# Product moments of order statistics.

Just a while ago I started studying order statistics and while studying the joint probability mass function I came across this statement:

$$\mu_{i,i+1:n} = \mu_{i:n}^{(2)} + \binom{n}{i} \sum_{x=0}^{\infty} \left( x \left[(F(x))^{i} - (F(x-1))^{i}\right] \sum_{y=x}^{\infty}(1-F(y))^{n-i}\right)$$ for $1\le i\le n-1$

The book I follow (A first course in order statistics) says it is a simplified expression for the product moments of order statistics when the support of the distribution consists of nonnegative integers by using the following theorem:

For $1<i_{1}<i_{2}< ...<i_{k} \le n$ the joint probability mass function of $x_{i_{l}:n}, x_{i_{2}:n}, x_{i_{k}:n}$ is given by:

$f_{i_{l},i_{2},...,i_{k}:n}(x_{i_{l}},x_{i_{2}}, ..., x_{i_{k}})=C(i_{l},i_{2},...,i_{k}:n)\times\int_{B} \left( \prod_{r=1}^{k} (u_{i_{r}}-u_{i_{r-1}})^{u_{i_{r}}-u_{i_{r-1}}-1}\right)(1-u_{i_{k}})^{n-u_{i_{k}}}du_{i_{1}}\cdot ... \cdot du_{i_{k}}$

where $i_{0}=0$ and $u_{0}=0$

$$C= \frac{n!}{(n-i_{k})! \prod_{r=1}^{k}(i_{r}-i_{r-1}-1)!}$$

B $= \{(u_{i_{1}},...,u_{i_{k}}):u_{i_{1}}\le u_{i_{2}}\le ...\le u_{i_{k}},\ F(x_{r}-)\le u_{r} \le F(x_{r}),\ r = i_{1}, i_{2}, ..., i_{k}\}$

a k-dimensional space

The thing is that I don't really understand how to get the simplified expression using the theorem, is it alright if I ask for a bit of help to verify the statement? Maybe something like a starting point?