Order statistics

• Feb 9th 2010, 02:25 PM
Azizi
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(Wink)
• Feb 11th 2010, 03:16 PM
matheagle
FIRST obtain the joint density of the random variables.
You will need f(x) and F(X).

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

Then obtain the joint density by the multinomial distribution.
The uniform can be found at wikipedia.
• Feb 11th 2010, 03:18 PM
Azizi
I don't understand... what happens if I have f(x) and F(x)...
And what is rvs?
• Feb 13th 2010, 02:47 PM
pgl1990
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..
• Feb 13th 2010, 02:55 PM
matheagle
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.
• Feb 13th 2010, 03:01 PM
pgl1990
Sorry I don't know how to do it..

Thanks for the information for the variances! I'm going to retry the problem
• Feb 13th 2010, 03:30 PM
matheagle
HIT the quote button to see how people type in TeX.
Then you can cut and paste long expressions.
• Feb 14th 2010, 05:38 AM
solvj
But once you find the joint distribution, how do you integrate $E(Y(k)*Y(j))$.
• Feb 15th 2010, 10:25 PM
matheagle
That's just basic probability.
It's the double integral with those two variables and their joint density.