(a) The probability density function of the random variable X is f(x) = 2e-2x for x> 0 Use the transformation Y = 2X to find the probability density function for the random variable Y (7 marks) (b) If a and b are constants and X is a random variable, prove that for moment generating function of a linear expression, Max+bj (t) = e"M x (at) (5 marks) (c) If Z ~ N(0, 1) has a moment generating function given by M, (t) -ez Show that M (1) -e 27 where X ~ N(u, o? ) and Z _ (8 marks)