1. ## Order statistics

Let Y1<Y2<Y3<Y4<Y5 denote the order statistics of a random sample of size 5 from a distribution having pdf f(x) = e^(-x), 0<x<inf, zero else where. Show that Z1=Y2 and Z2=Y4-Y2 are independent.(Hint: find the joint pdf of Y2 and Y4)

I found the joint pdf g(Y2,Y4) and also joint pdf h(Z1,Z2)
Now how should I proceed this?

2. Originally Posted by ninano1205
Let Y1<Y2<Y3<Y4<Y5 denote the order statistics of a random sample of size 5 from a distribution having pdf f(x) = e^(-x), 0<x<inf, zero else where. Show that Z1=Y2 and Z2=Y4-Y2 are independent.(Hint: find the joint pdf of Y2 and Y4)

I found the joint pdf g(Y2,Y4) and also joint pdf h(Z1,Z2)
Now how should I proceed this?
You are done.
If you have correctly found the joint density of Z1 and Z2
and it factors into two functions, g(Z1) and h(Z2), which are NOT necessarily the marginals,
AND the support is a rectangle, then the two rvs are independent. Most books leave out that fact.
BUT, you can get the marginal densities of Z1 and Z2 and show that their product gives you the joint density, then you have certainly proved that Z1 and Z2 are independent.

From your saying "which are NOT necessarily the marginals,
AND the support is a rectangle, then the two rvs are independent."
, you mean it's not necessary to find the marginals?
Because I've seen this from books and lectures and didnt know the reason. They just found joint density and it seems they assume when this joint density can be factored as h(z1)g(z2), then it's independent.
What's the reasoning for that?
What should I look it up if I do not understand this?
Thanks.

4. Originally Posted by ninano1205
From your saying "which are NOT necessarily the marginals,
AND the support is a rectangle, then the two rvs are independent."
, you mean it's not necessary to find the marginals?
Because I've seen this from books and lectures and didnt know the reason. They just found joint density and it seems they assume when this joint density can be factored as h(z1)g(z2), then it's independent.
What's the reasoning for that?
What should I look it up if I do not understand this?
Thanks.

Its weird but most books leave that out
You can see if the rvs are indep by just glancing at the joint density
I teach out of walpole and also wackerly's book
and both leave that out

5. ## Can anyone explain this?

Joint density h(z1,z2) can be factored out as g(z1)q(z2)
then z1 and z2 are indepent without finding marginals.
Why is that?