Let $X_i \sim \varepsilon (\lambda) \ 1 \leq i \leq n $ And $ N \sim Geo(\mu) $. Let $ Z = \sum_{i=1}^{N}X_i $, calculate the MGF

Question

Let $X_i \sim \varepsilon (\lambda) \ 1 \leq i \leq n $ And $ N \sim Geo(\mu) $. Let $ Z = \sum_{i=1}^{N}X_i $, calculate the MGF. Every $X_i$ is independent and $X_i$ is independent of $N$.

I got really stuck trying to do this one, and I guess it has something to be with Wald's equation:

$ E[Z] = E[N]E[X_1] $.

Answer 1

By definition of moment generating function:$$\begin{align}\mathsf M_Z(t)&=\mathsf E(\mathrm e^{tZ})\\[1ex]&=\mathsf E(\mathrm e^{t\sum_{i=1}^NX_i})\\[1ex]&=\mathsf E(\prod_{i=1}^N\mathrm e^{tX_i})\\[1ex]&=\mathsf E(\mathsf E(\prod_{i=1}^N\mathrm e^{tX_i}\mid N))\\[1ex]&=\end{align}$$

Now use that $(X_i)$ are iid (and also independent of $N$).

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PS:

$\mathsf M_{X_i}(s) = (1-s/\lambda)^{-1}$

Also

$\mathsf M_{N}(u) = \mu\mathrm e^{u} (1-(1-\mu)\mathrm e^{u})^{-1}$ for all $u<-\ln(1-\mu)$