Probability theory and random number generation А quick

3UREDELOLW\WKHRU\DQG
UDQGRPQXPEHUJHQHUDWLRQ
$TXLFNUHYLHZ
/iV]Oy6]LUPD\.DORV
3UREDELOLW\VSDFHWKURZLQJDGLH
6XEVHWVRIHOHPHQWDU\HYHQWV
HYHQWXQLRQDQGFRPSOHPHQW
RIDQHYHQWLVDOVRDQHYHQW
(OHPHQWDU\HYHQWV
SRVVLEOHRXWFRPHV
3SUREDELOLW\ PHDVXUHRQWKHHYHQWV
.ROJRPRURYD[LRPV
‡SUREDELOLW\LVLQ>@
‡PHDVXUHRIWKHHPSW\VHWLVRIWKHWRWDOVHWLV
‡3U$% 3U$3U%LI$DQG%DUHGLVMRLQW
5DQGRPYDULDEOHV
0DSVWKHHOHPHQWDU\
HYHQWVRQWRUHDOQXPEHUV
IHOHPHQWDU\HYHQW
3UREDELOLW\'LVWULEXWLRQ)XQFWLRQ3')[ 3^I[`
3UREDELOLW\'HQVLW\)XQFWLRQSGI[ 3^I [`
([SHFWHGYDOXH
(>I@ ΣI[Â SGI[
9DULDQFH
' >I@ (>I(>I@ @ (>I @( >I@
6WDQGDUGGHYLDWLRQ
' >I@
&RQWLQXRXVUDQGRPYDULDEOHV
FROG
0DSVWKHHOHPHQWDU\
HYHQWVRQWRUHDOQXPEHUV
IUHH]LQJ
KRW
I WHPSHUDWXUHLQ&HOVLXV
3UREDELOLW\'LVWULEXWLRQ)XQFWLRQ3')[ 3^I[`
3UREDELOLW\'HQVLW\)XQFWLRQSGI[ G3')[G[
([SHFWHGYDOXH
(>I@ ∫I[Â SGI[G[
9DULDQFH
' >I@ (>I(>I@ @ (>I @( >I@
6WDQGDUGGHYLDWLRQ
' >I@
&RQGLWLRQDOSUREDELOLW\DQG
LQGHSHQGHQFH
% $%
:HNQRZWKDWWKHRXWFRPHLVLQ$
:KDWLVWKHSUREDELOLW\WKDWLWLVLQ%"
$
3U%_$ 3U$%3U$
(YHQWVSDFH
,QGHSHQGHQFHNQRZLQJ$GRHVQRWKHOS3U%_$ 3U%
3U$% 3U$Â 3U%
2SHUDWLRQVRQUDQGRPYDULDEOHV
●
([SHFWHGYDOXHLVDOLQHDURSHUDWLRQ
± (>I I @ (>I @(>I @
± (>DI@ D(>I @
± (>I ÂI @ (>I @Â (>I @LIWKH\DUHLQGHSHQGHQW
●
9DULDQFHLVDTXDGUDWLFRSHUDWLRQ
± '>DI@ D'>I@
± '>I I @ '>I @'>I @LIWKH\DUHLQGHSHQGHQW
7KHRUHPVRIODUJHQXPEHUV
●
II«I0DUH´LQGHSHQGHQW´UDQGRPYDULDEOHV
RIWKH´VDPHGLVWULEXWLRQ´
± ZHDNDQGVWURQJODZV0Σ IP→(>I@
± WKHRUHPRILWHUDWHGORJDULWKP
VXS^0Σ IP(>I@` '√ORJORJ00
± &HQWUDOOLPLWWKHRUHP
0Σ IPLVQRUPDOGLVWULEXWLRQ
PHDQ(>I@VWDQGDUGGHYLDWLRQ'> I@√0
5DQGRPQXPEHUJHQHUDWLRQ
,WLVHQRXJKWRJHQHUDWHXQLIRUPO\GLVWULEXWHG
QXPEHUVLQ>@U
● 7UDQVIRUPDWLRQRIUDQGRPYDULDEOHV
I 3')U
● 3URRI
3U^I[` 3U^3')U[` 3U^U3')[` 3')[
●
5HDOUDQGRPQXPEHUJHQHUDWRUV
●
'HYLFHWRJHQHUDWHDVLQJOHGLJLWRIWKHQXPEHU
/6%
&ON
FRXQWHU
QRLVH
(
(
∆W
∆W
&RPSDUDWRU
5DQGRPGLJLWQXPEHURIFKDQJHVPRG
3VHXGRUDQGRPQXPEHUJHQHUDWLRQ
‡ GHWHUPLQLVWLFLWHUDWHGIXQFWLRQV
UQ )UQ
‡ EHKDYHOLNHUDQGRPVHTXHQFHVFKDRV
'HULYDWLYHRI)LVODUJH
‡ :KDWLVUDQGRP"""VWDWLVWLFDOWHVWV
&KDRVQXPEHURIUDEELWV
UQ &UQUQ
& & %DGUDQGRPQXPEHUJHQHUDWRUV
UQ )UQ
UQUQSDLUV
5HTXLUHPHQWVRI)
GHQVHO\ILOOVWKHUHFWDQJOH
● KDVODUJHGHULYDWLYHHYHU\ZKHUH
● VKRXOGEHLQ>@
●
SHULRGLFLW\
$SHULRGLFOHQJWK
&RQJUXHQWLDOJHQHUDWRU
)[ ^JÂ [F`
IUDFWLRQDOSDUWRIJÂ [F
JLVODUJH
6HOHFWLRQRIJDQGFDQG[
PDNLQJWKHDSHULRGLFOHQJWK
DQGWKHSHULRGODUJH
,VLW´UDQGRP´"
:KDWSURSHUWLHVRI´UDQGRPQHVV´DUHQHHGHG
P$0
● 6WDWLVWLFDOWHVWV
'LIP$0$
± XQLIRUPQHVV
LVVPDOO
.ROJRPRURY
●
6PLUQRY
± 6SHFWUDOSUREH
$
'LVWDQFHVEHWZHHQ
WKHSODQHVDUH
VPDOO