I have a question that is stated as follows:
Two random variable $P$ and $Q$ have MGF's:
$$M_P(s) = \left(\frac{1}{3} + \frac{2}{3}e^s \right) ^{10}$$ $$M_Q(s) = \frac{\frac{1}{5}4e^s}{1-\frac{4}{5}e^s}$$
Let $T = Q_1 + Q_2 + \ldots + Q_P$. The $Q$ variables are iid according to $M_Q(s)$. You want to find the MGF of $T$.
Hint: use iterated expectations
I know that since we have a sum of random variables, We can have a product of the moments of $Q_i$, but I am not sure how to incorporate the fact that the number of $Q$ variables is determined by the $P$.
\begin{align*} M_T(s) &= \mathbb{E}\big[e^{sT}\big] \\ &= \mathbb{E}\big[\mathbb{E}\big[e^{sT} | P\big]\big] \\ &= \mathbb{E}\big[\mathbb{E}\big[e^{s(Q_1 + \dots + Q_p)} | P\big]\big] \\ &= \mathbb{E}\big[\mathbb{E}\big[\prod_{i=1}^{P}e^{sQ_i} | P\big]\big] \\ &= \mathbb{E}\big[\prod_{i=1}^{P}\mathbb{E}\big[e^{sQ_i}\big]\big] \\ &= \mathbb{E}\big[M_Q(s)^P\big] \\ &= \mathbb{E}\big[e^{P\log M_Q(s)}\big] \\ &= \bigg(\frac{1}{3} + \frac{2}{3}M_Q(s)\bigg)^{10} \end{align*}