Mathematics - Introduction to Probability Theory

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Wkhq hdfk fdug lv dvvljqhg dq lqwhjhu rq wkh uhdo olqh1                      Vxssrvh zh dvvljq wklv pdsslqj wkh ixqfwlrq                    [ +
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zkhuh   
   lv dq| rxwfrph


Vxfk d pdsslqj lv fdoohg d        udqgrp yduldeoh1

Ghqlwlrq : , udqgrp yduldeoh/ [ + , lv d ghwhuplqlvwlf                                   ixqfwlrq      wkdw dvvljqv hdfk rxwfrph/         
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    41 Wkhuh lv qrwklqj udqgrp derxw lw
    51 Lw lv qrw d yduldeoh1 Udwkhu lw lv d vhw ixqfwlrq1
Krzhyhu/ lq shuiruplqj dqdo|vlv/ lw lv frqyhqlhqw wr wuhdw                        [ + ,    dv d yduldeoh1     Frqvlghu d vhfrqg h{dpsoh

ri d udqgrp yduldeoh1        vlqfh wkh udqgrp yduldeoh lv d ixqfwlrq ri dq rxwfrph/ zklfk kdv dq dvvrfldwhg

suredelolw|/ wkh udqgrp yduldeoh dovr kdv dq dvvrfldwhg suredelolw|1

                                                                                                
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H{dpsoh 7 Frqvlghu wkh vdph fdug h{shulphqw1 Vxssrvh zh ghqh d udqgrp yduldeoh                                               8 +
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                                                   li 
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Qrwh wkdw wkh rxwfrph fdq eh dvvljqhg wkh vdph qxpehu1     Wkh ixqfwlrq ghqhg e| wkh udqgrp yduldeoh
fdq e| dq| vhw ixqfwlrq surylghg wkdw wkh dvvrfldwhg suredelolwlhv dv wkh 	" lv }hur

Zkdw derxw wkh suredelolwlhv
,   udqgrp yduldeoh lv d ixqfwlrq ri wkh rxwfrphv ri dq h{shulphqw1                              Wkhuhiruh/ vlqfh hdfk rxwfrph kdv d

suredelolw|/ wkh qxpehu       dvvljqhg wr wkdw ixqfwlrq e| d udqgrp yduldeoh/                              dovr kdv d suredelolw|1          Wkh

suredelolw| ri d jlyhq udqgrp yduldeoh lv ghwhuplqhg dv iroorzv1

                                                                                                
                                               5 ^[ +  , @ { ` @ S           
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zkhuh   [ " +{, lv d pdsslqj iurp         wkh uhdo olqh/           ?/ wr wkh vhw ri doo rxwfrphv/ 
1
H{dpsoh 8         Frqvlghu wkh udqgrp yduldeoh [ +
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Fxpxodwlyh Glvwulexwlrq Ixqfwlrqv dqg Suredelolw| Ghqvlw| Ixqfwlrqv
Wzr zd|v wr dqdo|}h h{shulphqwv 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 Glvwul0
exwlrq Ixqfwlrq +FGI,         ri [ lv ghqrwhg e| wkh ixqfwlrq I[ +{! , dqg lv ghqhg dv iroorzv

                                                                 I[ +{! , @ S ^[  {! <                                               +54,

zkhuh wkh vhw   ^   [  {!   <   ghqhv dq hyhqw ru d froohfwlrq ri rxwfrphv1

H{dpsoh :       Frqvlghu wkh udqgrp yduldeoh lq H{dpsoh 61                                   Wkh FGI ri      [   lv




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H{dpsoh ;       Frqvlghu wkh udqgrp yduldeoh lq H{dpsoh 71                                   Wkh FGI ri      8   lv



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Ghqlwlrq <            Ohw   [ +
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1 Wkhq wkh Suredelolw| Ghqvlw|
Ixqfwlrq +sgi ,        ri   [ lv ghqrwhg e| wkh v|pero i[ +{, dqg lv ghqhg dv wkh ghulydwlyh ri lwv FGI
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Qrwh wkdw zh fdq uhfryhu wkh FGI iurp wkh SGI wkurxjk lqwhjudwlrq
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Wkhuhiruh    E[ +{, zloo dozd|v lqwhjudwh wr              4
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Ixuwkhupruh/ wkh suredelolw| wkdw [ + , lv ehwzhhq {"                                  dqg   {# / {#   A {"   dqg   {#   lqfoxvlyh/ fdq eh frpsxwhg

dv iroorzv
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Qrwh wkdw wkh dujxphqw/ 
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glvfxvvlrq/ zh zloo dvvxph wkdw wkh ghshqghqfh ri d udqgrp yduldeoh [ rq  lv lpsolflw dqg rplw wkh
dujxphqw iru frqyhqlhqfh


Prphqwv ri Udqgrp Yduldeohv
Prphqwv ri udqgrp yduldeohv lqfoxgh d qxpehu ri txdqwlwlhv vxfk dv dyhudjhv                                               /   yduldqfhv/ vwdqgdug
ghyldwlrqv/   hwf1      Wkh| duh sduwlfxoduo| xvhixo lq frppxqlfdwlrqv ehfdxvh wkh| surylgh ydoxdeoh lqirupdwlrq
derxw d udqgrp yduldeoh zlwk kdylqj wr nqrz lwv vwdwlvwlfv +l1h1/ wkh FGI ru sgi,


