exercise:1e002861f1: 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> Show that <math display="block"> P(S_1 \ge 0,\ S_2 \ge 0,\ \ldots,\ S_{2m} \ge 0) = u_{2m}\ . </math> '' Hint'': First explain why <math display="block"> \begin{eqnarray*} &&P(S_1 > 0,\ S_2 > 0,\ \ldots,\ S_{2m} > 0) \\ && \;\;\;\;\;\;\;\...") |
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\end{eqnarray*} | \end{eqnarray*} | ||
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Then use | Then use [[exercise:9e4fe88535 |Exercise]], together with the observation that if no equalization occurs in the | ||
first <math>2m</math> outcomes, then the path goes through the point <math>(1,1)</math> and remains on or above the | first <math>2m</math> outcomes, then the path goes through the point <math>(1,1)</math> and remains on or above the | ||
horizontal line <math>x = 1</math>. | horizontal line <math>x = 1</math>. | ||
Latest revision as of 00:31, 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]
Show that
[[math]]
P(S_1 \ge 0,\ S_2 \ge 0,\ \ldots,\ S_{2m} \ge 0) = u_{2m}\ .
[[/math]]
Hint: First explain why
[[math]]
\begin{eqnarray*}
&&P(S_1 \gt 0,\ S_2 \gt 0,\ \ldots,\ S_{2m} \gt 0) \\
&& \;\;\;\;\;\;\;\;\;\;\;\;\; = {1\over 2}P(S_1 \ne 0,\ S_2 \ne 0,\ \ldots,\ S_{2m} \ne 0) \ .
\end{eqnarray*}
[[/math]]
Then use Exercise, together with the observation that if no equalization occurs in the first [math]2m[/math] outcomes, then the path goes through the point [math](1,1)[/math] and remains on or above the horizontal line [math]x = 1[/math].