# Order Statistics

1. Nov 8, 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

I got part a, but now I'm stuck at part b.

I know that $$h(y_1,y_2,...,y_n) = n! e^{-y_1 - y_2 - ... - y_n}$$ and $$\Sigma a_i Y_i= a_1Y_1 + a_2Y_2 + ... + a_nY_n$$

Any hints/suggestions would be apprecited. Thank you.