Suppose that Y is a normal random variable with mean and variance 2, and suppose also that the conditional distribution of X, given that

Suppose that Y is a normal random variable with mean μ and variance σ2, and suppose also that the conditional distribution of X, given that Y = y, is normal with mean y and variance 1.
(a) Argue that the joint distribution of X, Y is the same as that of Y + Z, Y when Z is a standard normal random variable that is independent of Y.
(b) Use the result of part (a) to argue that X, Y has a bivariate normal distribution.
(c) Find E[X], Var(X), and Corr(X, Y).
(d) Find E[Y|X = x].
(e) What is the conditional distribution of Y given that X = x?