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_________________________________________________________________________________ §Transformation of Two Random Variables • Theorem: Let X and Y be continuous random variables with joint probability density function f(x,y). Let h1 and h2 be two real-valued functions of two variables, U = h1 (X,Y) and V = h2 (X,Y) to a set Q in the uv plane. If (a) the system of two equations of two unknowns h1(x,y) = u, h2(x,y) = v. has a unique solution for x and y in terms of u and v, i.e., x = w1(u,v) and y = w2(u,v) and (b) the function w1 and w2 have continuous partial derivatives, and the Jacobian of the transformation x = w1(u,v) and y = w2(u,v) is nonzero at all points (u,v); that is, the following 2 x 2 determinant is everywhere nonzero: J = |w1/u w2/v| |w1/u w2/v| = w1/u w2/v - w1/v w2/u 0, for all (u,v) Q, then the random variables U and V are jointly continuous with the joint probability density function g(u,v) given by g(u,v) = f(w1(u,v), w2(u,v)) |J| =0 (u,v) Q, otherwise. Proof: This theorem is the result of the change of variables theorem in double integrals. __________________________________________________________ © Shi-Chung Chang, Tzi-Dar Chiueh ___ _________________________________________________________________________________ * Example: Box-Muller Theorem (Generation of Normal r.v.) Let X and Y be two independent uniform random variables over (0,1); show that the random variables U = cos(2X)(-2lnY)0.5 and V = sin(2X)(-2lnY)0.5 are independent standard normal random variables. Solution: We have u2 + v2 = -2lny, y = w2 = e -(u2+v2)/2. cos(2x) = u/(u2+v2)0.5, sin(2x) = v/(u2+v2) 0.5 x = w1 = (1/2cos-1(u/(u2+v2) 0.5) . The first condition is satisfied. Next the Jacobian is w1/u w2/v - w1/v w2/u = {(-v)/[2(u2+v2)]}{ (-v) e -(u2+v2)/2} {u/[2(u2+v2)]}{ (-u) e -(u2+v2)/2} = (1/2) e -(u2+v2)/2 0. We have the joint pdf of U and V is given by g(u,v) = f (x,y)|J| = f (x,y) (1/2) e -(u2+v2)/2 = (1/2)e -(u2+v2)/2. Integrating over v and u, respectively, one obtains gU(u) = (1/2)0.5e -u2/2 and gV(v) = (1/2)0.5e -v2/2 . So U and V are independent standard normal random variables. __________________________________________________________ © Shi-Chung Chang, Tzi-Dar Chiueh ___