Much as Much as Dis Tri Buci Ones

7/28/2019 Much as Much as Dis Tri Buci Ones

1/75

A Concise Summary of @RISK ProbabilityDistribution Functions

Copyright 2002 Palisade Corporation

All Rights Reserved


2/75


3/75

Skewness:

21

21

21

12 1

22

++

++

Kurtosis:

( ) ( ) ( )( )( )( )32

6213

212121

21212

2121

++++

+++++

Mode:

2

1

21

1

+

1>1, 2>1

0 11

1 11, 21, 2=1

PDF - Beta(2,3)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

-0

.2

0.

0

0.

2

0.

4

0.

6

0.

8

1.

0

1.

2

CDF - Beta(2,3)

0.0

0.2

0.4

0.6

0.8

1.0

-0

.2

0.

0

0.

2

0.

4

0.

6

0.

8

1.

0

1.

2


4/75

Beta (Generalized)

RISKBetaGeneral(1, 2, min, max)

Parameters:

1 continuous shape parameter 1> 0


min continuous boundary parameter min < max

max continuous boundary parameter

Domain:

min x max continuous

Density and Cumulative Distribution Functions:

( ) ( )

( ) 12121

1211

minmax),(

xmaxminx)x(f

+

=

( )( )21z

21

21z ,I),(B

,B)x(F

= with

minmax

minxz

where B is theBetaFunction and Bz is theIncompleteBetaFunction.

Mean:

( )minmaxmin

21

1 +

+

Variance:

( ) ( )( ) 2

212

21

21 minmax1

+++


5/75

Skewness:

21

21

21

12 1

22

++

++

Kurtosis:

( ) ( ) ( )( )( )( )32

6213

212121

21212

2121

++++

+++++

Mode:

( )minmax2

1min

21

1 +

+ 1>1, 2>1

min 11

max 11, 21, 2=1

PDF - BetaGeneral(2,3,0,5)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-1 0 1 2 3 4 5 6

CDF - BetaGeneral(2,3,0,5)

0.0

0.2

0.4

0.6

0.8

1.0

-1 0 1 2 3 4 5 6


6/75

Beta (Subjective)

RISKBetaSubj(min, m.likely, mean, max)

Definitions:

2

maxminmid

+

( )( )( )( )minmaxlikely.mmean

likely.mmidminmean21

minmean

meanmax12

Parameters:


m.likely continuous parameter min < m.likely < max

mean continuous parameter min < mean < max


mean > mid if m.likely > mean

mean < mid if m.likely < meanmean = mid if m.likely = mean

Domain:


Density and Cumulative Distribution Functions:

( ) ( )

( ) 12121

1211

minmax),(

xmaxminx)x(f +

=

( )( )21z

21

21z ,I),(B

,B)x(F

= with

minmax

minxz

where B is theBeta Function and Bz is theIncomplete Beta Function.


7/75

Mean:

mean

Variance:

( )( )( )

likely.m3meanmid2

likely.mmeanmeanmaxminmean

+

Skewness:

( ) ( )( )

( )( )meanmaxminmean

likely.m3meanmid2likely.mmean

likely.m2midmean

meanmid2

+

+

Kurtosis:

( ) ( ) ( )

( )( )32

6213

212121

21212

2121

++++

+++++

Mode:

m.likely

PDF - BetaSubj(0,1,2,5)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

-1 0 1 2 3 4 5 6

CDF - BetaSubj(0,1,2,5)

0.0

0.2

0.4

0.6

0.8

1.0

-1 0 1 2 3 4 5 6


8/75

Binomial

RISKBinomial(n, p)

Parameters:

n discrete count parameter n > 0

p continuous success probability 0 < p < 1

Domain:

0 x n discrete

Mass and Cumulative Functions:

( ) xnx p1px

n)x(f

=

inix

0i

)p1(pi

n)x(F

=

=

Mean:

np

Variance:

( )p1np


9/75

Skewness:

( )

( )p1np

p21

Kurtosis:

( )p1np1

n

63

+

Mode:

(bimodal) ( ) 11np + and ( )1np + if ( )1np + is integral

(unimodal) largest integer less than ( )1np + otherwise

PMF - Binomial(8,.4)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

-1 0 1 2 3 4 5 6 7 8 9

CDF - Binomial(8,.4)

0.0

0.2

0.4

0.6

0.8

1.0

-1 0 1 2 3 4 5 6 7 8 9


10/75

Chi-Squared

RISKChiSq()

Parameters:

discrete shape parameter > 0

Domain:

0 x + continuous

Density and Cumulative Functions:

( )( ) 122x

2xe

22

1)x(f

=

( )

( )2

2)x(F

2x

=

where is the Gamma Function, and x is theIncomplete Gamma Function.

