus 4 120590 gt 2 (102)

120582119883 =2 (15120590

4+ 71205902+ 2)

120590 (1205902 minus 9)radic

1205902minus 4

(21205902 + 1)3 120590 gt 3 (103)


120581119883 =3 (81205908+ 212120590

6+ 95120590

4+ 33120590

2+ 12) (120590

2minus 4)

(1205902 minus 16) (1205902 minus 9) (21205902 + 1)2

120590 gt 4

(104)


120583(119905)

119883= 120583(119905)

119883plusmninfin

= 120583119905infin

sum

119896=0

(119905

120590)

2119896

119905 isin R (105)


120583(119905)

119883plusmninfin

=1

2120583119905[

infin

sum

119896=0

(119905

120590)

119896

+

infin

sum

119896=0

(minus119905

120590)

119896

]

=1

2120583119905(

120590

120590 minus 119905+

120590

120590 + 119905)

(106)



120583(119896)

119883= 1205831198961205902119896

sum

119899=0

(119896

119899)

(minus120583119883120583)119899




(2)

119883minus 1205832

119883


119883and 120583(4)




119880sim LU(119890

120583minus120590 119890120583+120590


119886 119890119887)


(119905)

LU = (119890119905119887minus 119890119905119886)(1119905(119887 minus 119886))



120583 1120590 1120590) due to


(119905)

LL = 119890119905120590(1 minus 11990521205902)minus1

4 Conclusion



Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120583+(12)1205902


120583(119896)

119883=

119896

sum

119899=0

(119896

119899) (minus1)

119899119890119896120583+(12)[119899+(119896minus119899)

2

]1205902

119896 isin N (72)

while

1205902

119883= Var [119883] = 120583

(2)

119883= 1198902120583+1205902

(1198901205902

minus 1) (73)





120583119883 =119890120583

Γ ((120574 minus 1) 120574)1198781 (74)

1205902

119883= minus 120583

2

119883+

1198902120583

Γ ((120574 minus 1) 120574)1198782 (75)

1198621198812

119883= Γ(

120574 minus 1

120574)1198782

11987821

minus 1 (76)

120582119883 = minus 119862119881minus3

119883minus 119862119881minus1

119883+

1198903120583

1205903119883Γ ((120574 minus 1) 120574)

1198783 (77)

120581119883 = minus 119862119881minus4

119883minus 6119862119881

minus2

119883minus 4

120582119883

119862119881119883

+1198904120583

1205904119883Γ ((120574 minus 1) 120574)

1198784

(78)



120583(1)


1205902

119883= 120583(2)

119883= 1205832

119883+ [Γ(

120574 minus 1

120574)]

minus1

(11989021205831198782 minus 2119890

1205831205831198831198781) (79)


(3)

1198831205903

119883


120582119883 = minus119862119881minus3

119883+ [1205903

119883Γ(

120574 minus 1

120574)]

minus1

times (11989031205831198783 minus 3119890

21205831205831198831198782 + 3119890

1205831205831198831198781)

(80)


(4)

1198831205904

119883 which


120581119883 = 119862119881minus4

119883+ [1205904

119883Γ(

120574 minus 1

120574)]

minus1

times (11989041205831198784 minus 4119890

31205831205831198831198783 + 6119890

21205831205832

1198831198782 minus 4119890

1205831205833

1198831198781)

(81)



radic120587119890(1198992

1205902


120583119883 = 119890120583+(12)120590

2

1205902

119883= 1198902120583+1205902

(1198901205902

minus 1)

119862119881119883 =radic1198901205902

minus 1

120582119883 = (1198901205902

+ 2)radic1198901205902

minus 1 120581119883 = 11989041205902

+ 211989031205902

+ 311989021205902

minus 3

(82)




120574

le

120583119883120574



Med119883120574 = (120583119892)119883120574

= 119890120583lt 120583119883

120574

(83)



Mode119883120574 = 119890120583minus120590120574

lt 119890120583= Med119883120574 (84)


Mode119883120574 = 119890120583= Med119883120574 (85)



Mode119883120574 = 0 lt 119890120583= Med119883120574 (86)





120583(119905)

119883=

119887119905minus 119886119905

119905 log (119887119886) 119905 isin R (87)

120583(119896)

119883=

(119886 minus 119887)119896

log119896 (119887119886)+

1

log (119887119886)

times

119896minus1

sum

119899=0

(119896

119899)(119886 minus 119887)

119899(119887119896minus119899

minus 119886119896minus119899

)

(119896 minus 119899) log119899 (119887119886) 119896 isin N

(88)


120583119883 =119887 minus 119886

log (119887119886) (89)

1205902

119883=

(119887 minus 119886)2

log2 (119887119886)+(119887 minus 119886) (119887 + 119886)

2 log (119887119886) (90)

119862119881119883 = radic1 +119887 + 119886

2 (119887 minus 119886)log 119887

119886 (91)

