exercise:Cf2779f01a: 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> An ergodic Markov chain is started in equilibrium (i.e., with initial probability vector <math>\mat{w}</math>). The mean time until the next occurrence of state <math>s_i</math> is <math>\bar{m_i} = \sum_k w_k m_{ki} + w_i r_i</math>. Show that...") |
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\newcommand{\mathds}{\mathbb}</math></div> An ergodic Markov chain is started in equilibrium | \newcommand{\mathds}{\mathbb}</math></div> An ergodic Markov chain is started in equilibrium (i.e., with initial probability vector <math>\mat{w}</math>). The mean time until the next occurrence of state <math>s_i</math> is <math>\bar{m_i} = \sum_k w_k m_{ki} + w_i r_i</math>. Show that <math>\bar {m_i} = z_{ii}/w_i</math>, by using the facts that <math>\mat {w}\mat {Z}= \mat {w}</math> and <math>m_{ki} = (z_{ii} - z_{ki})/w_i</math>. | ||
(i.e., with initial probability vector <math>\mat{w}</math>). The mean time until the | |||
next | |||
occurrence of state <math>s_i</math> is <math>\bar{m_i} = \sum_k w_k m_{ki} + w_i r_i</math>. | |||
Show that <math>\bar {m_i} = z_{ii}/w_i</math>, by using the facts that <math>\mat {w}\mat {Z} | |||
= | |||
\mat {w}</math> and <math>m_{ki} = (z_{ii} - z_{ki})/w_i</math>. | |||
Latest revision as of 01:28, 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]
An ergodic Markov chain is started in equilibrium (i.e., with initial probability vector [math]\mat{w}[/math]). The mean time until the next occurrence of state [math]s_i[/math] is [math]\bar{m_i} = \sum_k w_k m_{ki} + w_i r_i[/math]. Show that [math]\bar {m_i} = z_{ii}/w_i[/math], by using the facts that [math]\mat {w}\mat {Z}= \mat {w}[/math] and [math]m_{ki} = (z_{ii} - z_{ki})/w_i[/math].