Mean:

Variance:

2


11/75

Skewness:

8

Kurtosis:

+12

3

Mode:

-2 if 2

0 if = 1

PDF - ChiSq(5)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

-2 0 2 4 6 8

10

12

14

16

CDF - ChiSq(5)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-2 0 2 4 6 8

10

12

14

16


12/75

Cumulative (Ascending)

RISKCumul(min, max, {x}, {p})

Parameters:

min continuous parameter min < max

max continuous parameter

{x} = {x1, x2, , xN} array of continuous parameters min xi max

{p} = {p1, p2, , pN} array of continuous parameters 0 pi 1

Domain:



i1i

i1i

xx

pp)x(f

=

+

+ for xi x < xi+1

( )

+=

++

i1i

ii1ii

xx

xxppp)x(F for xi x xi+1

With the assumptions:

1. The arrays are ordered from left to right

2. The i index runs from 0 to N+1, with two extra elements : x0 min, p0 0 and xN+1 max, pN+1 1.

Mean:

No Closed Form

Variance:

No Closed Form


13/75

Skewness:

No Closed Form

Kurtosis:

No Closed Form

Mode:

No Closed Form

CDF - Cumul({1,2,3,4},{.2,.3,.7,.8})

0.0

0.2

0.4

0.6

0.8

1.0

-1 0 1 2 3 4 5 6

PDF - Cumul({1,2,3,4},{.2,.3,.7,.8})

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

-1 0 1 2 3 4 5 6


14/75

Cumulative (Descending)

RISKCumulD(min, max, {x}, {p})

Parameters:




{p} = {p1, p2, , pN} array of continuous parameters 0 pi 1

Domain:



i1i

1ii

xx

pp)x(f

=

+

+ for xi x < xi+1

( )

+=

++

i1i

i1iii

xx

xxppp1)x(F for xi x xi+1




Mean:

No Closed Form

Variance:

No Closed Form


15/75

Skewness:

No Closed Form

Kurtosis:

No Closed Form

Mode:

No Closed Form

PDF - CumulD({1,2,3,4},{.2,.3,.7,.8})

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

-1 0 1 2 3 4 5 6

CDF - CumulD({1,2,3,4},{.2,.3,.7,.8})

0.0

0.2

0.4

0.6

0.8

1.0

-1 0 1 2 3 4 5 6


16/75

Discrete

RISKDiscrete({x}, {p})

Parameters:

{x} = {x1, x2, , xN} array of continuous parameters

{p} = {p1, p2, , pN} array of continuous parameters

Domain:

}x{x discrete


ip)x(f = for ixx =

0)x(f = for }x{x

0)x(F = for x < x1

=

=

s

1i

ip)x(F for xs x < xs+1, s < N

1)x(F = for x xN

With the assumptions:1. The arrays are ordered from left to right

2. The p array is normalized to 1.

Mean:

=

i

N

1i

ipx


17/75

Variance:

Vp)x( i2

N

1i

i =

Skewness:

i3

N

1i

i23p)x(

V

1 =

Kurtosis:

i4

N

1i

i2p)x(

V

1 =

Mode:

The x-value corresponding to the highest p-value.

CDF - Discrete({1,2,3,4},{2,1,2,1})

0.0

0.2

0.4

0.6

0.8

1.0

0.

5

1.

0

1.

5

2.

0

2.

5

3.

0

3.

5

4.

0

4.

5

PMF - Discrete({1,2,3,4},{2,1,2,1})

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.

5

1.

0

1.

5

2.

0

2.

5

3.

0

3.

5

4.

0

4.

