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cse63146
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Homework Statement
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.
Homework Equations
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?