Question

Problem 1: Given the function g(x,y)-ke-xy-c)u(x-a)u(y-b) find the constant "k" in terms ofa, b, and c so that g(x,y) is a valid probability density function (15%). Are the random variables X and Y statistically independent (10%)? (Support your answer.)

Answer 1

PDF g(x, y) = ke^-(x + y + C) u(x - a) u(y - b) P(x, y) = Int g(x, y) dx dy P(x, y) = 1 Forall x, y elementof $ where $ elementof {x > a, y > b} integral _b^infinity integral _a^infinity ke^-(x + y + C) dx dy = 1 ke^-a e^-b e^-c = 1 k = 1/e^-(a + b + c) = e^(a + b + c) g(x, y) = e^a - x e^b - y P_x (x) = integral _b^infinity g(x, y) dy P_y (y) = integral _a^infinity g(x, y) dx P_x (x) = e^a - x P_y (y) = e^b - y g(x, y) = P_x (x) times P_y (y) = > x & y are independent variables.