5


18/75

Discrete Uniform

RISKDUniform({x})

Parameters:

{x} = {x1, x1, , xN} array of continuous parameters

Domain:

}x{x discrete


N

1)x(f = for }x{x

0)x(f = for }x{x

0)x(F = for x < x1

Ni)x(F = for xi x < xi+1

1)x(F = for x xN

assuming the {x} array is ordered.

Mean:

=

N

1ii

xN

1


19/75

Variance:

V)x(N

1 2N

1i

i =

Skewness:

3N

1i

i23)x(

NV

1 =

Kurtosis:

4N

1i

i2)x(

NV

1 =

Mode:

Not uniquely defined

PMF - DUniform({1,5,8,11,12})

0.00

0.05

0.10

0.15

0.20

0.25

0 2 4 6 810

12

14

CDF - DUniform({1,5,8,11,12})

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 810

12

14


20/75

Error Function

RISKErf(h)

Parameters:

h continuous inverse scale parameter h > 0

Domain:

- x + continuous


( )2hxeh

)x(f

=

( )hx2)x(F =

where is theError Function.

Mean:

0

Variance:

2h2

1


21/75

Skewness:

0

Kurtosis:

3

Mode:

0

PDF - Erf(1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

-2

.0

-1

.5

-1

.0

-0

.5

0.

0

0.

5

1.

0

1.

5

2.

0

CDF - Erf(1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-2

.0

-1

.5

-1

.0

-0

.5

0.

0

0.

5

1.

0

1.

5

2.

0


22/75

Erlang

RISKErlang(m,)

Parameters:

m integral shape parameter m > 0

continuous scale parameter > 0

Domain:

0 x < + continuous


= x

1

ex

)!1m(

1)x(f

( )

( )

( )

=

=

=

1m

0i

ixx

!i

xe1

m

m)x(F

where is the Gamma Function and x is theIncomplete Gamma Function.

Mean:

m

Variance:

2m


23/75

Skewness:

m

2

Kurtosis:

m

63+

Mode:

( )1m

PDF - Erlang(2,1)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-1 0 1 2 3 4 5 6 7

CDF - Erlang(2,1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-1 0 1 2 3 4 5 6 7


24/75

Exponential

RISKExpon()

Parameters:


Domain:

0 x < + continuous


=

xe)x(f

= xe1)x(F

Mean:

Variance:

2


25/75

Skewness:

2

Kurtosis:

9

Mode:

0

PDF - Expon(1)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-0

.5

0.

0

0.

5

1.

0

1.

5

2.

0

2.

5

3.

0

3.

5

4.

0

4.

5

5.

0

CDF - Expon(1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-0

.5

0.

0

0.

5

1.

0

1.

5

2.

0

2.

5

3.

0

3.

5

4.

0

4.

5

5.

0


26/75

Extreme Value

RISKExtValue(a, b)

Parameters:

a continuous location parameter

b continuous scale parameter b > 0

Domain:

- x + continuous


=

+ )zexp(ze

1

b

1)x(f

)zexp(e

1)x(F

= where

( )

b

axz

Mean:

( ) b577.a1ba +

where (x) is the derivative of the Gamma Function.

Variance:

6

b22


27/75

Skewness:

1.139547

Kurtosis:

5.4

Mode:

a

PDF - ExtValue(0,1)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-2

-1 0 1 2 3 4 5

CDF - ExtValue(0,1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-2

-1 0 1 2 3 4 5


28/75

Gamma

RISKGamma(, )

Parameters:

continuous shape parameter > 0


Domain:

0 < x < + continuous


( )

= x

1

ex1

)x(f

( )

( )

=

x)x(F

where is the Gamma Function and x is theIncomplete Gamma Function.

Mean:

Variance:

2


29/75

Skewness:

2

Kurtosis:

+

63

Mode:

( )1 if 1

Not Defined if < 1

CDF - Gamma(4,1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-2 0 2 4 6 8

10

12

PDF - Gamma(4,1)

0.00

0.05

0.10

0.15

0.20

0.25

-2 0 2 4 6 8

10

12


30/75

General

RISKGeneral(min, max, {x}, {p})

Parameters:




{p} = {p1, p2, , pN} array of continuous parameters pi 0

Domain:



( )i1ii1i

ii pp

xx

xxp)x(f

+= +

+

for xi x xi+1

( )( )( )

( )

