Problem 2Let (,,P) be a probability space.(1) For measurable events A,B,CE A with P(C) >0 and P(BnC)>0, show thatP(AnBnC)=P(A | BnC)P(B|C)P(C).(2) Let A be a measurable event and let B₁,...,B₁ € A as well as C₁,...,Cm E A be partitions ofthe sample space , i.e. B; nB;= for i #j; CnCe = fork #l; andShow thatn mP(A)=P(A|B₂nC;)P(B, C;)P(C;).m nnmQ=ÜB₁ =ÜC₁.i=1j=1i=1j=1if P(C;) >0 and P(B; nC;) >0 for all i = 1,...,n and j = 1,..., m.(3) Let X, Y, Z be random variables on (n,A,P) with possible values x₁,...,xe ER for X; y₁,...,ym €R for Y; and 2₁,...,Zn ER for Z. Suppose that P(Z = zh) >0 and P(Y = yj, Z = zk) > 0 for allk = 1,...,n and j = 1,...,m. Show that for every i = 1,...,.j=1k=1P{X = xi} = [[P{X=xi | Y=yj, Z = zh} P {Y=yj | Z = zk} P {Z = zk}.

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Problem 2 Let (,,P) be a probability space. (1) For measurable events A,B,CE A with P(C) >0 and P(BnC) > 0, show that P(AnBnC)=P(A | BOC) P(B | C)P(C). (2) Let A be a measurable event and let B₁,...,B, EA as well as C₁,...,Cm E A be partitions of the sample space 2, i.e. B, nB, = Ø for i #j; CnCe = for k‡ l; and Show that n m Q=UB₁=UC₁. j=1

Problem 2 Let (,,P) be a probability space. (1) For measurable events A,B,CE A with P(C) >0 and P(BnC) > 0, show that P(AnBnC)=P(A | BOC) P(B | C)P(C). (2) Let A be a measurable event and let B₁,...,B, EA as well as C₁,...,Cm E A be partitions of the sample space 2, i.e. B, nB, = Ø for i #j; CnCe = for k‡ l; and Show that n m Q=UB₁=UC₁. j=1

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 29E

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