Ghqlwlrq 43 Ohw [ eh d udqgrp yduldeoh zlwk wkh sgi i[ +{, Wkhq wkh h{shfwhg                                                           ydoxh    ru   dyhudjh   ri

[ / lv                                           ]"
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GhqlwlHG "" Ohw [ eh d udqgrp yduldeoh zlwk sgi i[ +{, Wkhq/ wkh yduldqfh                                                         ri   [   lv

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Ghqlwlrq 45         [ eh d udqgrp yduldeoh zlwk sgi E[ +{,
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dqg wkh Gwk fhqwudo prphqw r [     E        lv
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Ilqdoo|/ ohw   [ eh d udqgrp yduldeoh dqg ohw j +[, eh vrph ixqfwlrq ri wkdw udqgrp yduldeoh1                                              Lq prvw fdvhv
wklv   j +[ , zloo wudqvirup [ lqwr d qhz udqgrp yduldeoh/ zklfk zh zloo fdoo \
                                                  \ @ j +[ ,                                                                                            +66,

zklfk kdv lwv rzq   FGI dqg sgi/ zklfk pxvw eh irxqg wkurxjk wkh xvh ri udqgrp yduldeoh wudqvirupdwlrq

whfkqltxhv1    Krzhyhu/ li zh duh lqwhuhvwhg lq wkh                    FHFDGML       ri   \ / vxfk dv lwv dyhudjh/ wkhq nqrzohgjh derxw
\ *v FGI ru sgi duh qrw uhtxluhg1           Udwkhu zh fdq mxvw xvh wkh douhdg| nqrzq FGI dqg sgi ri                                             [   1


Wkhruhp 4 Ohw        [ eh d udqgrp yduldeoh zlwk sgi i[ +{,                              Ohw      \   @     j +[ ,   eh vrph wudqvirupdwlrq ri          [   1
Wkhq wkh    h{shfwhg ydoxh   ru   dyhudjh        ri   \   lv
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                                   \@OD ( 0 ^\ ` @ H ^j +[ ,` @                               j +{, i[ +{, g{                                           +67,
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Ixqfwlrqv ri d Udqgrp Yduldeoh
Ghqlwlrq 4 Frqvlghu d ixqfwlrq ri d udqgrp yduldeoh/ j +[ ,1 Wkh uhvxow lv dqrwkhu udqgrp yduldeoh/
zklfk zh frxog fdoo \ / \ @ j +[ ,1 Krzhyhu wkh FGI dqg sgi ri \ zloo eh glhuhqw wkdq wkh FGI dqg sgi
ri [ 1 Lq idfw zh zloo xvh lqirupdwlrq derxw wkh 1[ +{,/ i[ +{,/ dqg j +[ , wr ghwhuplqh 1\ +|, dqg i\ +|,

Wkh glvwulexwlrq ri t                   '        }Ef 
Wkh SGI ri wkh udqgrp yduldeoh/      \ / lv qrwklqj pruh wkdq wkh suredelolw| ri wkh hyhqw wkdw E  |j Wklv                               \
frqvlv                                                    \
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Wkh Suredelolw| Ghqvlw| Ixqfwlrq +sgi, ri                                             t       '       } Ef 
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H{dpsohv
H{dpsoh 43            Ohw [ eh d udqgrp yduldeoh1                  Ohw


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Ilqg   I\   +| ,   dqg i\ +|,1 Wr qg I\ +|,/ zh pxvw hydoxdwh wkh hyhqw E\                                      |j zkhuh \            lv wkh udqgrp yduldeoh
dqg   |   lv vlpso| d qxpehu1          Frqvlghu wkh iroorzlq             j judsk
                                                                              Y

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                                                {X ≤(y-1)/2}                                  x=(y-1)/2              X




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Wkh vkdghg yhuwlfdo olqh fruuhvsrqgv wr wkh hyhqw E
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Qrz vxssrvh [ zdv xqlirupo| glvwulexwhg ehwzhhq 3 dqg 41                                      Wkhq

                                                                                              
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H{dpsoh 44           Ohw [ eh d udqgrp yduldeoh1 Ohw

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Frpprq Udqgrp Yduldeohv
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Wkh Jdxvvldq udqgrp yduldeoh lv wkh prvw frpprq ri doo udqgrp yduldeohv1 Lw lv xvhg wr fkdudfwhul}h d
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Wkh H{srqhqwldo Udqgrp Yduldeoh
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                                                                                           46
                                                                       Exponential distributions for s=1,2,3,...,10
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Lq v|vwhpv ru vljqdov zkhuh wkh skdvh lv qrw nqrzq/ lw lv jhqhudoo| prghohg dv d udqgrp yduldeoh                                                                      / zklfk

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Wkh Fhqwudo Olplw Wkhruhp
Vxssrvh zh wdnh d odujh qxpehu ri udqgrp yduldeohv dqg vxp wkhp wrjhwkhu1                                                                 Wkh   fhqwudo olplw wkhruhp
vwdwhv wkdw wkh uhvxowdqw udqgrp yduldeoh zloo kdyh d Jdxvvldq glvwulexwlrq1                                                         Wklv lv rqh ri wkh uhdvrqv zk|

Jdxvvldq udqgrp yduldeohv duh vr frpprq1                            D pruh irupdo vwdwhphqw ri wkh wkhruhp iroorzv



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Ulfldq Udqgrp Yduldeohv
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                                                                                        47

						
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