++=

+

+

i1i

ii1iiii

xx2

xxpppxx)x(F)x(F for xi x xi+1



2. The {p} array has been normalized to give the general distribution unit area.


Mean:

No Closed Form

Variance:

No Closed Form


31/75

Skewness:

No Closed Form

Kurtosis:

No Closed Form

Mode:

No Closed Form

PDF - General(0,5,{1,2,3,4},{2,1,2,1})

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-1 0 1 2 3 4 5 6

CDF - General(0,5,{1,2,3,4},{2,1,2,1})

0.0

0.2

0.4

0.6

0.8

1.0

-1 0 1 2 3 4 5 6


32/75

Geometric

RISKGeomet(p)

Parameters:

p continuous success probability 0< p < 1

Domain:

0 x < + discrete


( )xp1p)x(f =

1x)p1(1)x(F +=

Mean:

1p1

Variance:

2p

p1


33/75

Skewness:

( )

p1

p2

Kurtosis:

( )2p1

p9

+

Mode:

0

PMF - Geomet(.5)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

-1 0 1 2 3 4 5 6 7

CDF - Geomet(.5)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-1 0 1 2 3 4 5 6 7


34/75

Histogram

RISKHistogrm(min, max, {p})

Parameters:



{p} = {p1, p2, , pN} array of continuous parameters pi 0

Domain:



ip)x(f = for xi x < xi+1

+=

+ i1i

iii

xx

xxp)x(F)x(F for xi x xi+1

where

+

N

minmaximinx i

The {p} array has been normalized to give the histogram unit area.

Mean:

No Closed Form

Variance:

No Closed Form


35/75

Skewness:

No Closed Form

Kurtosis:

No Closed Form

Mode:

Not Uniquely Defined.

PDF - Histogrm(0,5,{6,5,3,4,5})

0.00

0.05

0.10

0.15

0.20

0.25

0.30

-1 0 1 2 3 4 5 6

CDF - Histogrm(0,5,{6,5,3,4,5})

0.0

0.2

0.4

0.6

0.8

1.0

-1 0 1 2 3 4 5 6


36/75

Hypergeometric

RISKHyperGeo(n, D, M)

Parameters:

n the number of draws integer 0 < n < M

D the number of "tagged" items integer 0 < D < M

M the total number of items integer M > 0

Domain:

max(0,n+D-M) x min(n,D) discrete


=

n

M

xn

DM

x

D

)x(f

=

=

x

1i

n

M

xn

DM

x

D

)x(F

Mean:

MnD


37/75

Variance:

( )( )

( )

1M

nMDM

M

nD2

for M>1

Skewness:

( )( )

( ) )nM(DMnD

1M

2M

n2MD2M

for M>2

Mode:

(bimodal) xm and xm-1 if xm is integral

(unimodal) biggest integer less than xm otherwise

where( )( )

2M

1D1nxm

+

++

PMF - HyperGeo(6,5,10)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0 1 2 3 4 5 6

CDF - HyperGeo(6,5,10)

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5 6


38/75

Integer Uniform

RISKIntUniform(min, max)

Parameters:

min discrete boundary parameter min < max

max discrete boundary probability

Domain:

min x max discrete


1minmax

1)x(f

+=

1minmax

1minx)x(F

+

+=

Mean:

2

maxmin+

Variance:

+

1

2

minmax

6

minmax


39/75

Skewness:

0

Mode:


CDF - IntUniform(0,8)

0.0

0.2

0.4

0.6

0.8

1.0

-1 0 1 2 3 4 5 6 7 8 9

PMF - IntUniform(0,8)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

-1 0 1 2 3 4 5 6 7 8 9


40/75

Inverse Gaussian

RISKInvGauss(, )

Parameters:

continuous parameter > 0

continous parameter > 0

Domain:

x > 0 continuous


( )

=

x2

x

3

2

2

ex2

)x(f

+

+

= 1

x

x

e1x

x

)x(F 2


Mean:

Variance:

3


41/75

Skewness:

3

Kurtosis:

+153

Mode:

+

2

3

4

91

2

2

PDF - InvGauss(1,2)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-0

.5

0.

0

0.

5

1.

0

1.

5

2.

0

2.

5

3.

0

3.

5

4.

0

CDF - InvGauss(1,2)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-0

.5

0.

0

0.

5

1.

