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Math Help - Conditional expectation

  1. #1
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    Conditional expectation

    Let \xi_1,\xi_2,\xi_3 be three independent, identically distributed and bounded random variables with density p(x).

    What is the distribution of: \mathbb{E}\big(\max(\xi_1,\xi_2,\xi_3)|\min(\xi_1,  \xi_2,\xi_3)\big), as a function of p ?

    One can easily obtain the distribution of \max(\xi_1,\xi_2,\xi_3) and \min(\xi_1,\xi_2,\xi_3). But how do you get the joint distribution?
    Unless there is a better way to proceed.

    Thanks in advance for your help.
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  2. #2
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    Let M=\max(\xi_1,\xi_2,\xi_3) and m=\min(\xi_1,\xi_2,\xi_3). You can start from there:

    P(t\leq m\mbox{ and }M\leq u)=P(t\leq \xi_1\leq u)^3.

    This gives the distribution of (m,M).
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  3. #3
    MHF Contributor matheagle's Avatar
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    you can use the 'multinomial' formula for joint densities of order stats

    f(x_1,x_3)=3!f(x_1)[F(x_3)-F(x_1)]f(x_3)

    where I'm using x_1 as the first order stat and x_3 as the third.
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  4. #4
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    This is what you get using the joint probability suggested by Laurent.

    I am not familiar with the multinomial formula you mention. Is there any reference you could suggest where I can find it?

    Thanks for your answer.
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  5. #5
    MHF Contributor matheagle's Avatar
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    It's in many books, Hogg and Craig for one.
    I googled it the other day and I only found it online at another forum.
    It was incorrect initially but someone finally fixed it.
    sadly wikipedia only has this for uniforms...
    http://en.wikipedia.org/wiki/Order_statistic
    but from that you can see the full formula.
    You can obtain the joint density of any subset of order stats directly.
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