Hi,
I have a 3-part question from an exam I took and did not solve correctly. I'd love to find out what I should have done... (We already received the exam back, I'm "cleaning up" loose ends.) Thank you!
Assuming Y1, Y2, ..., Yn are iid Exp(1)
a) What are the mean, variance (sigma squared) and third and fourth central moments? (We were given the hint to prove )
b) Find and
c) Find
for the last part I wanted to use MGF of the kth order stat and let t=-1, but I'm not sure i can recognize the MGF of that order stat.
However.......
the density is
and using that as a valid density you can manipulate this expectation.
I've never done that with order stats, but it works......
NOW put that into the last term and let m=n+1, so are using a density of the kth order stat from a sample of size n+1.
you can pull the constants in and out of the integral and using the fact that a density integrates to one should work.
I get
Hi,
I am still hoping to follow up on this. Maybe I'll post what I did and you could point out where I went wrong?
Take iid sample of random variables that are Exp(1)
b) Find
I take that in probability.
I then can use Slutsky with
to give that in distribution
and thus in distribution and probability.
Then
What am I missing here???
Thanks a ton!