120582119883 =1

1205903119883

[1198873minus 1198863

3 log (119887119886)minus 3

(119887 minus 119886)2(119887 + 119886)

2 log2 (119887119886)+ 2

(119887 minus 119886)3

log3 (119887119886)]

(92)

120581119883 =1

1205904119883

[1198874minus 1198864

4 log (119887119886)minus 4

(119887 minus 119886) (1198873minus 1198863)

3 log2 (119887119886)

+3(119887 minus 119886)

3(119887 + 119886)

log3 (119887119886)minus 3

(119887 minus 119886)4

log4 (119887119886)]

(93)


120583(119905)

119883120574

=119890119905120583

119905120590Γ ((120574 minus 1) 120574 + 1)

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)(

120574

120574 minus 1)

2119898((120574minus1)120574)

times Γ((2119898 + 1)120574 minus 1

120574+ 1)

(94)


120583(119905)

119883= lim120574rarr1+

120583(119905)

119883120574

=119890119905120583

119905120590

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)=119890119905(120583+120590)

minus 119890119905(120583minus120590)

2119905120590

119905 isin R

(95)


119883= E[119883] = (12120590)

(119890120583+120590

minus 119890120583minus120590


120583(119896)

119883= lim120574rarr1+

120583(119896)

119883120574

= 119890119896120583119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898 + 1)

119896 isin N

(96)


120583(119896)

119883=119890119896120583

2120590

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 119890(119896minus119899)120590


(119896 minus 119899) 119896 isin N

(97)


Var[119883] = 120583(2)

119883minus 1205832


For 119896 = 3 and 119896 = 4 through 120583(3)119883

and 120583(4)119883 we obtain (92) and

(93) respectively


120583(119905)

119883=

1205831199051205902

1205902 minus 1199052gt 120583119905 120590 gt 119905 119905 isin R (98)

120583(119896)

119883= 120583119896119896

sum

119899=0

(119896

119899)

1205902(119899+1)


2] 120590 gt 119896 119896 isin N

(99)


120583119883 =1205831205902

1205902 minus 1gt 120583 120590 gt 1 (100)

1205902

119883=

12058321205902(21205902+ 1)

(1205902 minus 4) (1205902 minus 1)2 120590 gt 2 (101)

119862119881119883 =1

120590

radic21205902+ 1

1205902 minus 4 120590 gt 2 (102)

120582119883 =2 (15120590

4+ 71205902+ 2)

120590 (1205902 minus 9)radic

1205902minus 4

(21205902 + 1)3 120590 gt 3 (103)


120581119883 =3 (81205908+ 212120590

6+ 95120590

4+ 33120590

2+ 12) (120590

2minus 4)

(1205902 minus 16) (1205902 minus 9) (21205902 + 1)2

120590 gt 4

(104)


120583(119905)

119883= 120583(119905)

119883plusmninfin

= 120583119905infin

sum

119896=0

(119905

120590)

2119896

119905 isin R (105)


120583(119905)

119883plusmninfin

=1

2120583119905[

infin

sum

119896=0

(119905

120590)

119896

+

infin

sum

119896=0

(minus119905

120590)

119896

]

=1

2120583119905(

120590

120590 minus 119905+

120590

120590 + 119905)

(106)



120583(119896)

119883= 1205831198961205902119896

sum

119899=0

(119896

119899)

(minus120583119883120583)119899




(2)

119883minus 1205832

119883


119883and 120583(4)




119880sim LU(119890

120583minus120590 119890120583+120590


119886 119890119887)


(119905)

LU = (119890119905119887minus 119890119905119886)(1119905(119887 minus 119886))



120583 1120590 1120590) due to


(119905)

LL = 119890119905120590(1 minus 11990521205902)minus1

4 Conclusion



Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Mode119883120574 = 0 lt 119890120583= Med119883120574 (86)





120583(119905)

119883=

119887119905minus 119886119905

119905 log (119887119886) 119905 isin R (87)

120583(119896)

119883=

(119886 minus 119887)119896

log119896 (119887119886)+

1

log (119887119886)

times

119896minus1

sum

119899=0

(119896

119899)(119886 minus 119887)

119899(119887119896minus119899

minus 119886119896minus119899

)

(119896 minus 119899) log119899 (119887119886) 119896 isin N

(88)


120583119883 =119887 minus 119886

log (119887119886) (89)

1205902

119883=

(119887 minus 119886)2

log2 (119887119886)+(119887 minus 119886) (119887 + 119886)

2 log (119887119886) (90)

119862119881119883 = radic1 +119887 + 119886

2 (119887 minus 119886)log 119887

119886 (91)

120582119883 =1

1205903119883

[1198873minus 1198863

3 log (119887119886)minus 3

(119887 minus 119886)2(119887 + 119886)

2 log2 (119887119886)+ 2

(119887 minus 119886)3

log3 (119887119886)]

(92)

120581119883 =1

1205904119883

[1198874minus 1198864

4 log (119887119886)minus 4

(119887 minus 119886) (1198873minus 1198863)

