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Order Statistics/Change of Variable

  1. Nov 5, 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

    a)

    so [tex]y_1 = z_1/n[/tex] , [tex]y_2 = z_2/(n-1) +z_1/n[/tex] , [tex]y_3 = z_3/(n-2) + z_2/(n-1) +z_1/n[/tex] etc...


    So how would I find the jacobian?
     
  2. jcsd
  3. Nov 5, 2009 #2
    I got part a, but not sure how I would do part b.

    Thanks in advance.
     
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