The joint density for (X, Y) is given by f_{XY}(x,y) = frac{x^3 y^3}{16}; 0 ? x ? 2, 0 ? y ? 2. (a) Find the marginal densities of X and Y. (10 pts) (b) Are X and Y independent ? (5 pts) (c) Find P[X ? 1, Y ? 1]. (10 pts) (d) Find E[3X + 2Y]. (5 pts)
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Marginal densities of X and Y: To find the marginal density of X, we integrate the joint density over all possible values of Y: fx(x) = ∫0^2 fxy(x,y) dy = ∫0^2 1/13 dx = 1/13 * 2 = 2/13 Similarly, to find the marginal density of Y, we integrate the joint density Show more…
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