- Thread starter
- #1

$Z=X_1+\ldots+X_N$, where:

$X_i\sim_{iid}\,\text{Exponential}(\lambda)$

$N\sim\,\text{Geometric}_1(p)$

For all $i,\,N$ and $X_i$ are independent.

I need to find the probability distribution of $Z$:

$G_N(t)=\frac{(1-p)t}{1-pt}$

$M_X(t)=\frac{\lambda}{\lambda-t}$

$M_Z(z)=G_N(M_X(z))=\frac{(1-p)\left(\frac{\lambda}{\lambda-z}\right)}{1-p\left(\frac{ \lambda}{\lambda-z}\right)}$

$\Rightarrow Z\sim\,\text{Geometric}_1\left(p \frac{ \lambda}{\lambda-z}\right)$

Is that even correct? Should I be looking for $E[Z]$ and $V[Z]$ ?