3 log2 (119887119886)

+3(119887 minus 119886)

3(119887 + 119886)

log3 (119887119886)minus 3

(119887 minus 119886)4

log4 (119887119886)]

(93)


120583(119905)

119883120574

=119890119905120583

119905120590Γ ((120574 minus 1) 120574 + 1)

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)(

120574

120574 minus 1)

2119898((120574minus1)120574)

times Γ((2119898 + 1)120574 minus 1

120574+ 1)

(94)


120583(119905)

119883= lim120574rarr1+

120583(119905)

119883120574

=119890119905120583

119905120590

infin

sum

119898=0

(119905120590)2119898+1

(2119898 + 1)=119890119905(120583+120590)

minus 119890119905(120583minus120590)

2119905120590

119905 isin R

(95)


119883= E[119883] = (12120590)

(119890120583+120590

minus 119890120583minus120590


120583(119896)

119883= lim120574rarr1+

120583(119896)

119883120574

= 119890119896120583119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 infin

sum

119898=0

[(119896 minus 119899) 120590]2119898

(2119898 + 1)

119896 isin N

(96)


120583(119896)

119883=119890119896120583

2120590

119896

sum

119899=0

(119896

119899)(minus

120583119883

119890120583)

119899 119890(119896minus119899)120590


(119896 minus 119899) 119896 isin N

(97)


Var[119883] = 120583(2)

119883minus 1205832


For 119896 = 3 and 119896 = 4 through 120583(3)119883

and 120583(4)119883 we obtain (92) and

(93) respectively


120583(119905)

119883=

1205831199051205902

1205902 minus 1199052gt 120583119905 120590 gt 119905 119905 isin R (98)

120583(119896)

119883= 120583119896119896

sum

119899=0

(119896

119899)

1205902(119899+1)


2] 120590 gt 119896 119896 isin N

(99)


120583119883 =1205831205902

1205902 minus 1gt 120583 120590 gt 1 (100)

1205902

119883=

12058321205902(21205902+ 1)

(1205902 minus 4) (1205902 minus 1)2 120590 gt 2 (101)

119862119881119883 =1

120590

radic21205902+ 1

1205902 minus 4 120590 gt 2 (102)

120582119883 =2 (15120590

4+ 71205902+ 2)

120590 (1205902 minus 9)radic

1205902minus 4

(21205902 + 1)3 120590 gt 3 (103)


120581119883 =3 (81205908+ 212120590

6+ 95120590

4+ 33120590

2+ 12) (120590

2minus 4)

(1205902 minus 16) (1205902 minus 9) (21205902 + 1)2

120590 gt 4

(104)


120583(119905)

119883= 120583(119905)

119883plusmninfin

= 120583119905infin

sum

119896=0

(119905

120590)

2119896

119905 isin R (105)


120583(119905)

119883plusmninfin

=1

2120583119905[

infin

sum

119896=0

(119905

120590)

119896

+

infin

sum

119896=0

(minus119905

120590)

119896

]

=1

2120583119905(

120590

120590 minus 119905+

120590

120590 + 119905)

(106)



120583(119896)

119883= 1205831198961205902119896

sum

119899=0

(119896

119899)

(minus120583119883120583)119899




(2)

119883minus 1205832

119883


119883and 120583(4)




119880sim LU(119890

120583minus120590 119890120583+120590


119886 119890119887)


(119905)

LU = (119890119905119887minus 119890119905119886)(1119905(119887 minus 119886))



120583 1120590 1120590) due to


(119905)

LL = 119890119905120590(1 minus 11990521205902)minus1

4 Conclusion



Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120581119883 =3 (81205908+ 212120590

6+ 95120590

4+ 33120590

2+ 12) (120590

2minus 4)

(1205902 minus 16) (1205902 minus 9) (21205902 + 1)2

120590 gt 4

(104)


120583(119905)

119883= 120583(119905)

119883plusmninfin

= 120583119905infin

sum

119896=0

(119905

120590)

2119896

119905 isin R (105)


120583(119905)

119883plusmninfin

=1

2120583119905[

infin

sum

119896=0

(119905

120590)

119896

+

infin

sum

119896=0

(minus119905

120590)

119896

]

=1

2120583119905(

120590

120590 minus 119905+

120590

120590 + 119905)

(106)



120583(119896)

119883= 1205831198961205902119896

sum

119899=0

(119896

119899)

(minus120583119883120583)119899




(2)

119883minus 1205832

119883


119883and 120583(4)




119880sim LU(119890

120583minus120590 119890120583+120590


119886 119890119887)


(119905)

LU = (119890119905119887minus 119890119905119886)(1119905(119887 minus 119886))



120583 1120590 1120590) due to


(119905)

LL = 119890119905120590(1 minus 11990521205902)minus1

4 Conclusion



Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article On the Generalized Lognormal Distributiondownloads.hindawi.com/journals/jps/2013/432642.pdf · On the Generalized Lognormal Distribution ThomasL.TouliasandChristosP.Kitsos