Q1: Probability 1. Let X be a continuous random variable whose PDF is given by %+ %$ for a: E [3,0) f(a:)= w forwe[0,1] 0 otherwise. (a) Derive the GDP of X and draw the graph of it. (b) Calculate the probability Pr(1 g X). (c) Calculate the expected value of X. 2. Let D be a binary random variable with Pr(D = 1) = 0.8 and Pr(D = 1) = 0.2, and X and Y be normally distributed as N (0, 3) and N (0, 5), respectively. Further, let Z = DX +(1 D)Y. X, Y, and D are independent. (a) Calculate the expected value of (X Y)2. (b) Calculate the expected value of Z. (c) Calculate the variance of Z. (Hint: D2 = 1.)