Suppose Y is a R.V. which is equal to following other R.V. depending on parameter i, i.e.
\begin{equation} Y=\begin{Bmatrix} X_1 & i=1 \\ X_2 & i=2 \\ X_3 & i=3 \end{Bmatrix} \end{equation}
where MGF of $X_i$ is given as,$$M_{X_i}(t) = \Big(\frac{1}{3}\Big)^{i} (e^t + e^{2t} + e^{3t})^i e^{it}$$ How can one find the MGF of Y, given that R.V., i is equally likely?
Using the law of total expectation
$$ M_{Y}(t) = \mathbb{E}\left[e^{Yt}\right] = \sum_{j=1}^{3}\mathbb{E}\left[\left.e^{Yt}\right|i=j\right]P\left(i=j\right)$$
If each one of them is equally likely then
$$ M_{Y}(t) = \frac{1}{3}\sum_{i=1}^{3}M_{X_{i}}(t) = \frac{1}{3}\sum_{i=1}^{3}\left(\frac{1}{3}\right)^{i}\left(e^{t}+e^{2t}+e^{3t}\right)^{i}e^{it} = \left(\frac{1}{3}\right)\sum_{i=1}^{3}Z^{i}=\left(\frac{1}{3}\right)Z\frac{1-Z^{3}}{1-Z}$$ where $Z\equiv\left(\frac{1}{3}\right)\left(e^{t}+e^{2t}+e^{3t}\right)e^{t}$