Mathematics

1. Suppose X and Y are independent normal random variables, each with mean 0 and variances 9 and 16, respectively. Calculate the joint density of (X,X − Y ) and then the density of X − Y .

2. A point is generated on a unit disk in the following way: The radius, R, is uniform on (0, 1), and the angle Θ is uniform on (0, 2π) and is independent of R.

a. Find the joint density of X = R cos Θ and Y = R sin Θ.

b. Find the marginal densities of X and Y .

3. Suppose X and Y have joint density f(x, y) = λ2e−y, 0 < x < y <∞, and equal to 0 for all other (x, y).

(a) Calculate the joint density of U = Y −X, V = X. (b) Are U and V independent?

4. Show that f(x, y) = 2x+ 2y − 4xy, 0 < x, y < 1 and g(x, y) = 2 − 2x − 2y + 4xy, 0 < x, y < 1 are both densities. Then find

the marginal densities of X and Y in each case.

5. If X1 is uniform on (0, 1) and, conditionally given X!, X2 is uniform on (0, X1), find the joint and marginal distributions of X1 and X2.

6. Let P have a uniform distribution on (0, 1) and, conditional on P = p, let X have a Bernoulli distribution with parameter p. Find the conditional distribution of P given X.

7. Suppose that, conditional on N , X has a binomial distribution with N trials and probability p of success, and that N is a binomial random variable with m trials and probability r of success. Find the unconditional distribution of X.