# Range of i.i.d. normal random variables

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 $R$.

• You may find the answers to this question useful. Feb 20, 2012 at 9:51
• Maybe. I saw that question and answer but I don't see how that max central limit theorem would apply in some way to the range, since he max and min are clearly not independent. The Jensen trick might work to get an estimate for the expected value, didn't try yet.
– WimC
Feb 20, 2012 at 13:47

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)
}

• Thanks for the link. It is freely accessible on project euclid: goo.gl/4GTsT.
– WimC
Jun 9, 2012 at 18:18
• And this looks interesting too: goo.gl/qy7I1.
– WimC
Jun 9, 2012 at 18:36
• @WimC have you had any luck implementing the method in the second paper? I was wondering if it was any faster or more accurate than the code I listed... Jun 11, 2012 at 15:15