А.А. Кудрявцев - Пособие по теории вероятностей (1115312), страница 11
Текст из файла (страница 11)
. . < x n = bσ = f (y1 )[u(x1 ) − u(x0 )] + . . . + f (yn )[u(xn ) − u(xn−1 )], Öb!cWdef0g7iFjüUrm*N&'!!-P¿9IE*1]H+ @ *1}|h"'!.K'!!-P¿9D ø|h"'!.nnTO=Dst_-W.D&f[J& T]j\*Q§]^biNP]^J.W&bN^JYIOHNPkHOH_-d§FHGHNjm*Nj\¨FHGHQQHOoJNPUPG=D\*lOH_:V]^bRTR nG=D&WOH_d`kHQH]j\=b nJIO=Dst_-WmaxD&N^J]iK {xi − xi−1 } → 0IEI&JzZbOHYHHQHQσFHI5ZbOHYHHQHQ IQuIvI&stO=DLk=D&N^J]rK"I- h"I- xi−1 ≤ yi ≤ xi i = 1, .
. . , nf (x)u(x)I=Zbf (x) du(x).0HE!=E!'<^ f (x) '!!-P¿'H\ 7QA¾0HE!=E!'v' u(x) O=D&¾\7 "!'U>E=<'N 9F8!')!!'N BEH<>!1*1B!'.[><'!!$ ξ !;+$+^¾><'\a GHQ|^JIR¿UPIWIG.KJn¦kJII&JGHN^stYN[a, b]¤EÁ Ä ¦Ã aCÖ Eξ =+∞Zx dFξ (x),(8.2)DP¿P )+9 F (x)R 2¯0EH!E=!<'L^A+)B!<':^\EH<>!1!*F1\!'<>!'!$RB4'!'!.ξ%*)!'U^C,#P`<!^C<>"7 \7 "!'U>=#PG9F8!')v<'^ØÓBÒûHE?-+HR]PFHI\*l&sPIW.D&OHQHN:IFHGHNjm*Nj\*NPOHQK Ö [ mH\=KSO=DV.ILM5m*NPOHQKRaDJN^RDJQHkHNt]PYIUPIIM9QomD&OHQKKHW\=KHN^J]rK)W9vI\*l&pQHOH]^JWN]j\=bk=D&NPW"vI\*NPNXbLm*IvOH_-R-nOHN^MNj\*Q+QH]PFHI\*l&stIoW.D&OHQHNIFHGHNjm*Nj\*NPOHQHN Ö RaDJNPRaDJQHkHNP]PYIUPI5IM9QmD&OHQK"kHNPGHN^sXQHOJNPUPG=D\ÃXNPvNPUjD¤FHGHNjm*Nj\*NPOHQK Ö EQ Ö [ KHW\=KHf[J]rKu|PYHWQHW.D\*NPOJOH_-RTQED&YQstWNP]^JOHIHnGHQHRaD&OHIW]PYHQHdQHOJNPUPG=D\gQHRTNPN^J)]PRT_-]j\gF\*I&iDm*Q`FHILmYHGHQo−∞ξWIdnKHW\=KHf[iNPdH]rK'UPG=D&ZEQHYIR§FHItm*_-OJNPUPG=D\*lOHIdyZbOHYHHQHQ¦D&YId9MN[IkHNPWQmHoOH_-d"UPNPIRTN^JGHQHkHNP]tYHQHd']PRT_-]j\yQHRTNPN^JQHOJNPUPG=D\n]^JILKiQHd"WSFHG=D&WIdzk=D&]^JQ"]PII&JoOHI&pNPOHQK¯ Ö jÄSNPdH]^JWQJN^\*lOHIHn¦FHI]^JGHIQHR¿QHOJNPUPG=D\*lObf¼]^bRTR¦b mH\=K`QHOJNjoUPG=D\D Ö j=RTNPNPR[[σ = y1 [u(x1 ) − u(x0 )] + .
