X is a poisson random variable with parameter Y, and Y itself is a poisson Random variable with parameter $\lambda$ how can I find the moment generating function of X + Y.
At first I tried to find the distribution of X using the law of total probability, but I couldn't calculate the series and got stuck.
the series I tried to calculate
$\sum_{b=0}^{\infty} P(X = a| Y = b) P(Y = b) = \frac{b^a e^{-b}}{a!} \cdot \frac{\lambda^{b}e^{-\lambda}}{b!}$
Using the law of total expectation (tower rule) and the fact that the MGF of a poisson distribution with mean $\mu$ is $t \mapsto e^{\mu (e^t-1)}$,
\begin{align} E[e^{t(X+Y)}] &= E[E[e^{t(X+Y)} \mid Y]] \\ &= E[e^{tY} E[e^{tX} \mid Y]] \\ &= E[e^{tY} e^{Y(e^t - 1)}]. \end{align} Combine the two terms and apply the MGF of $Y$ to finish.