2. The continuous random variables x and y have known joint probability density function f(x,y) given by x Osysxsl fxy(x,y) = y +1 0 otherwise where is a positive constant. a) Determine the value of the constant c that makes the above a valid joint probability density function. b) Using the result of part a), obtain now the marginal probability density function fx(x) of the random variable x. c) Using the result of part a), obtain now the marginal probability density function f(x) of the random variable y. d) Verify the results of parts b) and c) to make sure that the two probability density functions are valid. e) Are the two random variables statistically independent? f) Obtain the expected value E(XY).

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.]