Moment generating function of summation of random variables

Question

I have difficulty to understand the reason of following equality.

Can any one show me the reason of last equality ?

The thing that ı did not understand is that how did they write as power of n because x are different than each other?

Picture is from Sheldon M. Ross stochastic process page 22.

Answer 1

It is just the property $e^{a+b}=e^ae^b$. For any given $n$ \begin{align}\exp{\left(t\sum_{i=1}^nX_i\right)}&=\exp\left(tX_1+tX_2+\dots+ tX_n\right)\\[0.2cm]&=\exp\left(tX_1\right)\exp\left(tX_2\right)\cdots\exp\left(tX_n\right)\\[0.2cm]&=\psi_{X_1}(t)\psi_{X_2}(t)\cdots\psi_{X_n}(t)=^{X_i's \text{ are identically distributed}}\\[0.2cm]&=\psi_{X}(t)\psi_{X}(t)\cdots\psi_{X}(t)=\left(\psi_X(t)\right)^n\end{align} Hence $$\mathbb E\left[\exp{\left(t\sum_{i=1}^NX_i\right)}\mid N\right]=\mathbb E\left[\left(\psi_X(t)\right)^N\mid N\right]=\left(\psi_X(t)\right)^N$$

Answer 2

By definition of the conditional expectation

$E[X|Y]=\phi(Y)$

where

$\phi(y)=E[X|Y=y]$