Independent identically distributed continuous random variables
independent and identically distributed continuous random variables X1, X2, ..., Xn with
cumulative distribution function G(x) and
probability density function g(x). Let T denote the range of a sample of size n from a population with distribution function G(x).
The range has cumulative distribution function
Gumbel notes that the "beauty of this formula is completely marred by the facts that, in general, we cannot express G(x + t) by G(x), and that the numerical integration is lengthy and tiresome."
If the distribution of each Xi is limited to the right (or left) then the asymptotic distribution of the range is equal to the asymptotic distribution of the largest (smallest) value. For more general distributions the asymptotic distribution can be expressed as a
The mean range is given by
where x(G) is the inverse function. In the case where each of the Xi has a
standard normal distribution, the mean range is given by