Let [math]X_1[/math] and [math]X_2[/math] be independent random variables and for [math]i = 1, 2[/math], let [math]Y_i = \phi_i(X_i)[/math], where [math]\phi_i[/math] is strictly increasing on the range of [math]X_i[/math]. Show that [math]Y_1[/math] and [math]Y_2[/math] are independent. Note that the same result is true without the assumption that the [math]\phi_i[/math]'s are strictly increasing, but the proof is more difficult.