Section 13.1

Maxima and Minima of Dice and Cards

E[max] and E[min] of discrete samples via the tail-sum formula or direct counting.

Toolkit

For a random variable $X$ taking values in $\{1,\dots,n\}$ (or $\{1,\dots,N\}$ ), the tail-sum formula converts an expectation into a sum of survival probabilities, avoiding the need for the full pmf:

\mathbb{E}[X]=\sum_{k\ge 1}\mathbb{P}(X\ge k) = \sum_{k\ge 0}\mathbb{P}(X>k).

For order statistics of i.i.d. samples, $\mathbb{P}(\max\le k)=F(k)^n$ and $\mathbb{P}(\min\ge k)=(1-F(k-1))^n$ , which combine with tail-summing to turn extremal expectations into sums of powers (squares, cubes, etc.) that can be evaluated in closed form. For sampling without replacement from $\{1,\dots,N\}$ , order-statistic probabilities become hypergeometric, $\mathbb{P}(\max=k)=\binom{k-1}{n-1}/\binom{N}{n}$ , and the identity $k\binom{k-1}{n-1}=n\binom{k}{n}$ together with the hockey-stick identity $\sum_k \binom{k}{n}=\binom{K+1}{n+1}$ collapses the resulting sum. Symmetry of the underlying distribution (e.g. $x\mapsto n+1-x$ for a die) can give medians or means for free, without computing any probabilities.

Problems