(30 points) Suppose that the support S of a continuous random vector (X1, X2) is given...

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(30 points) Suppose that the support S of a continuous random vector (X1, X2) is given by S = {(x1, x2): 0 < x < 1, 0 x 1}. Suppose also that the joint cdf, F(x1, x2), of (X1, X2) satisfies F(x1, x2) = x1x2 for (x1, x2) S, where c is a constant. Let Y = X + X2 and Y2 = X2. (a) Describe the joint pdf, f(x1, x2), of X1 and X2. (b) Determine P(X2 X1). (c) Calculate E(X1), E(X1X2), and E(X + 3X1X2). (d) Describe the joint pdf, fy,Y2 (91, 92), of Y and Y2. (e) Describe the pdf, fy, (y1), of Y. (30 points) Suppose that the support S of a continuous random vector (X1, X2) is given by S = {(x1, x2): 0 < x < 1, 0 x 1}. Suppose also that the joint cdf, F(x1, x2), of (X1, X2) satisfies F(x1, x2) = x1x2 for (x1, x2) S, where c is a constant. Let Y = X + X2 and Y2 = X2. (a) Describe the joint pdf, f(x1, x2), of X1 and X2. (b) Determine P(X2 X1). (c) Calculate E(X1), E(X1X2), and E(X + 3X1X2). (d) Describe the joint pdf, fy,Y2 (91, 92), of Y and Y2. (e) Describe the pdf, fy, (y1), of Y.