Let f(x, y) = 2 be a joint probability density function of two random variables X and Y defined on the region R = {(x, y) : 0 < x < y < 1}. (a) Calculate Z Z R f(x, y) dA (b) Consider the change of variables u = x y and v = y. Sketch the region S in the uv-plane that corresponds to the region R in the xy-plane under this transformation. (c) In one or two sentences, explain why the region S is preferred to the region R. (d) What is the Jacobian of this transformation?