Suppose that X1, X2, . . . ,X5 are five random variables for which the joint p.d.f. can be factored in the following form for all points (X1, X2, . . . , X5) ∈ R5: f (x1, x2, . . . , x5) = g(x1, x2)h(x3, x4, x5), where g and h are certain nonnegative functions. Show that if Y1 = r1 (X1, X2) and Y2 = r2 (X3, X4, X5), then the random variables Y1 and Y2 are independent.