0

1.

5

2.

0

2.

5

3.

0

3.

5

4.

0


42/75

Logistic

RISKLogistic(, )

Parameters:

continuous location parameter


Domain:

- x + continuous


=4

x

2

1hsec

)x(f

2

2

x

2

1tanh1

)x(F

+

=

where sech is theHyperbolic Secant Function and tanh is theHyperbolic Tangent Function.

Mean:

Variance:

3

22


43/75

Skewness:

0

Kurtosis:

4.2

Mode:

PDF - Logistic(0,1)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

-5

-4

-3

-2

-1 0 1 2 3 4 5

CDF - Logistic(0,1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-5

-4

-3

-2

-1 0 1 2 3 4 5


44/75

Log-Logistic

RISKLogLogistic(, , )

Parameters:


continous scale parameter > 0


Definitions:

Domain:

x + continuous


( )21

t1

t)x(f

+

=

+

=

t

11

1)x(F with

xt

Mean:

( ) + csc for > 1


45/75

Variance:

( ) ( )[ ] 22 csc2csc2 for > 2

Skewness:

( ) ( ) ( ) ( )

( ) ( )[ ] 23

2

32

csc2csc2

csc2csc2csc63csc3

+ for > 3

Mode:

++

1

11 for > 1

0 for 1

PDF - LogLogistic(0,1,5)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

-0

.5

0.

0

0.

5

1.

0

1.

5

2.

0

2.

5

3.

0

CDF - LogLogistic(0,1,5)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-0

.5

0.

0

0.

5

1.

0

1.

5

2.

0

2.

5

3.

0


46/75

Lognormal (Format 1)

RISKLognorm(, )

Parameters:

continuous parameter > 0


Domain:

0 x + continuous


2xln

2

1

e2x

1)x(f

=

=

xln)x(F

with

+

22

2

ln and

+

2

1ln


Mean:

Variance:

2


47/75

Skewness:

+

3

3

Kurtosis:

332234 ++ with

+ 1

Mode:

( ) 23224

+

PDF - Lognorm(1,1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-1 0 1 2 3 4 5 6

CDF - Lognorm(1,1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-1 0 1 2 3 4 5 6


48/75

Lognormal (Format 2)

RISKLognorm2(, )

Parameters:

continuous parameter


Domain:

0 x + continuous


2xln

2

1

e2x

1)x(f

=

=

xln)x(F


Mean:

2

2

e

+

Variance:

( )1e2 with2

e


49/75

Skewness:

( ) 12 + with2

e

Kurtosis:

332 234 ++ with2

e

Mode:

2e

PDF - Lognorm2(0,1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-2 0 2 4 6 8

10

12

CDF - Lognorm2(0,1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-2 0 2 4 6 8

10

12


50/75

Negative Binomial

RISKNegBin(s, p)

Parameters:

s the number of successes discrete parameter s > 0

p probability of a single success continuous parameter 0 < p < 1

Domain:

0 x n discrete


( )xs p1px

1xs)x(f

+=

ix

0i

s)p1(

i

1isp)x(F

+=

=

Where ( ) is theBinomial Coefficient.

Mean:

( )

p

p1s

Variance:

( )2

p

p1s


51/75

Skewness:

( )p1s

p2

Kurtosis:

( )p1s

p

s

63

2

++

Mode:

(bimodal) z and z + 1 integer z > 0

(unimodal) 0 z < 0

(unimodal) smallest integer greater than z otherwise

where( )

p

1p1sz

PDF - NegBin(3,.6)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

-1 0 1 2 3 4 5 6 7 8 9

CDF - NegBin(3,.6)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-1 0 1 2 3 4 5 6 7 8 9


52/75

Normal

RISKNormal(, )

Parameters:



Domain:

- x + continuous


2x

2

1

e2

1)x(f

=

=

x)x(F


Mean:

Variance:

2


53/75

Skewness:

0

Kurtosis:

3

Mode:

PDF - Normal(0,1)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

-3

-2

-1 0 1 2 3

CDF - Normal(0,1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-3

-2

-1 0 1 2 3


54/75

Pareto (First Kind)

Pareto(, a)

Parameters:


a continuous scale parameter a > 0

Domain:

a x + continuous


1x

a)x(f

+

=

=

x

a1)x(F

Mean:

