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Maximum Order Statistic Question

  1. Jan 28, 2014 #1
    1. The problem statement, all variables and given/known data
    Let Yi∼iid,uniform[0,θ]. Let U=max{Yi}. Derive the distribution of U and give the value of any associated parameters. Also calculate E(U) and Var(U).



    2. Relevant equations
    f(y)=1/Θ and F(y)=y/Θ



    3. The attempt at a solution
    Since we have a product of iid random variables, we can multiply the cdf's a total of n times, giving us F(yn)=[F(y)]^n=(y/Θ)^n, so f(u)=n(y/Θ)^n-1, meaning U~Be(n,1) with α=n and β=1.

    I'm stuck on the E(u) part. This is what I have, ∫(from 0 to Θ)of u*n(y/Θ)^n-1 du. Please help.
     
  2. jcsd
  3. Jan 28, 2014 #2

    Ray Vickson

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    You cannot hope to get a correct answer if you are careless. From ##F_n(y) = (y/\theta)^n## we have the density ##f_n(y) = dF_n(y)/dy = (n/\theta) (y/\theta)^{n-1},## which is not what you wrote. I don't know why you write f(u) instead of f(y).

    Anyway, the answer is given by a simple, calculus 101 integral. Just write it out and think about it.
     
  4. Jan 28, 2014 #3
    Ray, so we would have ∫from 0 to Θ of (y*n/Θ)(y/Θ)^(n-1) then correct?
     
  5. Jan 28, 2014 #4
    If I'm doing things correctly here, I get E(U) = (nΘ)/n+1 and with the usual calculations, Var(U)=(nΘ^2)/(n+2)-((nΘ)/(n+1))^2
     
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