Let X and Y be two random variables with given joint PMF,

QUESTION:

Let X and Y be two random variables with given joint PMF, and let 9 and h be two functions of X and Y, respectively. Show that if X and Y are independent, then the same is true for the random variables g{X) and h{Y). Solution. Let U = g{X) and V = h{Y). Then, we have PU,v{u,v) = L PX,y{x, y) { (x,y) I g(x)=u, h(y)=v} - L PX (x)py{y) { (x,y) I g(x)=u, h(y)=v } - L px{x) L py{y) {x I g(x)=u} {x I h(y)=v} = Pu {u)pv (v), so U and V are independent.