Question: 4. Let X1, X2, ... be independent identically distributed random variables with mean / and variance o'. Let N be a random variable following a

4. Let X1, X2, ... be independent identically distributed random variables with mean / and variance o'. Let N be a random variable following a poisson distribution with mean A and variance A. Let Y = X1 + X2+ ... + XN. . (a) Show that E(Y | N = n) = nu and var(Y [ N = n) = no? . (b) Let Z = E(Y | N), the conditional expectation of Y given N. Find the mean and variance of Z. . (c) Show that E(Y) = M . (d) Find var(Y)

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