1. ## Order statistics

Hey I am having difficulties with this problem, can anyone help please?

Let Y1, Y2,...,Yn be independent, uniformly distributed random variables on the interval [0, θ]. Find the joint density function of Y(j) and Y(k) where j and k are integers 1≤ j < k ≤ n. Find V(Y(k) - Y(j)), the variance of the difference between two order statistics.

Thanks

2. FIRST obtain the joint density of the random variables.
You will need f(x) and F(X).

$\displaystyle f(x)={1\over \theta}$ and $\displaystyle F(x)={x\over \theta}$ on the interval $\displaystyle (0,\theta)$

Then obtain the joint density by the multinomial distribution.
The uniform can be found at wikipedia.

3. I don't understand... what happens if I have f(x) and F(x)...
And what is rvs?

4. Hi I found the joint distribution function to be:

factorial(n)*(((Yj/theta)^(j-1))(Yk/theta-Yj/theta)^(k-j-1))(1-Yk/theta)^(n-1)/(factorial(j-1)*factorial(k-j-1)*factorial(n-k)*theta^2)

But then to find the variance between the two order statistics I don't now how to do..

5. It would be nice if you wrote this in tex.
I won't read the math otherwise.
To get the variance you have two options.
You can transform the random variables.
Let R=Y(k)-Y(j), W= either one you wish.
Then integrate out W and you have R, which I call your range.
Then get the variance of R.
The second way is easier.
Just compute V(Y(k))-2Cov(Y(k),Y(j))+V(Y(j)).

NOTE that if you take these rvs and divide by theta, I believe you get Beta's.

6. Sorry I don't know how to do it..

Thanks for the information for the variances! I'm going to retry the problem

7. HIT the quote button to see how people type in TeX.
Then you can cut and paste long expressions.

8. But once you find the joint distribution, how do you integrate $\displaystyle E(Y(k)*Y(j))$.

9. That's just basic probability.
It's the double integral with those two variables and their joint density.