[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

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$'s are strictly increasing, but the proof is

more difficult.