Let $X_1,...,X_n$ be i.i.d Bernoulli(q) random variables. If we let $T_n$ denote the total number of occurrences to a chosen set V inside of our state space, what is the best way to go about determining the m.g.f of $T_n$? Would it be something like below? Where $[z^k]$ denotes the coefficient of $z^k$ in some polynomial. (i.e. $[z^2](1-3z^2) = -3$)
$\mathbb E[z^{t_n}] = \sum_{k=0}^n P(T_k=k)*(z^k)$