Xn:n and X1:n are random variables so I'm not quite sure how to interpret the statement....? Do you mean that Pr(Xn:n - X1:n>a) is greater than some value ....?Xn:n - X1:n>a

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

.... (1).

When r = 1 and s = n the interval becomes the range W and equation (1) reduces to:

.... (2).

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:

.... (3).

Note by the way the interesting result that and E(W) = 1 in the limit n --> oo ....