# Mean and variance of the order statistics of a discrete uniform sample without replacement

In answering calculate the mean and variance of the highest number drawn on lottery based on the lowest number drawn, I couldn't find the mean and variance of the order statistics of a discrete uniform sample without replacement anywhere, so I figured I'd derive them here for future reference.

Draw $k$ distinct numbers uniformly from $1$ to $n$ and order them as $X_{(1)}\lt\cdots\lt X_{(k)}$. What are the mean and variance of the $X_{(i)}$?

Uniformly randomly permute $n-k$ white balls and $k$ red balls and take the positions of the red balls as the numbers drawn. Consider the indicator variables $Y_{im}$ that are $1$ if the $m$-th white ball is to the left of the $i$-th red ball and $0$ otherwise. Then $X_{(i)}=\sum_mY_{im}+i$.
To find the mean, we need $\def\xp#1{\mathbb E\left[#1\right]}\xp{Y_{im}}$, the probability that a given white ball is to the left of the $i$-th red ball. This happens if among the $k$ red balls and the $1$ given white ball the rank of the white ball is at most $i$, with probability $\frac i{k+1}$. Thus we have
To find the variance, we need $\xp{Y_{il}Y_{im}}$, the probability that two particular white balls are both to the left of the $i$-th red ball. This happens if among the $k$ red balls and the $2$ particular white balls the ranks of the white balls are at most $i+1$, with probability $\frac{\binom{i+1}2}{\binom{k+2}2}=\frac{i(i+1)}{(k+1)(k+2)}$. Thus we have
As expected, we have symmetry under the transformation $i\to k+1-i$.