2. The joint probability density function of the random variables X and Y is if 0 < x < 4, 1 < y< 5, otherwise fx, y (x, y) = сху 0 with c = 1/96. (a) Sketch the region in the (x, y)-plane where fx,y (x, y) > 0. Next, show that the constant c has to have the value 1/96. (b) Compute P(X ≥ 3, Y ≤ 2). (c) Find: i) fx, the marginal density for X. ii) fy, the marginal density for Y. (d) Are X and Y are independent? Motivate your answer, then compute the conditional density of X given Y = y and use this to compute the conditional probability P(X ≥ 2 | Y = 3). (e) Find P(X+Y < 4). [Total2

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