Let X and Y be jointly continuous with joint probability density function f (x, y) and marginal densities fX (x) and fY (y). Suppose that f (x, y) = g(x)h(y) where g(x) is a function of x alone, h(y) is a function of y alone, and both g(x) and h(y) are nonnegative.

a. Show that there exists a positive constant c such that f_X (x) = cg(x) and f_Y (y) = (1/c)h(y).

b. Use part (a) to show that X and Y are independent.