[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]

For the gambler's ruin problem, assume that the

gambler starts with [math]k[/math] dollars. Let [math]T_k[/math] be the time to reach 0 for the first time.

• Show that the generating function [math]h_k(t)[/math] for [math]T_k[/math] is the [math]k[/math]th power of the generating function for the time [math]T[/math] to ruin starting at 1. Hint: Let [math]T_k = U_1 + U_2 +\cdots+ U_k[/math], where [math]U_j[/math] is the time for the walk starting at [math]j[/math] to reach [math]j - 1[/math] for the first time.
• Find [math]h_k(1)[/math] and [math]h_k'(1)[/math] and interpret your results.