Chapter 7, Problem 7.56TE

Suppose that Y is a normal random variable with mean $\mu$ and variance ${\sigma }^2$, 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 $E\left[X\right],\mathrm{var}\left(X\right)$, and $Corr\left(X,Y\right)$.

d. Find $E\left[Y|X=x\right]$.

e. What is the conditional distribution of Y given that $X=x$?