Shorter Die Battles (and Order Statistics)

Félix Balado

November 28, 2021

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

\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

\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

\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}

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