# Characteristic Function $\phi_{S_N}$ where $N$ is a random variable

Let $$X_1, X_2$$, ... be i.i.d random variables all whose characteristic functions are $$\phi_{X_i} = \phi(t), i = 1, 2, ...$$

If $$N$$ is a random variable taking values in the positive integers, and if $$N$$ is independent of $$X_1, X_2$$, ..., determine $$\phi_{S_N}$$ where $$S_N = X_1 + X_2 + ... + X_N.$$

Attempt:

$$\phi_{S_N}(t) = E(e^{itS_N}) = E(exp\{it(X_1 + X_2 + ... + X_N)\}$$

$$= E(e^{itX_1}e^{itX_2}...e^{itX_N})$$

$$= E(e^{itX_1})...E(e^{itX_N})$$

$$= \phi_{X_1}\phi_{X_2}...\phi_{X_N}$$

$$= [\phi(t)]^N$$

Official Answer: $$E\Big([\phi(t)]^N\Big)$$

The problem in your answer is that $$\phi(t)^N$$ is a random variable which may not be constant.
First observe that $$e^{itS_N}=\sum_{n\geqslant 1}e^{itS_n}\mathbf 1\{N=n\}$$. Then take the expectation on both sides. In order to compute $$\mathbb E\left[e^{itS_n}\mathbf 1\{N=n\}\right]$$, first use the independence of $$N$$ with $$S_n$$ to write it as $$\mathbb E\left[e^{itS_n}\right]\mathbb P\{N=n\}$$. Then use what you did for a fixed $$n$$.