. . + yn [u(xn ) − u(xn−1 )],rU m*NnCED'GHQH]jbOHYN Ö 5QstIvG=DMNPO=D']jbRaRaDmH\=Kxi−1 ≤ yi ≤ xi i = 1, . . . , nσOHNPYI&JIGHI&d5ZbOHYHHQHQ5G=D&]PFHGHNjm*Nj\*NPOHQKGHQFHNPGHNjVItm*N-YFHGHNjm*Nj\=bFHGHQmaxi {xi −ntbkHQJ_-W.DKnkJI[FHGHQXI&JGHQH=DJNj\lOH_:V ]j\D&UjD&NPRT_-Nxi−1 } → 0KHW\=KHf[J]KqI&JGHQH=DJNj\*lOH_%RTQnFHI\=bk=D&NPR-nkJI xnyUeim*[u(xN i )−u(xNP]^JluFi−1\*I)]oEξ = S1 − S2S1iDm*l)RTN^M5m=b`UPG=D&ZQHYD&RTQZbOHYHHQHdQFHGHQn¦D{F\*I&iDm*lSFHILm"UPG=D&ZEQHYIR`ZbOHYHHQHQ y = 1 FHyGHQ = Fξ (x)]PR-GHQHx]& ≥Ö 0 j¦SDYH2 QHRIvG=DstIR-n.QHOJNPUPG=D\+çJQ\=JlNP]tDy Ö y ]P=W&KFsLξD&(x)O9][QHOJNPxUPG=<D\*0IR§ÁTQHRaD&O=DE]PII&JOHI&pNjoOHQHNPR[[Zx dFξ (x) = −Z0−∞Fξ (x) dx ++∞Z0(1 − Fξ (x)) dx.(8.3)s ê<t!ý Ú ¤:«¥z9²¿Ù w Æß.ÞàHv² Î ª« w ®ß&®HÖF (x)1F (x)1S1S20dÏ »¼0±º7ë¾ £ xÏd»¼±ºWë¾ ¤ x¾xSaxCE Ö [ )\*NPWIdk=D&]^JQ]PII&JOHI&pNPOHQKu Ö @ RT_OHNbYDsLD\*QFHGHNjom*Nj\*_QHOJNPUPGHQHG=IW.D&OHQK¦D&YDK9]PQJ&b.D&HQK'KHW\=KHN^J]rKyIv_-kHOHIdzmH\=K9\*QJNPG=DJ&bG=_FHIJNPIGHQHQWNPGHILKJOHI]^JNPd0T]j\*QFHGHNjm*Nj\*_äQHOJNPUPGHQHGHIW.D&OHQKOHNzbYDst_-W.D&f[J]rKnJI9|jJI"IstO=Dk=D&N^Jn*kJI9GHNPkHlzQm*N^J"OHNXIzOHNPIFHGHNjm*Nj\*NPOHOHIRQHOJNPUPG=D\*N&nD5IvzQHOJNjoxi1 yixiUPG=D\*NcFHIXW]PNPd"Iv\D&]^JQ"WI&stRTILM9OH_:V5stO=DLkHNPOHQHd'FHItm*_-OJNPUPG=D\*lOHIUPIF=D&G=D&RTN^JG=D¨mD&OHOHIR¨]j\=bk=D&NEsLD&FHQH]PQZ+∞ZQx dFξ (x)x dFξ (x)−∞KHW\=KHf[J]rKu|PYHWQHW.D\*NPOJOH_-RTQ¾x axCE Ö @ =+DJNPRaDJQHk=NP]PYINEIM9QmD&OHQHNcKHW\=KHN^J]K"VD&G=D&YJNPGHQH]PJQoYId'G=D&]PFHGHNjm*Nj\*NPOHQKu]j\=bk=D&dHOHId'WNj\*QHkHQHOH_n*DSOHN]tD&RTId']j\=bk=D&dHOHId'WNj\*QHkHQHOH_nJIzNP]^Jln=NP]\*QdEξ = Eηξ=η¾x axCE Ö £H T]j\*Q]^biNP]^J.W&b.N^Jn=Qy]j\=bk=D&dHO=DKyWNj\*QHkHQHO=D QHRTNPN^JEξm*QH]PYHGHN^JOHINcG=D&]PFHGHNjm*Nj\*NPOHQHNcQ5FHGHQHOHQHRaD&N^JXstO=DLkHNPOHQK]%WNPGHILKξJOHI]^JKox1 , x 2 , .
