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

a)

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

So how would I find the jacobian?