What is the covariance between two binomial distributions?

[Matteo_]

Hi everybody!

I have a random iid sample Xi, i=1, ..., n

The empirical cdf of the sample at poin s is

$$\hat{F}\left(s\right)=n^{-1}\sum_{i=1}^{n}{\textbf{1}_{\left(-\infty, s\right)}\left(x_{i}\right)}$$

Clearly $$\hat{F}\left(s\right)$$ is binomially distributed with parameters n and p=F(t) (true cdf).

Now I need to find the covariance between $$\hat{F}\left(s\right)$$ and $$\hat{F}\left(t\right)$$ for s<t.

I know that the result is $$n^{-1}F\left(s\right)\left(1-F\left(t\right)\right)$$

Any help is very appreciated...

Thanks!

 

[EnumaElish]

http://en.wikipedia.org/wiki/Binomial_distribution#Covariance_between_two_binomials