The problem with the MGF is that you cannot recognize the final distribution. With n=2 you can play with this to get an F.
Question: Let for . For , define . Compute
(Note that the constants can be negative,thus its not exactly non-central chi square)
My Attempt: I tried computing the pdf. I tried to use the M.G.F trick. We know that . Clearly
I am stuck here. I have forgotten Fourier Transforms, I think
The Fourier transform is something like
If we had a closed form for this, we would be able to use the inverse Fourier transform. But even then, it's not sure we'd get a closed form
I've thought of seperating the positive c_i's and the negative ones.
And work on the probability from this.. But I didn't really try.
Where did this come from?
That may shed some light on this problem.
If you know something about the c's we can tranfrom this to a sum of gamma's. But having the c's negative messes that up.
It just looks like a nasty multivariate calculus problem where the you have the joint density which is your integrand
and you need to integrate over a region in
Lol I was going to ask the same question : where does this come from ?
We can redefine Y :
Suppose (if =0, there is no sequence) are positive. The other ci's are negative.
There is no problem of permutation since the Wi's are iid.
This may simplify a bit the pdf.
It looks like it's worse than that.It just looks like a nasty multivariate calculus problem
If we want the pdf of Y, we'd have to make a substitution which, as far as I can see, wouldn't lead to a closed form...
Actually I am no longer working on this problem. I changed the model and technique to make computations easier.
So thanks again. This discussion convinced me to change the technique
*choosing all c_i's negative is trivial