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Math Help - Order statistics

  1. #1
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    Exclamation 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
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  2. #2
    MHF Contributor matheagle's Avatar
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    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.
    You can read the first page of.... http://www.springerlink.com/content/6njl7x1l5n48m431/
    Last edited by matheagle; February 11th 2010 at 02:53 PM.
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  3. #3
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    I don't understand... what happens if I have f(x) and F(x)...
    And what is rvs?
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  4. #4
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    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..
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  5. #5
    MHF Contributor matheagle's Avatar
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    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.
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  6. #6
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    Sorry I don't know how to do it..

    Thanks for the information for the variances! I'm going to retry the problem
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  7. #7
    MHF Contributor matheagle's Avatar
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    HIT the quote button to see how people type in TeX.
    Then you can cut and paste long expressions.
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  8. #8
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    But once you find the joint distribution, how do you integrate E(Y(k)*Y(j)).
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  9. #9
    MHF Contributor matheagle's Avatar
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    That's just basic probability.
    It's the double integral with those two variables and their joint density.
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