Be that as it may, the following might help ...... I'm changing the notation to represent Xn:n by and X1:n by .
Let be a sequence of i.i.d. random variables with pdf denoted by f(x), cdf denoted by F(x) and order statistics .
It can be shown that the pdf of the interval (where ) is given by
When r = 1 and s = n the interval becomes the range W and equation (1) reduces to:
When ~ , f(x) = 1 for and zero elsewhere and equation (2) becomes:
where the upper integral terminal is because f(x + w) = 0 for . Therefore the pdf of the range of n numbers selected at random from the interval (0, 1) is:
Note by the way the interesting result that and E(W) = 1 in the limit n --> oo ....