The last couple posts have touched on order statistics for normal random variables. I wrote the posts quickly and didn’t go into much detail, and more detail would be useful.

Given two integers *n* ≥ *r* ≥ 1, define *E*(*r*, *n*) to be the *r*th order statistic of *n* samples from standard normal random variables. That is, if we were to take *n* samples and then sort them in increasing order, the expected value of the *r*th sample is *E*(*r*, *n*).

We can compute *E*(*r*, *n*) exactly by

where φ and Φ are the PDF and CDF of a standard normal random variable respectively. We can numerically evaluate *E*(*r*, *n*) by numerically evaluating its defining integral.

The previous posts have used

*d*_{n} = *E*(*n*, *n*) – *E*(1, *n*) = 2 *E*(*n*, *n*).

The second equality above follows by symmetry.

We can compute *d*_{n} in Mathematica as follows.

Phi[x_] := (1 + Erf[x/Sqrt[2]])/2 phi[x_] := Exp[-x^2/2]/Sqrt[2 Pi] d[n_] := 2 n NIntegrate[ x Phi[x]^(n - 1) phi[x], {x, -Infinity, Infinity}]

And we can reproduce the table here by

Table[1/d[n], {n, 2, 10}]

Finally, we can see how *d*_{n} behaves for large *n* by calling

ListPlot[Table[d[n], {n, 1, 1000}]]

to produce the following graph.

See the next post for approximations to *d*_{n}.