[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]

Suppose that [math]X[/math] and [math]Y[/math] are continuous random variables with

density functions [math]f_X(x)[/math] and [math]f_Y(y)[/math], respectively. Let [math]f(x, y)[/math] denote the joint density function of [math](X, Y)[/math]. Show that

[[math]] \int_{-\infty}^\infty f(x, y)\, dy = f_X(x)\ , [[/math]]

and

[[math]] \int_{-\infty}^\infty f(x, y)\, dx = f_Y(y)\ . [[/math]]