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 ($\displaystyle X_{1},...,X_{N} $) follow a Gamma distribution: $\displaystyle f_{X}(x)=x^{k_{1}-1} \frac {e^{-x}}{\Gamma(k_{1})}, for \text{ } x>0 $

1 variable ($\displaystyle Y $) follows a 'negative' Gamma distribution: $\displaystyle f_{Y}(x)=(-x)^{k_{2}-1} \frac {e^{x}}{\Gamma(k_{2})}, for \text{ } x<0 $

$\displaystyle Z=\alpha+\frac{1}{N}\sum_{i=1}^{N} \beta_{1} X_i+\beta_{2} Y$

I need to find the pdf of the sum:

where $\displaystyle \alpha,\beta_{1},\beta_{2}$ - 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 $\displaystyle \frac{1}{N}\sum_{i=1}^{N} \beta_{1} X_i $ and the pdf of $\displaystyle \beta_{2} Y$

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 $\displaystyle Z$ ? Thank you!