Suppose that Y is a normal random variable with mean µ and variance σ2, and suppose also t...

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.

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(b) Use the result of part (a) to argue that X, Y has a bivariate normal distribution.

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(c) Find E[X],Var(X), and Corr(X, Y).

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(d) Find .

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(e) What is the conditional distribution of Y given that X = x?