exercise:F1914d2351: 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 walk on a circle of circumference <math>n</math>. The walker takes one unit step clockwise with probability <math>p</math> and one unit counterclockwise with probability <math>q = 1 - p</math>. Modify the program ''' Ergodic...") |
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<math>n</math> and <math>p</math> and compute the basic quantities for this chain. | <math>n</math> and <math>p</math> and compute the basic quantities for this chain. | ||
<ul><li> For which values of <math>n</math> is this chain regular? ergodic? | <ul style="list-style-type:lower-alpha"><li> For which values of <math>n</math> is this chain regular? ergodic? | ||
</li> | </li> | ||
<li> What is the limiting vector <math>\mat{w}</math>? | <li> What is the limiting vector <math>\mat{w}</math>? | ||
Latest revision as of 01:25, 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]
Consider a random walk on a circle of circumference [math]n[/math].
The walker takes one unit step clockwise with probability [math]p[/math] and one unit counterclockwise with probability [math]q = 1 - p[/math]. Modify the program ErgodicChain to allow you to input [math]n[/math] and [math]p[/math] and compute the basic quantities for this chain.
- For which values of [math]n[/math] is this chain regular? ergodic?
- What is the limiting vector [math]\mat{w}[/math]?
- Find the mean first passage matrix for [math]n = 5[/math] and [math]p = .5[/math]. Verify that [math]m_{ij} = d(n - d)[/math], where [math]d[/math] is the clockwise distance from [math]i[/math] to [math]j[/math].