1

a

for > 1

Variance:

( ) ( )21a2

2

for > 2


55/75

Skewness:

+ 2

3

12 for > 3

Kurtosis:

( )

( )( )43

2323 2

++for > 4

Mode:

a

PDF - Pareto(2,1)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 1 2 3 4 5 6 7 8 910

11

CDF - Pareto(2,1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 1 2 3 4 5 6 7 8 910

11


56/75

Pareto (Second Kind)

RISKPareto2(b, q)

Parameters:

b continuous scale parameter b> 0

q continuous shape parameter q > 0

Domain:

0 x + continuous


( ) 1q

q

bx

qb)x(f

++=

( )q

q

bx

b1)x(F

+=

Mean:

1q

b

for q > 1

Variance:

( ) ( )2q1q

qb

2

2

for q > 2


57/75

Skewness:

+

3q

1q

q

2q2 for q > 3

Mode:

0

PDF - Pareto2(3,3)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-2 0 2 4 6 8

10

12

CDF - Pareto2(3,3)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-2 0 2 4 6 8

10

12


58/75

Pearson Type V

RISKPearson5(, )

Parameters:


continous scale parameter > 0

Domain:

0 x < + continuous


( ) ( ) 1

x

x

e1)x(f

+

=

F(x) Has No Closed Form

Mean:

1

for > 1

Variance:

( ) ( )212

2

for > 2


59/75

Skewness:

3

24

for > 3

Kurtosis:

( )( )( )( )43

253

+ for > 4

Mode:

1+

PDF - Pearson5(3,1)

0.0

0.5

1.0

1.5

2.0

2.5

-0

.5

0.

0

0.

5

1.

0

1.

5

2.

0

2.

5

CDF - Pearson5(3,1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-0

.5

0.

0

0.

5

1.

0

1.

5

2.

0

2.

5


60/75

Pearson Type VI

RISKPearson6(1, 2, )

Parameters:




Domain:

0 x < + continuous


( )

( )21

1

x1

x

,B

1)x(f

1

21+

+

=

F(x) Has No Closed Form.

where B is theBeta Function.

Mean:

12

1

for2 > 1

Variance:

( )

( ) ( )21

1

22

2

2112

+for2 > 2


61/75

Skewness:

( )

+

+

3

12

1

22

2

21

211

2 for2 > 3

Kurtosis:

( )

( )( )

( )

( )( )

++

+

5

1

12

43

232

211

22

22

2 for2 > 4

Mode:

( )1

1

2

1

+

for1 > 1

0 otherwise

PDF - Pearson6(3,3,1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-1 0 1 2 3 4 5 6 7 8 9

CDF - Pearson6(3,3,1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-1 0 1 2 3 4 5 6 7 8 9


62/75

Pert (Beta)

RISKPert(min, m.likely, max)

Definitions:

6

maxlikely.m4min ++

minmax

min61

minmax

max62

Parameters:


m.likely continuous parameter min < m.likely < max


Domain:



( ) ( )

( ) 12121

1211

minmax),(

xmaxminx)x(f

+

=

( )( )21z

21

21z ,I),(B

,B)x(F

= with

minmax

minxz

where B is theBeta Function and Bz is theIncomplete Beta Function.

Mean:

6

maxlikely.m4min ++


63/75

Variance:

( )( )7

maxmin

Skewness:

( )( )+

maxmin

7

4

2maxmin

Kurtosis:

( ) ( ) ( )( )( )32

6213212121

21212

2121++++

+++++

Mode:

m.likely

PDF - Pert(0,1,3)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-0

.5

0.

0

0.

5

1.

0

1.

5

2.

0

2.

5

3.

0

3.

5

CDF - Pert(0,1,3)

0.0

0.2

0.4

0.6

0.8

1.0

-0

.5

0.

0

0.

5

1.

0

1.

5

2.

0

2.

5

3.

0

3.

