exercise:F3789f52e8: Difference between revisions
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(Created page with "<div class="d-none"><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></div> 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. <ul><li> Show that the generating function <math>h_k(t)</math> for <math>T_k</math> is the...") |
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gambler starts with <math>k</math> dollars. Let <math>T_k</math> be the time to reach 0 for the | gambler starts with <math>k</math> dollars. Let <math>T_k</math> be the time to reach 0 for the | ||
first time. | first time. | ||
<ul><li> Show that the generating function <math>h_k(t)</math> for <math>T_k</math> is the <math>k</math>th | <ul style="list-style-type:lower-alpha"><li> 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 | 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 | = 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 | ||
Latest revision as of 00:55, 15 June 2024
[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.