# Thread: Problem of the week :)

1. ## Problem of the week :)

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

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

I need to find the pdf of the sum:
$Z=\alpha+\frac{1}{N}\sum_{i=1}^{N} \beta_{1} X_i+\beta_{2} Y$
where $\alpha,\beta_{1},\beta_{2}$ - 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 $\frac{1}{N}\sum_{i=1}^{N} \beta_{1} X_i$ and the pdf of $\beta_{2} Y$

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 $Z$ ? Thank you!

2. how about just using the MGFs?
I would split it into two parts....

$Z=\alpha+\frac{\beta_1}{N}\sum_{i=1}^{N} X_i+\beta_{2} Y$

Pull out the beta1 and use MGFs on the sum of the gammas and call that W.
Then write Z as

$Z=\alpha+\frac{\beta_1}{N}W+\beta_{2} Y$

This might not be recognizable as any known distribution, via MGFs
but you can use Calc 3 to obtain the density here.

AND you better have independence here.

3. Yes, I've split it into two parts then obtained the characteristic function of Z. (I think it's easier to obtain density from CHF than from MGFs). Now, to find the pdf of Z we must evaluate the integral:

$
f_{Z}(x)=\int^{\infty}_{-\infty} \frac{e^{-i u (x-\alpha)}}{(1+i u \beta_{1})^{k_{1}}(1-i u \frac{\beta_{2}}{N})^{N k_{2}}}du
$

the density will then consist of 2 parts: for $x>\alpha$ : $\int^{\infty}_{0}$ and for $x<\alpha$ : $\int^{0}_{-\infty}$

From Calculus 3, any suggestion to solve this integral ? I'm having some trouble with that. Thank you the Beagle.

4. Originally Posted by stokastik
any suggestion to solve this integral ? I'm having some trouble with that.
I mean you could just use Wolfram.......

5. I've tried but Wolfram and Maple couldn't do it, so...