(c) [3 marks] Plot the cumulative distribution function of X. 2. [12 marks] Let X be a continuous random variable X with pdf f (x) = clacle-121 , -00 1. (a) [5 marks] Show that Y has pdf fy(y) = aya-le-ya, y > o . (b) [6 marks] Show that the mof of Y exists (you don't have to derive the mef). 4. [20 marks] Consider two fair four-sided dice, each engraved with 1, 2, 3, 4 on the four sides. Let X1 and X2 denote the respective outcomes of their (independent) rolling, and X be the larger outcome of the two. (a) [4 marks] Find P(X1 + X2 = 5). (b) [4 marks] Find P(X = X1). (c) [4 marks] Find E(X] | X > 2). (d) [4 marks] Find Var(X,|X > 2). (e) [4 marks] Show that the probability function of X is f (a) = 2x - 1 16 x = 1, 2, 3, 4