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    Joint distributions for Order Statistics




    I have a problem with part b of this question. I found the joint distribution and seem to have problem integrating it. I seem to have gotten a double integral that I cannot integrate. It appears I need part b for part c and d of this question.

    So any help with the integration would be appreciated.

    Thanks.
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    Re: Joint distributions for Order Statistics

    When you're about to integrate, be careful how you set the values which you are going to integrate over. E.g. order statistic j goes from order statistic i to 1.

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    Re: Joint distributions for Order Statistics

    can someone give me a hint on evaluating the integral. I tried converting factorials to gamma functions and can't seem to do anything either.

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    Re: Joint distributions for Order Statistics

    As what Englund mentioned, when you obtain the joint pdf in part a) you should be aware and state the support accordingly:

    f_{U_{(i)},U_{(j)}}(u, v) = \frac {n!} {(i-1)!(j-i-1)!(n-j)!} u^{i-1}(v-u)^{j-i-1}(1-v)^{n-j}, 0 < u < v < 1

    as 1 \leq i < j \leq n

    http://en.wikipedia.org/wiki/Order_s...m_distribution

    Do you mean you want to verify the normalizing constant \frac {n!} {(i-1)!(j-i-1)!(n-j)!},
    i.e. the usual identity for a joint pdf

    \int_0^1 \int_0^v f_{U_{(i)},U_{(j)}}(u, v)dudv = 1

    This one will not be hard; you can try the substitution u = wv in the inner integral and you should figure out the beta integral.

    For part b) you just need to apply this identity again to evaluate the cross moments

    E[U_{(i)}U_{(j)}] = \int_0^1 \int_0^v uvf_{U_{(i)},U_{(j)}}(u, v)dudv

    by adjusting the coefficients appropriately.

  5. The Following User Says Thank You to BGM For This Useful Post:

    bk123 (04-28-2013)

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    Re: Joint distributions for Order Statistics

    Thanks. It came out very easily.

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    Re: Joint distributions for Order Statistics

    Can someone give me a hint on part ii? I tried using the same substitution on the integral, but it doesn't seem to fall out as expected.
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    Re: Joint distributions for Order Statistics

    f(yi|yj)=f(yi, yj)/f(yj)

    Edit: Sorry, I thought you were asking about part i.
    Last edited by Englund; 04-28-2013 at 12:18 PM.

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    Re: Joint distributions for Order Statistics

    @BGM can you please make it more specific for the inner integral part. thx

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    Re: Joint distributions for Order Statistics


    Sorry can you ask your question again, more precisely? Not exactly sure which part is not clear.

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