# Order Statistics/Change of Variable

1. Nov 5, 2009

### cse63146

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 $$f(x) = e^{-x}$$ 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.

2. Relevant equations

3. 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?

2. Nov 5, 2009

### cse63146

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