[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]p[/math] be the probability distribution

[[math]] p = \pmatrix{ 0 & 1 & 2 \cr 0 & 1/3 & 2/3 \cr}\ , [[/math]]

and let [math]p_n = p * p * \cdots * p[/math] be the [math]n[/math]-fold convolution of [math]p[/math] with itself.

• Find [math]p_2[/math] by direct calculation (see Definition).
• Find the ordinary generating functions [math]h(z)[/math] and [math]h_2(z)[/math] for [math]p[/math] and [math]p_2[/math], and verify that [math]h_2(z) = (h(z))^2[/math].
• Find [math]h_n(z)[/math] from [math]h(z)[/math].
• Find the first two moments, and hence the mean and variance, of [math]p_n[/math] from [math]h_n(z)[/math]. Verify that the mean of [math]p_n[/math] is [math]n[/math] times the mean of [math]p[/math].
• Find those integers [math]j[/math] for which [math]p_n(j) \gt 0[/math] from [math]h_n(z)[/math].