Let $X_1, X_2, . . . , X_n$ be mutually independent copies of X. Calculate the characteristic function of $S_n = X_1 + X_2 + . . . + X_n$. Determine how $S_n$ is distributed.
I am trying to solve the above question. However, I do not know how should I start. I understand that the definition of a characteristic function is
$$ϕ(t) = Ee^{itX} = \int e^{itX(w)}P(dw) $$
I have tried to find $E(S_n) = \bar{X}, $ but I do not know how can I continue.
Can anyone help please? Thanks!
Note that $$ Ee^{itS_n}=E\prod_{i=1}^ne^{itX_i}=\prod_{i=1}^nEe^{tX_i}=\prod_{i=1}^nEe^{tX_1}=\phi_{X_1}(t)^n $$ where in the second equality we use independence and in the third use identically distributed. Here $\phi_{X_1}(t)$ is the characteristic function of $X_1$.