\(\newcommand{\footnotename}{footnote}\)
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\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\let \LWRorighspace \hspace \)
\(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\)
\(\newcommand {\TextOrMath }[2]{#2}\)
\(\newcommand {\mathnormal }[1]{{#1}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
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\(\newcommand {\addtolength }[2]{}\)
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\(\newcommand {\number }[1]{}\)
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\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)
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\(\def \LWRabsorbquotenumber "#1 {}\)
\(\newcommand {\LWRabsorboption }[1][]{}\)
\(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\)
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\(\def \AA {\unicode {x00C5}}\)
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\( \newcommand {\multicolumn }[3]{#3}\)
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\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
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Shorter Die Battles (and Order Statistics)
We rederive the main result in [1] in a shorter way, without combinatorics. If we have \(m\) independent and identically distributed
(i.i.d.) random variables (r.v.’s) \(X_1,\dots ,X_m\), with \(X_i=X\) having cumulative distribution function (cdf) \(F_X(x)=\Pr (X\le x)\), then the r.v. \(X_{(m)}=\max _i X_i\) has cdf
\(\seteqnumber{0}{}{0}\)
\begin{equation*}
F_{X_{(m)}}(x)=\Pr (X_{(m)}\le x)=\Pi _{i=1}^m \Pr (X_i\le x)=\left (\Pr (X\le x)\right )^m=F_X^m(x).
\end{equation*}
Let \(X\) be a uniform r.v. with support \(\mathcal {X}=\{1,2,\dots ,k\}\) modelling a fair die with \(k\) faces. For \(x\in \mathcal {X}\) the cdf of \(X\) is
\(\seteqnumber{0}{}{0}\)
\begin{equation*}
\label {eq:cdf} F_X(x)=\frac {x}{k}.
\end{equation*}
If \(Y_{(n)}\) is the maximum of the i.i.d. r.v.’s \(Y_1,\dots ,Y_n\) with \(Y_i=X\) and independent of \(X_1,\dots ,X_m\), then
\(\seteqnumber{0}{}{0}\)
\begin{align}
\label {eq:prob} \Pr (X_{(m)}>Y_{(n)})&= \sum _{a>b}\Pr (X_{(m)}=a,Y_{(n)}=b)\nonumber \\ &=\sum _{a=2}^k \Pr (X_{(m)}=a)\Pr (Y_{(n)}<a)\nonumber \\ &=\sum _{a=2}^k \left (F_{X_{(m)}}(a)-F_{X_{(m)}}(a-1)\right
)F_{Y_{(n)}}(a-1)\nonumber \\ &=\sum _{a=2}^k \left (F_{X}^m(a)-F_{X}^m(a-1)\right )F_{X}^n(a-1)\nonumber \\ &=\frac {1}{k^{m+n}}\sum _{a=2}^k\left (a^m-(a-1)^m\right )(a-1)^n.\nonumber
\end{align}
References
Last updated: 2021/10/28 12:23:33
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