Problem 2 Let (,,P) be a probability space. (1) For measurable events A,B,C E A with P(C)>0 and P(BC) > 0, show that P(AnBnC)=P(A | BnC)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 , i.e. B, nB, Ø for i #j; CnCe = Ø for k #l; and
Problem 2 Let (,,P) be a probability space. (1) For measurable events A,B,C E A with P(C)>0 and P(BC) > 0, show that P(AnBnC)=P(A | BnC)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 , i.e. B, nB, Ø for i #j; CnCe = Ø for k #l; and
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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Transcribed Image Text: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 | 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 of
the sample space , i.e. B; nB;= for i #j; CnCe = fork #l; and
Show that
n m
P(A)=P(A|B₂nC;)P(B, C;)P(C;).
m n
n
m
Q=ÜB₁ =ÜC₁.
i=1
j=1
i=1j=1
if 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 all
k = 1,...,n and j = 1,...,m. Show that for every i = 1,...,.
j=1k=1
P{X = xi} = [[P{X=xi | Y=yj, Z = zh} P {Y=yj | Z = zk} P {Z = zk}.
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