exercise:C2bcd1e04d: 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> Toss a fair die repeatedly. Let <math>S_n</math> denote the total of the outcomes through the <math>n</math>th toss. Show that there is a limiting value for the proportion of the first <math>n</math> values of <math>S_n</math> that are divisible...") |
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Toss a fair die repeatedly. Let <math>S_n</math> denote the total of the outcomes through the <math>n</math>th toss. Show that there is a limiting value for the proportion of the first <math>n</math> values of <math>S_n</math> that are divisible by 7, and compute the value for this limit. '' Hint'': The desired limit is an equilibrium probability vector for an appropriate seven state Markov chain. | |||
of | |||
the outcomes through the <math>n</math>th toss. Show that there is a limiting value for | |||
the | |||
proportion of the first <math>n</math> values of <math>S_n</math> that are divisible by 7, and | |||
compute | |||
the value for this limit. '' Hint'': The desired limit is an equilibrium | |||
probability vector for an appropriate seven state Markov chain. | |||
Latest revision as of 21:56, 17 June 2024
Toss a fair die repeatedly. Let [math]S_n[/math] denote the total of the outcomes through the [math]n[/math]th toss. Show that there is a limiting value for the proportion of the first [math]n[/math] values of [math]S_n[/math] that are divisible by 7, and compute the value for this limit. Hint: The desired limit is an equilibrium probability vector for an appropriate seven state Markov chain.