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Order Statistics

  1. Nov 8, 2009 #1
    1. The problem statement, all variables and given/known data

    Let Y1<Y2<...<Yn be the order statistics of a random sample of size n from the pdf [tex]f(x) = e^{-x}[/tex] x ranging from 0 to infinity.

    a) Show that Z1=nY1, Z2 = (n-1)(Y2 - y1) Z3= (n-2)(Y3-Y2)... Zn = Yn - Y_(n-1) are independent and that each Z has the exp distribution.

    b) Demonstrate that all linear functions of Y1, Y2,...,Yn such as [tex] \Sigma a_i Y_i[/tex] can be expressed as a linear function of independent random variables.


    2. Relevant equations



    3. The attempt at a solution

    I got part a, but now I'm stuck at part b.

    I know that [tex]h(y_1,y_2,...,y_n) = n! e^{-y_1 - y_2 - ... - y_n}[/tex] and [tex] \Sigma a_i Y_i= a_1Y_1 + a_2Y_2 + ... + a_nY_n[/tex]

    Any hints/suggestions would be apprecited. Thank you.
     
  2. jcsd
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