Let $X_1, \dotsc, X_n$ be i.i.d. standard normal random variables. Define the range $R \in \mathbb{R}_{\geq 0}$ as $R = \max \{X_1, \dotsc, X_n\} – \min \{X_1, \dotsc, X_n \}$. I am looking for a simple expression that is a good approximation of the density function $r(x)$ of $R$. For my application the number $n$ is fairly large ($n=128$ in this particular case). I get the following exact form of $r$ where $\Phi$ is the CDF for each $X_i$ and $x \geq 0$:
$$
r(x) = \frac{ n(n-1) e^{-x^2/4}}{2 \pi} \int_{\mathbb{R}} e^{-s^2} \left( \Phi(s + x/2) – \Phi(s – x/2) \right)^{n-2} ds
$$
I've tried to estimate the integral in this expression, for example by using
$$
\Phi(s + x/2) – \Phi(s – x/2) \leq \Phi(x/2) – \Phi(-x/2) = \textrm{erf}(\frac{x}{2 \sqrt{2}})
$$
but this seems too coarse, certainly for small values of $x$. Any pointers would be appreciated, also for partial results like estimating the expected value and variance of $R$.
Best Answer
I think this paper may help. Someone may have expanded on this in the past 50 years, but it seems like a good place to start:
Tables of range and studentized range, HL Harter
http://www.jstor.org/stable/2237810
Edit: In case you're interested and familiar with R, here is some code that seems to work (for me, at least):