. .RTQ]PII&JWN^J]^JWNPO=OHIHn=JISS vp1 , p 2 , . . .SEξ = vXxi p i .(8.4)i¾ x axCE]^biNP]^J.W&b.N^Jn=Qy]j\=bk=D&dHO=DKyWNj\*QHkHQHO=D QHRTNPN^JÖ Ô¢ T]j\*QEξξD&v]PI\*f[JOHIzOHNPFHGHNPGH_-WOHINXG=D&]PFHGHNjm*Nj\*NtOHQHNS]EF\*I&JOHI]^JlfnJIf (x)Eξ =Zxf (x) dx.(8.5)u_æG=D&]P]PRTI&JGHNj\*Q¯UPN^IRTN^JGHQHk=NP]PY=QHd¯]PRT_-]j\1RaDJNPRaDJQHkHNP]PYIUPIgILM9QmD&OHQK NPGHNPdm*NPRJNPFHNPGHlyY+G=D&]P]PRTI&JGHNPOHQ=f;NPUPI'ZEQstQHkHNP]PYIUPIy]PRT_-]j\DED&Y+QstWNP]^JOHIQspYI\*lOHIUPI5YbGH]tD5ZEQstQHYHQnHYIIGm*QHO=DJD5HNPOJG=D5RaD&]P]]PQH]^JNPRT_n*]PI]^JILKiNPdQs JIkHNPYuO=D5FHG.KHRTIdn=QHRTNPN^J9WQmnxþ ï ÿï =Pnxi m i,i=1 mii=1Pn(8.6)b!cWdef0g7iFjüÖ rU m*N]^bJlEYIIGm*QHO=DJ_§JIkHNPYn.WYI&JIGH_:V]PI]PGHNjm*I&JIkHNPO=_¨RaD&]P]P_]PII&JWN^Joximi]^JWNPOHOHIH T]j\*QMNJGHNPv&b.N^J]KO=DdJQ`YIIGm*QHO=DJ&b HNPOJG=D+RaD&]P]zvNP]PYIOHNPkHOHIUPIJIOHYIUPI]^JNPG.M9OKnHF\*I&JOHI]^Jl5WYI&JIGHIR§IFHGHNjm*Nj\=KHN^J]KyZbOHYHHQHNPdnJIWm(x)|^JIR¸]j\=bk=D&NXQH]PFHI\*l&sPb.N^J]KZEIGHR¦bL\Dhxþ ï ÿï =RRxm(x) dx.m(x) dx(8.7):]PFHIRTOHQHR1JNPFHNPGHlHnkJI+WNPGHILKJOHI]^JlKHW\=KHN^J]K J&D&YId M5N5RTNPGHIdnYD&YQRaD&]jo]tDn]XJIdu\*Qpl"G=DstOHQHHNPdnkJI"RaD&]P]tDoWNPGHILKJOHI]^Jl'W]PNPd+]PQH]^JNPRT_ÇJIkHNPYgQ\*QW]PNPUPIy]^JNPG.M9OK* G=D&WOKHN^J]rKqNjm*QHOHQHHN&GHQq|^JIRb]j\*IWQHQqsPO=D&RTNPO=DJNj\*QIvNPQVm*GHIvNPduW Ö . :Q Ö Ô :]^JD&OHIW&KJ]KuG=D&WOH_-RTQuNjm*QHOHQHHN& T]j\*QuJNPFHNPGHlz]PG=D&WOHQJl]PII&JOHI&pNPOHQK Ö £ ¦Q) Ö Ô¢ ¦][]PII&JOHI&pNPOHQKHRTQq Ö . ¦Q) Ö Ô ¦]PII&JWN^J]^J.WNPOHOHIHnJI]^J&D&OHIW&QJ]K)KH]POHIHnkJI'RaDJNPRaDJQHkHNP]PYINILM9QmD&OHQHNSNP]^JlyOHNkJI"QHOHIN&nYD&Y)YIoIG.m*QHO=DJD"HNPOJG=D'RaD&]P]S]PQH]^JNPRT_EnW"YI&JIGHId)Njm*QHOHQHkHO=DK)RaD&]P]tD9G=D&]PFHGHNjm*Nj\*NPO*DWz]PII&JW&N^J]PJWQHQ+]sLD&YIOHIRgG=D&]PFHGHNjm*Nj\*NtOHQKWNPGHILKHJOHI&]^JNtd¤EÁ Ä ¦Ã aC Ö @ @Ä`\=*10EH!E=!!'"1!+;$+"^)1+!'\EI.!'!*^L\0HEIE!=E !/ '<`^ \g-: IR=*1 −→.B9IRP#C / / $)6 ?"*^-*1 Û'U;9 B'< "'*"*.σBnng −1 (B) ≡ {x ∈ IRn | g(x) ∈ B} ∈ Bn)U^[6 / P# B ∈ B R¾xSa xCE Ö ¦ :IGHNj\*NPW]PYDK oÓD\*UPNPvG=D NP]^Jlm*NPOHO=DKW]PNPRTQI&JYHGH_J._-RTQêRTOHILMNP]^JW.σD&RTQFHGHI]^JG=BD&OH]^J.W.DnoÓD\*UPNPvG=DnFHIGHIM5oσ-uILM9OHI`FHIYDoIRnsLDJlHnkJIEmD&OHO=DK oÓD\*UPNPvG=DKHW\=KHN^J]rK9O=D&QHRTNPOHl&pNPd oÓD\*UPNPvGHIdn]PItm*NPG.M5DiNPdσW]PNERTOHIMNP]^JW.D5WQσmDnHUrm*NnB1 × .
. . × B nBi ∈ B i = 1, . . . , n¾x axCEÖ Ô* :IGHNj\*NPW]PYDKZbOHYHHQKuKHW\=KHN^J]rK]j\=bk=D&dHOHId+WNj\*QHkHQoOHIduO=DQstRTNPGHQHRTIR¸FHGHI]^JG=D&OH]^J.W&N(IRn , Bn ) ¤Á ux Ö S 2 / @` \- =E*^O0HE!Y=EV!'<H^"R@½!I ξPD)- "IM=*`*^BHE><1!!*^B'!><'!!CEg(ξ) ≡Zx dFg(ξ) (x) =Zg(x)g(x) dFξ (x).)+\!'<^SBC{!HE><;+$1!+*1[^ .0'!HE><!¤'!=E!E!$'<Á^Mξ )+ÄA¦1'(ÃHa B+C'!+'!\'UÖIECq><£H*YP°¥"@BA#PD*+H+'U ;**1Sk!0"HE7 !-=E !/ '$01'! BAÒ P(B)> 0ÓxF (x| B) = P(ξ < x| B).s ê<t!ý Ú¤:«¥z9²¿Ù w Æß.ÞàHv² Î ª« w ®ß&®H¤EÁĦÃaCE Ö Ô¢°\+!$& n 7\ 7"'!>U-=E'< Ü98F'!)!' ÜBHE><1!.;+$*1+B'!^¾><'!><!'$ \ξ A'H\+'!'UC><(A*'U;*kG- / $0"' B Ò P(B) > 0Ó C!.E(ξ| B) = x dF (x| B),PD)+ F (x| B) -"I,HB+!*^Q0HE!=E!'<^(A)+\!'<^OBHE><1!*1QB'!><'!!$ ξ R[ÖZ b]^Jl{¥FHI\*O=DK5UPGbFHF=DX]PIv_:JQHd9Q{B1 , .
