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Math Help - expected value of order stats for uniform distribution

  1. #1
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    expected value of order stats for uniform distribution

    Question.

    Let Y_1, ..., Y_n be a random sample from uniform distribution U[0,\theta], where \theta>0 is an unknown parameter. Let Y_{(k)} be the k-th smallest value among Y_1, ..., Y_n for 1\leqk\leqn.
    (a) find density function of Y_{(k)}
    (b) Find E[Y_{(k)}].

    You may use the identity \int_0^1u^i(1-u)^jdu=\frac{i!j!}{(i+j+1)!} for integers i,j>0.

    Answer.

    (a) can be found in a textbook so I'll skip it here.

    (b)

    E[Y_{(k)}]=\int_0^{\theta}yf_{Y_{(k)}}(y)dy=\int_0^{\theta}y  \frac{n!}{(k-1)!(n-k)!}\frac{1}{\theta}(\frac{y}{\theta})^{k-1}(1-\frac{y}{\theta})^{n-k}dy

    =\int_0^{\theta}y\frac{n!}{(k-1)!(n-k)!}\frac{1}{\theta}(\frac{y}{\theta})^k(\frac{y}{  \theta})^{-1}(1-\frac{y}{\theta})^{n-k}dy=\int_0^{\theta}\frac{y{\theta}}{{\theta}y}\fr  ac{n!}{(k-1)!(n-k)!}(\frac{y}{\theta})^k(1-\frac{y}{\theta})^{n-k}dy

    I will now integrate not y but \frac{y}{\theta} which will (1) change the upper bound of integration to 1 and (2) d(\frac{y}{\theta})=\frac{1}{\theta}dy so I need to add one more theta into the integrand. I then take all non-y items outside the integral, and apply the 'identity' given above to what's left under the integral sign

    \theta\frac{n!}{(k-1)!(n-k)!}\int_0^1(\frac{y}{\theta})^k(1-\frac{y}{\theta})^{n-k}d(\frac{y}{\theta})=\theta\frac{n!}{(k-1)!(n-k)!}\frac{k!(n-k)!}{(k+n-k+1)!}=\theta\frac{n!(k-1)!k}{(k-1)!n!(n+1)}=\theta\frac{k}{n+1}

    Now, does it make sense as an expected value of k-th order statistic? It is between 0 and theta (since k/(n+1) is between 0 and 1). But I cannot say anything else, I cannot visualise a distribution where any value of y between 0 and theta has equal probability to happen.
    Last edited by Volga; February 11th 2011 at 03:04 PM.
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  2. #2
    MHF Contributor harish21's Avatar
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    \theta\dfrac{k}{n+1} looks good

    See here
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  3. #3
    MHF Contributor matheagle's Avatar
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    you have a factorial missing in the beta density in the identity
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  4. #4
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    Thank you!
    Put back missing factorial too.
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