I've got this problem on multivariable distributions I'm stuck on...

Let \(\displaystyle X_1;X_2;X_3\)denote a random sample of size n = 3 from a distribution with the geometric pdf

\(\displaystyle P(X=k)=(\frac{3}{4}) ( \frac{1}{4} )^{k-1}\) and given the distribution \(\displaystyle Z = X_1 + X_2 + X_3\)

Question find the pdf of \(\displaystyle Z\) then find \(\displaystyle P(Z=15)\).

Now I considered using the moment generating function for 3 reasons: (a) \(\displaystyle X_1, X_2, \text{and} X_3\) are a random sample size \(\displaystyle \rightarrow \ X_1, X_2, \text{and } X_3\) are independent (b) they have the same pdf thus mgf, and (c) well, mgf can be used for this.

So i started out like this:

\(\displaystyle M_Z (t) = E(e^{tZ})=E(e^{(X_1+X_2+X_3 )\cdot t})=E(e^{X_1 t} \cdot e^{X_2 t} \cdot e^{X_3 t})= ( E(e^{X_i t}) )^3 \)\(\displaystyle = (\frac{p e^t}{1-(1-p) e^t})^3\)

Then I got stuck there... I presume I don't know how to "transform" variables in descrete distributions unlike continuous ones.

So any help is much appreciated!