Order Statistics/Change of Variable

[cse63146]

Homework Statement

Let Y1<Y2<...<Yn be the order statistics of a random sample of size n from the pdf $$f(x) = e^{-x}$$ 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 $$\Sigma a_i Y_i$$ can be expressed as a linear function of independent random variables.

Homework Equations

The Attempt at a Solution

a)

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


So how would I find the jacobian?

 

[cse63146]

I got part a, but not sure how I would do part b.

Thanks in advance.