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.
February 11th 2010, 02:16 PM
FIRST obtain the joint density of the random variables.
You will need f(x) and F(X).
But then to find the variance between the two order statistics I don't now how to do..
February 13th 2010, 01:55 PM
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.
February 13th 2010, 02:01 PM
Sorry I don't know how to do it..
Thanks for the information for the variances! I'm going to retry the problem
February 13th 2010, 02:30 PM
HIT the quote button to see how people type in TeX.
Then you can cut and paste long expressions.
February 14th 2010, 04:38 AM
But once you find the joint distribution, how do you integrate .
February 15th 2010, 09:25 PM
That's just basic probability.
It's the double integral with those two variables and their joint density.