[Math] Range of i.i.d. normal random variables

probability distributions

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):

r<-function(x,n){
inner.int<-function(s){
exp(-s^2)*(pnorm(s+x/2)-pnorm(s-x/2))^(n-2)
}
return(n*(n-1)*exp(-x^2/4)/(2*pi)*integrate(inner.int,-Inf,Inf)$value)
}
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