[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[/math] and [math]Y[/math] be two random variables defined on the

finite sample space [math]\Omega[/math]. Assume that [math]X[/math], [math]Y[/math], [math]X + Y[/math], and [math]X - Y[/math] all have the same distribution. Prove that [math]P(X = Y = 0) = 1[/math].