А.А. Кудрявцев - Пособие по теории вероятностей (1115312), страница 10
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=1−1FHGHQFHGHQ u ≥ v;u < v.CED&dJQD. 5ZbOHYHHQHfÅG=D]PFHGHNjm*Nj\*NtOHQKQ¸F\*I&JOHI]^Jl§]j\=bk=D&dHOHIdWNj\*QHkHQHOH_~v %ZbOHYHHQHf;G=D&]PFHGHNjm*Nj\*NPO=QKqQ)k=D&]^JOH_-NstO=DLkHNPOHQKqWNPGHIKJOHI]^JQ+mH\=K)]j\=bk=D&ξd1oOHIdyWNj\*QHkHQHOH_ξ2ÁaCE -b]^JlnJIUrmDom b]^JlT]j\*Qx>1P(ξ1 < x) = P{(u, v)| u < x, (u, v) ∈ Ω} = P(Ω) = 1.x≤0nJIUrmDP(ξ1 < x) = P{(u, v)| u < x, (u, v) ∈ Ω} = P(∅) = 0.0<x≤1nHJIz]PIv_JQHNA(x) = {(u, v)| u < x, (u, v) ∈ Ω}IvG=DsPbN^J"FHG.KHRTI&bUPI\*lOHQHY]PIz]^JIGHIO=D&RTQn*G=D&WOH_-RTQ Q]PR-=GHQH]^bOHIY)I5IFHGHNjm*Nj\*NPOHQHf¥U^NPIRTN^JGHQ=kHNP]PYIdWNPGHILKJOHI]^JQ+QHRTNP1NPR xP(ξ1 < x) =mes A(x)= x.mes Ω[m[ jb!cWde(fhgi]𢾠D&RTN^JQHR-n:kJI§FHI\=bkHNPOHO=DKZbOHYHHQKG=D&]PFHGHNjm*Nj\*NPO=QK]PIWF=DmD&N^J]+ZbOHYHHQoNPdG=D&]PFHGHNjm*Nj\*NPOHQKêG=D&WOHIRTNPGHOHIUPI`sLD&YIO=D`]+F=D&G=D&RTN^JG=D&RTQQ]PR-GHQH]^b.oOHIY" t£ j.:I]PFHI\*l&stIW.D&W&pQH]PlXIFHGHNjm*Nj\*NPOHQHNPRg Q\*QzFHGHILm*QH0ZEZEN^1GHNPOHHQHGHIW.D&WbM5NFHI\=bkHNPOHObfZbOHYHHQHfG=D&]PFHGHNjm*Nj\*NtOHQKn.FHI\=bk=D&NPRqW_-G=DM5NPOHQHNmH\=KF\*I&JohOHI]^JQξ1FHGHQ1FHGHQ x ∈ [0, 1];f (x) =[x∈/ [0, 1].0xcO=D\*IUPQHkHOHI¸FHI\=bk=D&NPR¼ZbOHYHHQHf\*NPOHQK]j\=bk=D&dHOHI&dWNj\*QHkHQHOH_hv10G=D&]PFHGHNjm*Njoξ2Ïd»¼±ºä¼¾ ¤¤x1u0P(ξ2 < x) = 1/21FHI]PYI\*lYbFHGHQFHGHQ x ≤ −1;FHGHQ − 1 < x ≤ 1;x > 1,{ξ2 < x} = {ξ2 = −1} = {(u, v)|FHGHQ]PR GHQH]^b.ou < v, (u, v) ∈ Ω}−1 < x ≤ 1[m[OHIY9 j&CED&FHIRTOHQHR-nkJIEZbOHYHHQK5G=D&]PFHGHNjm*Nj\*NjoHO QKSm*QH]PYHGHN^JOHIUPIsLD&YIO=DcQHRTNPN^JE]PYDkHYHQWcJIkHYDVn&]PII&JWN^J]^J.W&bf[iQV5k=D&]^JOH_-RstO=DkHNPOHQKHR¸]j\=bk=D&dHOHIdWNj\*QHkHQHOH_nD5WNj\*QHkHQHOH_]PYDLkHYIW9G=D&WOKHf[J]rKk=D&]^JOH_-RstO=DkHNPOHQKHR1WNPGHILKJOHI]^JQnI&JYbLmDuFHI\=bk=D&NPR-nkJI[ÁD&]PFHGHNjm*Nj\*NPOHQ=N]j\=bk=D&dHOHId¯WNj\*QHkHQHOH_O=DP(ξst_-W.2 D&=N^J]−1)K = P(ξ2 = 1) =1/2ξ2À]j\=bk=D&N&nYIUrmD+m*QH]PYHGHN^JO*DK¸]j\=bk=D&dHO=DK¨WNj\*QHkHQHO=D FHGHQHOHQHRaD&N^J 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