Question

1. Suppose that the joint density of X and Y is given by exp(-y) (1- exp(-x)), if 0 S y,0 syS oo exp(-x) (1- exp(-y)), if 0SyS ,0 oo (e,y)exp(-y) Then . The marginal density of X (and also that of Y), ·The conditional density of Y given X = x and vice versa, Cov(X, Y) . Are X and Y independent? Explain with proper justification.

Answer 1

Suppose that the joint of x and y is given by f(x, y) = {ext(-y) (1 - exp(-x)) ext(-x0 (1 - exp(-x)) if 0 lessthanorequalto x lessthanorequalto y, 0 lessthanorequalto y < infinity if 0 lessthanorequalto y lessthanorequalto x, 0 lessthanorequalto x lessthanorequalto infinity Marginal density of x (also that of y) f_1 (x) = integral^infinity_0 f(x, y) dy = [integral^x_0 f(x, y) dy + integral^infinity_x f(x, y) dy] = integral^x_0 e^-x (1 - e^-y) dy + integral^infinity_x e^-y (1 - e^-x) dy = e^-x [y + e^-y]^x_0 - (1 - e^-x) (e^-y)^infinity_x = e^-x [y + e^-x - 1] + (1 - e^-x) e^-x = xe^-x + e^-2x - e^-x + e^-x - e^-2x = xe^-x f_2 (y) = integral^infinity_0 f(x, y) dx = integral^y_0 e^-y (1 - e^-x) dx + integral^infinity_y e^-x (1 - e^-y) dx = e^-y [x + e^-x]^y_0 + (1 - e^-y) [e^-y] = e^-y [y + e^-y - 1]