[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]\Omega[/math] be the sample space

[[math]] \Omega = \{0,1,2,\dots\}\ , [[/math]]

and define a distribution function by

[[math]] m(j) = (1 - r)^j r\ , [[/math]]

for some fixed [math]r[/math], [math]0 \lt r \lt 1[/math], and for [math]j = 0, 1, 2, \ldots[/math]. Show that this is a distribution function for [math]\Omega[/math].