exercise:195b345783: 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> A ''lead change'' in a random walk occurs at time <math>2k</math> if <math>S_{2k-1}</math> and <math>S_{2k+1}</math> are of opposite sign. <ul><li> Give a rigorous argument which proves that among all walks of length <math>2m</math> that have an...") |
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<math>2k</math> if | <math>2k</math> if | ||
<math>S_{2k-1}</math> and <math>S_{2k+1}</math> are of opposite sign. | <math>S_{2k-1}</math> and <math>S_{2k+1}</math> are of opposite sign. | ||
<ul><li> Give a rigorous argument which proves that | <ul style="list-style-type:lower-alpha"><li> Give a rigorous argument which proves that | ||
among all walks of length <math>2m</math> that have an equalization at time <math>2k</math>, exactly half have a lead | among all walks of length <math>2m</math> that have an equalization at time <math>2k</math>, exactly half have a lead | ||
change at time <math>2k</math>. | change at time <math>2k</math>. | ||
Latest revision as of 01:31, 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]
A lead change in a random walk occurs at time
[math]2k[/math] if [math]S_{2k-1}[/math] and [math]S_{2k+1}[/math] are of opposite sign.
- Give a rigorous argument which proves that among all walks of length [math]2m[/math] that have an equalization at time [math]2k[/math], exactly half have a lead change at time [math]2k[/math].
- Deduce that the total number of lead changes among all walks of length [math]2m[/math] equals
[[math]] {1\over 2}(g_{2m} - u_{2m})\ . [[/math]]
- Find an asymptotic expression for the average number of lead changes in a random walk of length [math]2m[/math].