. . , B nF (x| B1 ), . . . , F (x| Bn )]PII&JWN^J]^J.W&bf[iQ=Ny|^JQHR¿]PIv_JQKHRÇb.]j\*IW&OH_-NyZbOHYHHQHQ§G=D&]PFHGHNjm*Nj\*NPOHQHK§]j\=b.ok=D&dHOHId)WNj\*QHkHQHOH_ IUemD9FHI'ZEIGHR¦bL\*NFHI\*OHIdqW&NPGHIKJOHI]^JQ)mH\=K)ZbOHYHHQHQξG=D&]PFHGHNjm*Nj\*NPOHQK]j\=bk=D&dHOHIdWNj\*QHkHQHOH_]PFHG=D&WNjmH\*QHWI9FHGHNjm*]^JD&W\*NPOHQHNF (x)ξF (x) =nXP(Bi )F (x| Bi ).i=1 s%FHI&]j\*Njm*OHNPUPISG=D&WNPOH]^JW.DQ5IFHGHNjm*Nj\*NPOHQK Ö Ô¢cW_JNPYD&N^JS]j\*Njm=bf[iNtN[FHGHNjm*]^JD&Wo\*NPOHQHNXRaDJNPRaDJQHkHNP]PYIUPI9ILM9QmD&OHQKy]j\=bk=D&dHOHIdWNj\*QHkHQHOH_ hξEξ =nXP(Bi )E(ξ| Bi ).(8.8)SÖ Ö ÁD&WNPOH]^JWIu Ö Ö aO=Dst_W.D&N^J]KM!9P#D m` 7H$nA.*1B^!""')U^ 7 7"'!>U-=P#@98F'!)!'<^=+pJD%ZEIGHR¦bL\D-I]^J&D&N^J]rK]PFHG=D&WNjmH\*QHWIdQWc]j\=bk=D&N&nYIUemDcG=D&]P]PRaDJGHQHW.D&NPRaDK]PIWIYbFHOHI]^Jl]PIv_JQHdi=1¾x axCEKHW\=KHN^J]rK+vNP]PYIOHNPkHOHIdnFHGHQ+|^JI&RWN^G.VOHQHd+FHGHNjm*Nj\ ]^bRTRTQHGHIW.D&OHQKsLD&RTNPOBKo inN^J]rKO=D∞ç:NPdHk=D&]"RT_¥mDm*QHRÇI&FHGHNjm*Nj\*NPOHQHN"J&D&Y`O=Dst_-W.D&NPRT_:V`OHN^sLD&WQH]PQHRT_:V]j\=bk=D&doOH_:V WNj\*QHkHQHO :I\*NPNzFHItm*GHIvOHIuIyOHN^sLD&WQH]PQHRT_:V]j\=bk=D&dHOH_:VW&Nj\*QHkHQHO=DV GHNPkHlFHIdm*N^J+W " o ÄD&OHOHINzIFHGHNjm*Nj\*NPOHQHN9JGHNPv&bN^J.]K`O=D&RêmH\=K`ZEIGHR¦bt\*QHGHIW.DoOHQKOHNP]PYI\*lYHQV`WDM9OH_:V ]PWIdH]^JW+RaDJNPRaDJQHkHNP]PY=QV`ILM9QmD&OHQHdQqm*QH]PFHNPGH]PQHd]j\=bk=D&dHOH_:VWNj\*QHkHQHO¤EÁ Ä ¦Ã aCE Ö OT-T[+''< 7$& 7 'UC5 -\ 'M)U^[ 6 / !$&|q% HEB><∈1!!B$C 5i=\1,'!><.'!.!.$ , nξ C , .
. . , ξ !;+$+6^"D;+.1niP(ξ1 ∈ B1 , . . . , ξn ∈ Bn ) = P(ξ1 ∈ B1 ) · . . . · P(ξn ∈ Bn ).DJNPRaDJQHkHNt]PYIN-IM9QmD&OHQHN:]j\=bk=D&dHOHIdzWNj\*QHkHQHOH_gIv\DmD&N^J]j\*Njm=bf[iQHRTQI]POHIWOH_-RTQu]PWIdH]^JW.D&RTQh [ E(a + bξ) = a + bEξ,Urm*Na, b ∈ IR~n NP]j\*Q]^biNP]^J.W&bf[J\*fcv_-Nm*W.DyQs5bk=D&]^JW&bf[iQVWG=D&WNPOH]^JWNXRaDJNPRaDJQHkHNP]PYHQVILM9QmD&OHQHd~E(ξ + η) = Eξ + Eηb!cWdef0g7iFjü@ "NP]j\*Q£ |Eξ| ≤ E|ξ|¢ "NP]j\*Q. "NP]j\*Q P(a ≤ ξ ≤ b) = 1~P(ξ ≤ η) = 1ξ≥0QnHJIEξ = 0P(A) = EIIω (A)Ö "Fb]^JlnJInnHUem*Nna ≤ Eξ ≤ b a, b ∈ IREξ ≤ EηnHJIÖ@ξ ár=â ãrâ 0IIω (A) = 1n~~~n=NP]j\*Qω∈An=QIIω (A) = 0nHNP]j\*Qω 6∈ A~]^bJlävIGHNj\*NPW]PYHQHNZbOHYHHQHQ¼JD&YHQHN&n"kJIg(x, y) g1 (x) g2 (y)~NP]j\*QnQ\*Qg(x, y) = g1 (x)g2 (y)P(g1 (ξ) ≥ 0) = 1 P(g2 (η) ≥ 0) = 1NP]j\*Qn]^biNP]^JW&bf[JnTJIn¦FHGHQHkHNPREg1 (ξ) Eg2 (η)Eg(ξ, η) = Eg1 (ξ)Eg2 (η)mH\=K"]^biNP]^J.WIW.D&OHQKOHNPIvtVItm*QHRTIQzm*I]^JDJIkHOHI]^biNP]PJWIW.D&OHQHNEg(ξ, η)Q~Eg1 (ξ) Eg2 (η). "NP]j\*QQ]^bJlzOHN^sLD&WQH]PQHRT_-NS]j\=bk=D&dHOH_-NXWNj\*QHkHQHOH_n=JI9" 8F'<^ ¤'!)EÁ!R '<^nÄ!¦Ã;+a$C+ Ö ^nÔ'!q!å)HE'!!=E=E!'<`^ II*1n(A)0HE'U!;V=E-!'*11'+'n '!!7)'!=E\ 7`"'! Ñ>U--=/ $0P# .AξηEξη = EξEηω¾ x%ÄSxTx Ö HCED&dJQ'RaDJNPRaDJQHkHNt]PYHQHNXILM9QmD&OHQK"]j\=bk=D&dHOH_:VyWNj\*QHkHQHOξ2Q nHIFHGHNjm*Nj\*NPOHOH_VWsLDmDLk=DV ££5Q+ £¢]PII&JWN^J]jJ.WNPO=OHIHηQv_:\*Q O=D&dm*NPOH_ÁaCE -Á¦D&]PFHGHNjm*Nj\*NPOHQK`]j\=bk=D&dHOH_:VWNj\*QHkHQHOξ2 ηFHGHQGHN^pNPOHQHQsLDmDLk5 ££cQ5 £¢tÄE\=KSGHN^pNPOHQK5FHI]jJ&D&W\*NPOHOHIdsLDmDLkHQWI]PFHI\*losPb.NPRT]rKZEIGHR¦bt\D&RTQ` Ö £ :Q` Ö Ô¢ j=RTNPNPRom Eξ2 = −1 ·11+ 1 · = 0.22ÄE\=K'JIUPIHn=kJIv_¯O=D&dJQRaDJNPRaDJQHkHNP]PYINXILM9QmD&OHQHN]j\=bk=D&dHOHIdWNj\*QHkHQHOH_ nJGHNPv&b.N^J]KyO=D&dJQyF\*I&JOHI]^JlHGHItm*QHZEZENPGHNPOHHQHGHIW.D&WZbOHYHHQHfG=D&]PFHGHηNjofη (x)√ FHGHQm*Nj\*NPOHQKnFHI\=bk=D&NPRQFHGHQFη (x)fη (x) = 1/(2 x)x ∈ [0, 1] fη (x) = 0=¦D&YHQHRgIvG=DstIR-nx∈/ [0, 1][Eη =Z1011x √ dx = .2 x3¾x%ÄSxTx Ö CED&dJQ¸RaDJNPRaDJQHkHNP]PYIN+IMzQmD&OHQHN]j\=bk=DdHOHIdWNj\*QHkHQHOH_n=Uem*NQ IFHGHNjm*Nj\*NPOHOH_WsLDmDLk=DV ££5Q+ £¢]PII&JWN^J]^JWNtOHOHIHξ2 + ηξ2 ηÁaCE -ÄE\=KzJIUPIHnkJIv_GHN^pQJ.l|^J&bzsLDmDLkbnRTIM9OHIXO=D&dJQ"G=D&]PFHGHNjom*Nj\*NPOHQHN%]^bRTRT_¨]j\=bk=D&dHOH_:V5WNj\*QHkHQHOQ .