STAT 333 - Conditional Probability and Conditional Expectation - Part 1.pdf

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2Conditional Probability and Conditional Expecta-tion2.1DefinitionsJointly Discrete Case:IfX1andX2are both discrete random variables with jointpmfp(x1, x2) and marginal pmfsp1(x1) andp2(x2), respectively, then the conditionaldistribution ofX1givenX2=x2, denoted byX1|(X2=x2), is defined via itscondi-tional pmfp1|2(x1|x2) =P(X1=x1|X2=x2) =P(X1=x1, X2=x2)P(X2=x2)=p(x1, x2)p2(x2),provided thatp2(x2)>0. Similarly, the conditional distribution ofX2|(X1=x1) isdefined via its conditional pmfp2|1(x2|x1) =P(X2=x2|X1=x1) =p(x1, x2)p1(x1),provided thatp1(x1)>0.Remarks:
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