Let X and Y be two random variables. Suppose X|Y ~ Binomial(n, Y ) where n is fixed and Y stands for the probability of success. Suppose the success probability Y is a random variable distributed as a Beta distribution, Y ~ Beta(a, B). The PDF of Beta(a, ß) is defined by the following f(y) = { S Bay*-(1 – y)%-1, 0 0, ß > 0, otherwise, 0, where B(a, B) T(a)I"(3) T(a+B) (1) Find E(X|Y) and V(X|Y). (2) Find E(X). (3) Find Cor(X, Y). Hint: E(X|Y) is a linear function of Y. (4) Find the conditional PDF of Y given X = x.

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