Problem 4 (15 points). Let X and Y be independent random variables taking values in X {-1,1}. We have P(X = 1) = q while

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Problem 4 15 Points Let X And Y Be Independent Random Variables Taking Values In X 1 1 We Have P X 1 Q While 1 (80.16 KiB) Viewed 78 times

Problem 4 (15 points). Let X and Y be independent random variables taking values in X {-1,1}. We have P(X = 1) = q while Y is uniformly distributed on {-1,1}. Let Z = XY. (a) (5 points) Find the conditional distribution P(Y|Z). Note that conditional probabilities are a function of Z = z. (b) (5 points) The conditional mean of Y given Z = z is E[Y|Z = z) = P(y|z). YEX Find E[Y|Z = z) as a function of z. 3 (c) (5 points) The conditional mean of Y given Z (where the latter is viewed as a random variable) is HY Z E[Y\2] = y P(y\Z). YEX Since HY Z is a function of the random variable Z, it too is a random variable. Compute the probability distribution of HYZ