Please answer the following question:

(a) Let X be a random variable with probability mass function fx(:c) = 2q'\" for a: = 2,3,... where q is a constant. i. Show that q = 1/2. [2 marks] ii. Find the cumulative distribution function of X. [2 marks] iii. Work out the cumulant-generating function of X and, hence, compute IE(X). [6 marks] (b) Suppose that X is normally distributed with mean p = 0 and variance 02 = 2, and define Y = X2. i. Show that the moment-generating function of Y is given by Mt) = (1 40-1/2, and determine the values of t for which this function is well defined. [5 marks] ii. Use the Markov inequality to show that, for all a > 0, P(Y Z a) S Vie\"g . [Hint: Notice that My(1/8) = ] [5 marks]