I've been working on this for about 10 days. I know I need some book revising...

So, there are N+1 r.v.

N variables ( ) follow a Gamma distribution:

1 variable ( ) follows a 'negative' Gamma distribution:

I need to find the pdf of the sum:

where - just parameters

What I did:

My first method was by characteristic function. I found the ch.f. of the sum and tried to inverse-Fourier-transmorm it, so that it gives the pdf. But I got a cumbersome integral to evaluate (I posted it here: http://www.mathhelpforum.com/math-he...tegration.html ), still no help .

Convolution method is my second try. I found the pdf of and the pdf of

Now I'm working to find the convolution of these 2 parts. Not very simple (to me), not sure If I can make it.

Any suggestions to help with my methods or to find the pdf of ? Thank you!