1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Science Advisor
    Homework Helper

    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Threads - Maximum Order Statistic Date
2nd order non-linear pde Today at 7:31 AM
Maximum/minimum problem Yesterday at 4:00 AM
Minimum/Maximum problem Thursday at 1:55 PM
Maximum/minimum problem Mar 11, 2018
Finding maximum curvature on lnx Dec 3, 2017