MATH20701 and place signat PDF File SECTION A Answer ALL of the four questions. A1. Suppose that X is a discrete random variable such that X ~ Geom(p) where 0 < p < 1, i.e. its probability mass function is given by where q = 1-p. Px (k) =pq-1 for k = 1, 2,... (px (x) = 0 otherwise), (a) Prove that the probability generating function of X is given by Px(t) = pt/(1-qt) for |t| <1/q. (b) Hence, or otherwise, find the mean and standard deviation of X. [10 marks] A2. Let (X, Y) be a continuous bivariate random variable with joint probability density function fxx(x,y) = {hry for 0<<r<l, 0 otherwise, where kЄ R. (a) Find the number k. (b) Find P(X+Y < 1). (c) Find P(X > 1/2 | X+Y < 1). [10 marks]