Let Y be a discrete r.v., A be an event with 0 < P(A) < 1, and IA be the indicator r.v
Chapter 9, Problem 20(choose chapter or problem)
Let Y be a discrete r.v., A be an event with 0 < P(A) < 1, and IA be the indicator r.v. for A. (a) Explain precisely how the r.v. E(Y |IA) relates to the numbers E(Y |A) and E(Y |Ac). (b) Show that E(Y |A) = E(Y IA)/P(A), directly from the definitions of expectation and conditional expectation. Hint: First find the PMF of Y IA in terms of P(A) and the conditional PMF of Y given A. (c) Use (b) to give a short proof of the fact that E(Y ) = E(Y |A)P(A)+E(Y |Ac)P(Ac).
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