I'm having some difficulty finding the mean and the variance of the following question:

Let

be independent uniformly dist. r.v. on the interval

I found my density function for

*k*th order stat between 1 and n, which is:

where

now using what I found if I try to find the

I get:

where

=

where

now the integral looks a lot like a

*Beta Function*, but what puzzles me is that the beta function goes from 0 to 1, and I have 0 to

. If I were to make the substitution. I would get:

where

And for the

I get:

where

=

where

Again the first part of the integral looks like the

*Beta Function*, but since there's an additional y within it, I'm a little confused on how to integrate this out, since I don't think that a closed form of this function exists.