For an event A, let I_{A} equal 1 if A occurs and let it equal 0 if A does not occur. For a random variable X, show that E[X|A] = E[XI_{A}]/P(A)
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this is trivial try and do it yourself.
is it correct to do in this way? E(XIA)/P(A) =[E(XIA|IA=0)P(IA=0)+E(XIA|IA=1)P(IA=1)]/P(A) =[E(X|A)P(A)]/P(A) =E(X|A)
Yes, that is correct. You have used the formula which is known as "total probability" or "the law of iterated expectations".