# distribution of the range of N independent standard normal variables

#### AndyMC

##### New Member
Let x1, x2, ... xn be independent standard normal variables. (standard normal: normal with 0 mean and variance 1).

Let the range R be defined as R=max(x_i)-min(x_i)

I would like to find the distribution of the range R.

I found a procedure but I end up with an integral I can't solve analytically:

Let fi(x) be the distribution of a standard normal, Fi(x) the cumulative distribution, and g(R) the distribution of the range R , and G(R) the cumulative distribution of the range R.

G(R)=integral (x from -infinity to +infinity) (n*((Fi(x+R)-Fi(x))^(n-1))*fi(x)dx)

It means I choose randomly one of the n variables to be the minimum x, and impose that the other n-1 are between x and x+R, and multiply it by n to consider the n possible choices of minimum. Then I integrate for all possible values of the minimum x.

g(R)=integral (x from -infinity to + infinity) (n*(n-1)*((Fi(x+R)-Fi(x))^(n-2))*fi(x)*fi(x+R)dx)

It means that if I want the probability of the range being exactly R, i choose two variables randomly out of n to be respectively the maximum and the minimum, then I impose the other n-2 are between them, multiply by all the possible choices (n*(n-1)) (because I can switch the roles of the two I've chosen between maximum and minimum), and integrate for all possible values of the minimum x.

You can verify that G(0)=0, lim for R to infinity G(R)=1, g(R) is non negative.

If I can't find analytically the distribution, can I at least find E[R] and var[R], as a function of n?

Thank you!

#### AndyMC

##### New Member
Thanks for the link, but unfortunately I can't derive the Range distribution as the distribution of the difference of maximum-minimum, because these two random variables are not independent. That is, the conditional probability that y=max(x_i) given z=min(x_i) is not the same distribution of the unconditional probability that y=max(x_i). I tested my formula for n=2 and it gives the correct distribution of the range. Unfortunately I can't solve the integral for n greater than 2

#### fed1

##### TS Contributor
distribution of the range of N independent standard normal variables

This is the tag line of your post. So they are independant by assumption?

#### Dragan

##### Super Moderator
Unfortunately I can't solve the integral for n greater than 2
I would suggest that you look at the techniques that the author used in this article:

Renner, R. M. (1976). Evaluation by power series of the means of normal order statistics of samples of sizes six and seven. Mathematical Chronicle 4, 141–147.