5


64/75

Poisson

RISKPoisson()

Parameters:

mean number of successes continuous > 0

Domain:

0 x < + discrete


!x

e)x(f

x =

=

=

x

0n

n

!ne)x(F

Mean:

Variance:


65/75

Skewness:

1

Kurtosis:

+

13

Mode:

(bimodal) and +1 (bimodal) if is an integer

(unimodal) largest integer less than otherwise

PMF - Poisson(3)

0.00

0.05

0.10

0.15

0.20

0.25

-1 0 1 2 3 4 5 6 7 8 9

CDF - Poisson(3)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-1 0 1 2 3 4 5 6 7 8 9


66/75

Rayleigh

RISKRayleigh(b)

Parameters:

b continuous scale parameter b> 0

Domain:

0 x < + continuous


2

b

x

2

1

2e

b

x)x(f

=

2

b

x

2

1

e1)x(F

=

Mean:

2b

Variance:

2

2b2


67/75

Skewness:

( )

( )6311.0

4

32

23

Kurtosis:

( )2451.3

4

332

2

2

Mode:

b

PDF - Rayleigh(1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-0

.5

0.

0

0.

5

1.

0

1.

5

2.

0

2.

5

3.

0

3.

5

CDF - Rayleigh(1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-0

.5

0.

0

0.

5

1.

0

1.

5

2.

0

2.

5

3.

0

3.

5


68/75

Students t

RISKStudent()

Parameters:

the degrees of freedom integer > 0

Domain:

- x + continuous


2

1

2x

2

2

1

1)x(f

+

+

+

=

+=

2,

2

1I1

2

1)x(F s with 2

2

x

xs

+

where is the Gamma Function and Ix is theIncomplete Beta Function.

Mean:

0 for > 1*

*even though the mean is not well defined for = 1, the distribution is still symmetrical about 0.

Variance:

2

for > 2


69/75

Skewness:

0 for > 3*

*even though the skewness is not well defined for 3, the distribution is still symmetric about 0.

Kurtosis:

4

23 for > 4

Mode:

0

PDF - Student(3)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-5

-4

-3

-2

-1 0 1 2 3 4 5

CDF - Student(3)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-5

-4

-3

-2

-1 0 1 2 3 4 5


70/75

Triangular

RISKTriang(min, m.likely, max)

Parameters:


m.likely continuous mode parameter min m.likely max


Domain:



( )

min)min)(maxlikely.m(

minx2)x(f

= min x m.likely

( )

min))(maxlikely.m(max

xmax2)x(f

= m.likely x max

( )

( )( )minmaxminlikely.mminx

)x(F2

= min x m.likely

( )

( )( )minmaxlikely.mmax

xmax1)x(F

2

= m.likely x max

Mean:

3

maxlikely.mmax ++


71/75

Variance:

( )( ) ( )( ) ( )( )

18

minmaxminlikely.mlikely.mmaxmaxlikely.mmax 222 ++

Skewness:

( ) 2322

3f

9ff

5

22

+

where

( )1

minmax

minlikely.m2f

Kurtosis:

2.4

Mode:

m.likely

PDF - Triang(0,3,5)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

-1 0 1 2 3 4 5 6

CDF - Triang(0,3,5)

0.0

0.2

0.4

0.6

0.8

1.0

-1 0 1 2 3 4 5 6


72/75

Uniform

RISKUniform(min, max)

Parameters:



Domain:



minmax

1)x(f

=

minmax

minx)x(F

=

Mean:

2

minmax

Variance:

( )12

minmax2


73/75

Skewness:

0

Kurtosis:

1.8

Mode:


CDF - Uniform(0,1)

0.0

0.2

0.4

0.6

0.8

1.0

-0

.2

0.

0

0.

2

0.

4

0.

6

0.

8

1.

0

1.

2

PDF - Uniform(0,1)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-0

.2

0.

0

0.

2

0.

4

0.

6

0.

8

1.

0

1.

2


74/75

Weibull

RISKWeibull(, )

Parameters:



Domain:

0 x < + continuous


( )

= x

1

ex

)x(f

( )= xe1)x(F

Mean:

+

11b

where is the Gamma Function.

Variance:

+

+

11

21

22



75/75

Skewness:

23

2

3

1121

112

11

213

31

+

+

++

+

++

+


Mode:

11

1 for >1

0 for 1

PDF - Weibull(2,1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

-0

.5

0.

0

0.

5

1.

0

1.

5

2.

0

2.

5

CDF - Weibull(2,1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-0

.5

0.

0

0.

5

1.

0

1.

5

2.

0

2.

5

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