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2. (Functions of random variables. Independence) a) Let Y1 and Y; be two non negative continuous random variables. Suppose their joint pdf faotorizee as firsts) = minimise}, for some functions 31 and 92. Prove that Y1 and Y: are independent random variables. b) Let X1 and X2 be two independent exponential random variables Elm} Consider the following transformations: i} Y1 = X1 + X2, Y2 = lexe; ii) Y1 = X1 + X2, Y2 =X1. Find the joint pdf f(y1?y2] of the transformed random variables Y1 and Y2 in both cases i) and ii). Are Y1 and Y2 independent? c) Find the marginal pdf'e of Y1 and Y2, for both transformations i} and ii)