exercise:C39474ce8e: 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> Consider a random walker who moves on the integers 0, 1, \ldots, <math>N</math>, moving one step to the right with probability <math>p</math> and one step to the left with probability <math>q = 1 - p</math>. If the walker ever reaches 0 or <math...") |
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\newcommand{\mathds}{\mathbb}</math></div> Consider a random walker who moves on the integers 0, 1, | \newcommand{\mathds}{\mathbb}</math></div> Consider a random walker who moves on the integers 0, 1, ..., <math>N</math>, moving one step to the right with probability <math>p</math> and one step to the left with probability <math>q = 1 - p</math>. If the walker ever reaches 0 or <math>N</math> he stays there. (This is the Gambler's Ruin problem of [[exercise:8ae7bbfa06 |Exercise]].) | ||
the | |||
left with probability <math>q = 1 - p</math>. If the walker ever reaches 0 or <math>N</math> he | |||
stays | |||
there. (This is the Gambler's Ruin problem of | |||
If <math>p = q</math> show that the function | If <math>p = q</math> show that the function | ||
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f(i) = i | f(i) = i | ||
</math> | </math> | ||
is a harmonic function (see | |||
is a harmonic function (see [[exercise:Bc093aec03 |Exercise]]), and if <math>p \ne q</math> then | |||
<math display="block"> | <math display="block"> | ||
f(i) = \biggl(\frac {q}{p}\biggr)^i | f(i) = \biggl(\frac {q}{p}\biggr)^i | ||
</math> | </math> | ||
is a harmonic function. Use this and the result of | |||
to show that the probability <math>b_{iN}</math> of being absorbed in state <math>N</math> starting | is a harmonic function. Use this and the result of [[exercise:Bc093aec03 |Exercise]] to show that the probability <math>b_{iN}</math> of being absorbed in state <math>N</math> starting in state <math>i</math> is | ||
in | |||
state <math>i</math> is | |||
<math display="block"> | <math display="block"> | ||
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\mbox{if}\,\,p \ne q.\cr}\right. | \mbox{if}\,\,p \ne q.\cr}\right. | ||
</math> | </math> | ||
For an alternative derivation of these results see | |||
For an alternative derivation of these results see [[exercise:087f70d94a |Exercise]]. | |||
Latest revision as of 00:53, 16 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]
Consider a random walker who moves on the integers 0, 1, ..., [math]N[/math], moving one step to the right with probability [math]p[/math] and one step to the left with probability [math]q = 1 - p[/math]. If the walker ever reaches 0 or [math]N[/math] he stays there. (This is the Gambler's Ruin problem of Exercise.)
If [math]p = q[/math] show that the function
[[math]]
f(i) = i
[[/math]]
is a harmonic function (see Exercise), and if [math]p \ne q[/math] then
[[math]]
f(i) = \biggl(\frac {q}{p}\biggr)^i
[[/math]]
is a harmonic function. Use this and the result of Exercise to show that the probability [math]b_{iN}[/math] of being absorbed in state [math]N[/math] starting in state [math]i[/math] is
[[math]]
b_{iN} = \left \{ \matrix{
\frac iN, &\mbox{if}\,\, p = q, \cr
\frac{({q \over p})^i - 1}{({q \over p})^{N} - 1}, &
\mbox{if}\,\,p \ne q.\cr}\right.
[[/math]]
For an alternative derivation of these results see Exercise.