[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]

In the gambler's ruin problem, assume that the gambler initial stake is 1 dollar, and assume that her probability of success on any one game is [math]p[/math]. Let [math]T[/math] be the

number of games until 0 is reached (the gambler is ruined). Show that the generating function for [math]T[/math] is

[[math]] h(z) = \frac{1 - \sqrt{1 - 4pqz^2}}{2pz}\ , [[/math]]

and that

[[math]] h(1) = \left \{ \begin{array}{ll} q/p, & \mbox{if $q \leq p$}, \\ 1, & \mbox{if $q \geq p,$} \end{array} \right. [[/math]]

and

[[math]] h'(1) = \left \{ \begin{array}{ll} 1/(q - p), & \mbox{if $q \gt p$}, \\ \infty, & \mbox{if $q = p.$} \end{array} \right. [[/math]]

Interpret your results in terms of the time [math]T[/math] to reach 0. (See also Example.)