# Question: Suppose that Y is a normal random variable with mean

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?

(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?

**View Solution:**## Answer to relevant Questions

In the text, we noted that when the Xi are all nonnegative random variables. Since an integral is a limit of sums, one might expect that whenever X(t), 0 ≤ t < ∞, are all nonnegative random variables; and this result is ...A set of 1000 cards numbered 1 through 1000 is randomly distributed among 1000 people with each receiving one card. Compute the expected number of cards that are given to people whose age matches the number on the card. For a group of 100 people, compute (a) The expected number of days of the year that are birthdays of exactly 3 people: (b) The expected number of distinct birthdays. Gambles are independent, and each one results in the player being equally likely to win or lose 1 unit. Let W denote the net winnings of a gambler whose strategy is to stop gambling immediately after his first win. Find (a) ...The random variables X and Y have a joint density function given by Compute Cov(X, Y).Post your question