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Problem 2 Let (2,,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 be partitions of the sample space 2, i.e. BnB, =ø for i #j; CnCe = for k #l; and Show that m n=ÜB₁ =ÜC₁. j=1 n m P(A)=P(A|B₂NC,)P(B, C,)P(C₂). i=1j=1 if P(C) >0 and P(B; nCj) >0 for all i = 1,...,n and j = 1,..., m. (3) Let X, Y, Z be random variables on (2, 4, P) with possible values x₁,...,xe € R for X; y1,...,ym € R for Y; and 21,...,Zn ER for Z. Suppose that P(Z = z) >0 and P(Y = yj, Z = zk) > 0 for all k = 1,...,n and j = 1,...,m. Show that for every i = 1,...,l m n P{X = x₁}= [[P{X = x₁ | Y = yj, Z = zk} P {Y=yj | Z = zk} P {Z = zk}. j=1k=1

Problem 2 Let (2,,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 be partitions of the sample space 2, i.e. BnB, =ø for i #j; CnCe = for k #l; and Show that m n=ÜB₁ =ÜC₁. j=1 n m P(A)=P(A|B₂NC,)P(B, C,)P(C₂). i=1j=1 if P(C) >0 and P(B; nCj) >0 for all i = 1,...,n and j = 1,..., m. (3) Let X, Y, Z be random variables on (2, 4, P) with possible values x₁,...,xe € R for X; y1,...,ym € R for Y; and 21,...,Zn ER for Z. Suppose that P(Z = z) >0 and P(Y = yj, Z = zk) > 0 for all k = 1,...,n and j = 1,...,m. Show that for every i = 1,...,l m n P{X = x₁}= [[P{X = x₁ | Y = yj, Z = zk} P {Y=yj | Z = zk} P {Z = zk}. j=1k=1

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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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 | 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}.
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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