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5

85

9

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S ^V +, @ 4` @ S @ V4 +4,

@ S ^ @ iD> >5> 6>7> 8>9> :>;> <>43> M>T>Nj` @4

7

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@

4

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@ V4 +5=8,

@ S ^!` @ 3

S ^5 V +, 6` @ S @ V4 ^5> 6`

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v dqg doo j` @4

5

Fxpxodwlyh Glvwulexwlrq Ixqfwlrqv dqg Suredelolw| Ghqvlw| Ixqfwlrqv

Wzr zd|v wr dqdo|h hshulphqwv yld udqgrp yduldeohv lv wkurxjk wkh Fxpxodwlyh Glvwulexwlrq Ixqfwlrq+FGI, dqg suredelolw| ghqvlw| ixqfwlrq +sgi,

Ghqlwlrq ; Ohw [ +, eh d udqgrp yduldeoh ghqhg rq d vdpsoh vsdfh 1 Wkhq wkh Fxpxodwlyh Glvwul0exwlrq Ixqfwlrq +FGI, ri [ lv ghqrwhg e| wkh ixqfwlrq I[ +3, dqg lv ghqhg dv iroorzv

I[ +3, @ S ^[ 3` +54,

zkhuh wkh vhw ^[ 3` ghqhv dq hyhqw ru d froohfwlrq ri rxwfrphv1

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I[ +3, @ S ^[ 3` @4

85

85[

n@4

x +3 n,

1

1 5 2 x

F (x)X

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IV +v3, @4

7x +v3 4, .

4

7x +v3 5, .

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Vr

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;AAAA?AAAA=

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6

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+56,

:

s o

1

1 2 3 4

F (s )S o

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i[ +, @g

gI[ +, @

g

gS ^[ +, ` +57,

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I[ +, @

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4

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4

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dv iroorzv

S ^4 ? [ 5` @ S ^[ 5` S ^[ 4` @ I[ +5, I[ +4, @

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4

i[ +, g +5:,

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Prphqwv ri Udqgrp Yduldeohv

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[/ lv

[dyj @ H ^[` /

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5

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l/

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[ @

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pq

[@

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q

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\ @ j +[, +66,

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f (x )X

1

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ri [1 Lq idfw zh zloo xvh lqirupdwlrq derxw wkh I[ +,/ i[ +,/ dqg j +[, wr ghwhuplqh I\ +|, dqg i\ +|,

Wkh glvwulexwlrq ri t ' Ef

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i\ +|, @g

g|I\ +|, +6:,

Hdpsohv

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\ @ j +[, @ 5[ .4

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X

Y

y

Y y≤

Y= 2X+1

x=(y-1)/2X (y-1)/2≤

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hyhqw = [ +, 4

5+| 4,

1 Erwk hyhqwv kdyh wkh hdfw vdph rxwfrphv1 Wkhuhiruh

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= [ +,

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[

4

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@[ j

4 +|,

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4

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4 | A 6+74,

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i\ +|, @g

g|I\ +|, +75,

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i\ +|, @

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54 | 6

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g|I\ +|, @

g

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@

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X

Y

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I\ +|, @ 3 | ? 3

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I\ +|, @

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44

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xµ

σ

Iljxuh 4= sgi ri d Jdxvvldq Udqgrp Yduldeoh

Frpprq Udqgrp Yduldeohv

Wkh Jdxvvldq Udqgrp Yduldeoh

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qxpehu ri udqgrp skhqrphqrq vxfk dv qrlvh prvw frppxqlfdwlrq v|vwhp wr wkh udqgrp xfwxdwlrqv ri

wkh ghvluhg uhfhlyhg yrowdjh lq qrq0olqh0ri0vljkw zluhohvv frppxqlfdwlrqv v|vwhpv vxfk dv fhooxodu v|vwhpv

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ri 5

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i[ +, @4

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h

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I[ +, @

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T +, @4s5

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hui +, @5s

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h

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S ^4 ? [ 5` @4

s5

5

[

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4

h

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5 p[

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T

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[@

5

\@

51

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] @

s[5 . \ 5

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uhsuhvhqwv wkh uhfhlyhg hqyhorsh/ qdpho|sY 5

[. Y 5

\1 ] lv fdoohg d Ud|olhjk udqgrp yduldeoh1 Wkh sgi ri ]

lv

i] +, @

5h

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[@

5

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5

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0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

Fx(x

)

Rayleigh distributions for s=1,2,3,...,10

1

2

3

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I] +, @

]

4

i] +, g @

]

3

i] +, g @ 4 h

5

55 +8:,

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Wkh Hsrqhqwldo Udqgrp Yduldeoh

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Z @ ]5 +8;,

Li ] uhsuhvhqwv wkh yrowdjh hqyhorsh ri d vljqdo/ wkh Z uhsuhvhqwv wkh srzhu lq d vljqdo/ wr zlwklq d

frqvwdqw1 Wkhq Z kdv wkh iroorzlqj sgi

iZ +z, @4

5h

z

5x +z, +8<,

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46

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Fx(x

)

Exponential distributions for s=1,2,3,...,10

1

2

3

4

Wkh FGI ri Z lv

IZ +z, @4 h

5

5

x +z, +93,

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Xqlirup Udqgrp Yduldeoh

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lv xqlirupo| glvwulexwhg ehwzhhq 3 dqg 51 Wkh sgi ri vxfk d udqgrp yduldeoh lv

i +, @4

5^x +, x + 5,` @

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53 5

3 rwkhuzlvh+94,

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I +, @

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;?=

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53 5

4 A 5

+95,

Wkh Fhqwudo Olplw Wkhruhp

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vwdwhv wkdw wkh uhvxowdqw udqgrp yduldeoh zloo kdyh d Jdxvvldq glvwulexwlrq1 Wklv lv rqh ri wkh uhdvrqv zk|

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d yduldqfh ri 41 Ghqh d qhz udqgrp yduldeoh/ \ / dv iroorzv

\ @ olpQ$4

4

Q

Q[

l@4

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\@ 41

Ulfldq Udqgrp Yduldeohv

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iroorzv d Ud|olhjk glvwulexwlrq1 Krzhyhu/ li olqh0ri0vljkw lv hvwdeolvkhg/ wkhq wkh vljqdo iroorzv d Ulfldq

glvwulexwlrq/ zklfk kdv wkh iroorzlqj sgi

i] +, @

5h

5.y5

55 L3

y

5

x +, +97,

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ghjhqhudwhv wr d Ud|olhjk glvwulexwlrq1

47