¤RTN^JILm*N-O=DVILM5m*NPOHQKG=D&]PFHGHNjm*Njoξ2 η\*NPOHQKq]^bRTRT_ÇOHN^sLD&WQH]PQHRT_:Vq]j\=bk=D&dHOH_:VqWNj\*QHkHQHO)RT_ÇFHIUPIWIGHQHRW Iom*IvOHIUPIyGHItmD"sLDLmDkHQqW'Iv&iNPRê]j\=bk=D&NKHW\=KHf[J]K+m*I]^JDJIkHOHIyJGbLm*INPRTYHQHRTQom STs ê<t!ý Ú ¤:«¥z9²¿Ù w Æß.ÞàHv² Î ª« w ®ß&®HÖ £[C ImH\=KzGHN^pNPOHQK"|^JId9sLDmDLkHQ9O=D&Rqm*I]^JDJIkHOHIWI]PFHI\*l&stIW.DJl]rK9]PWIdH]^JWIRRaDJNPRaDJQHkHNP]PYIUPI9ILM9QmD&OHQK=RTNPNPRE(ξ2 + η) = Eξ2 + Eη = 1/3¾x%ÄSxax Ö @ EDM5m*_-d]^J&bLm*NPOJXUPGbFHFH_n.]PI]^JILKiNPdzQskHNj\*IWNPYnFHNPGHNjm24IkHNPGHNjm*OH_-R ]PNPRTQHO=D&GHIR FHIJNPIGHQHQ5WNPGHILKJOHI]^JNPdzOHN^sLD&WQH]PQHRTIEI&J]PWIQVYI\H\*NPUm*Nj\D&N^J%m*IRaDpOHNPNsLDLmD&OHQHN]W&NPGHIKJOHI]^JlfnLNP]j\*QS]^JIQJ[V.IGHI&pDKXFHIUPItmDnQ0.4]TWNPGHILKJOHI]^JlfntNP]j\*QXQm*N^J-m*ILM5m*l-Q\*QE]POHNPUç:YI\*lYI%W-]PGHNjm*OHNPR+]^J&btm*NPOHJIWUPI&JIW_¸Y9IkHNPGHNjm*0.8OHIR¦b5]PNPRTQHO=D&GbnNP]j\*Q"WX]PNPRTNP]^JGHNOHNPO=D&]^JOH_:Vzm*OHNPd9WX]PGHNjm*OHNPRWm*W.DG=DsLD5vI\*l&pN&n=kHNPR¨FHIUPILM9QV=wÁaCE -H¤vI&stO=DLkHQHR`kHNPGHN^s YI\*QHkHNP]^JWI]^J&bLm*NPOJ.IWHnUPI&JIW_:V9Y5sLD&OKoJQKHR-nkHNPGHN^sQ]PIv_JQKnsLD&Y\*fcξ k=D&f[iQHNP]KqWzJIR-nkJI'FHNPGHNjm ]PNPRTQHO=D&GHIRA B]^JILK\D)V.IGHI&pDK¨FHIUPItmDqQgOHNPO=D&]^JO=DK¸FHIUPItmDq]PII&JWN^J]^J.WNPOHOHIH:Iqb]j\*IWQHfsLDmDLkHQ¦I]PYI\*lYb]^J&bLm*NPOJ._}UPI&JIW&KJ]rK§Y`sLD&OKJQKHRP(B) = 2P(A) = 2/3OHN^sLD&WQH]PQHRTI'm*GbU5I&J'm*GbUjDnRTIM9OHI']PkHQJDJlHnkJIuRT_¿QHRTNPNPRm*Nj\*Iu]PIy]rV.NPRTId :NPGHObt\H\*QnWuYI&JIGHIdqºb]PFHNjV.IRzKHW\=KHN^J]rK`]PIv_JQHN ]^J&btm*NPOHJ)v&btm*N^J)UPI&JIWYnnvQHOH{IRTQ=D\*lObf¥]j\=bk=D&dHObf]PNPRTQHO=D&Gb ¤vI&stO=DLkHQHRkHNPGHN^sn = 1, .
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