Let X and Y be independent Exponential(A) random variables. Let M = min{X,Y} min{X,Y} and U = max{X, Y}. 1. Compute the distribution function and probability density of M. 2. Compute the distribution function and probability density of U. 3. Compute the joint distribution function F(M,U) (x, y) of M and U, and thus deduce the joint density fM,U (x, y).

7. Let X and Y denote two continuous random variables. Let f(x,y) denote the joint probability density function and fx(x) and fy (y) the marginal probability density functions for X and Y, respectively. Finally let Z = aX + bY, where a and b are non-zero real numbers. (e) Derive an expression for Cov(Z) as a function of Var (X), Var(Y) and Cov(X,Y). [You may use standard results relating to variance and covariance without proof, but these should be clearly stated.]

Suppose that X and Y are statistically independent and identically distributed uniform random variables on (0,1). (a) Write down the joint probability density function fx,y(x,y) of X and Y on its support.

3. Let X and Y be two independent random variables with identical probability density function given by e- for x > 0 f(x) = elsewhere. What is the probability density function of W= max{X, Y } ?