Let [math]X_1[/math] and [math]X_2[/math] be independent random variables and let [math]Y_1 = \phi_1(X_1)[/math] and [math]Y_2 = \phi_2(X_2)[/math].

• Show that
  [[math]] P(Y_1 = r, Y_2 = s) = \sum_{\phi_1(a) = r \atop \phi_2(b) = s} P(X_1 = a, X_2 = b)\ . [[/math]]
• Using (a), show that [math]P(Y_1 = r, Y_2 = s) = P(Y_1 = r)P(Y_2 = s)[/math] so that [math]Y_1[/math] and [math]Y_2[/math] are independent.