moment-generating function is well defined

Question

Let $X$ be a random variable, $t \in R, M_X(t) = E(exp(t*i*X))$ the moment-generating function. Proof that $M_X$ is well defined.

I assume I have to proof that if the random variables $X_1$ and $X_2$ have the same distribution, then $E(exp(t*i*X_1)) = E(exp(t*i*X_2))$ should hold. But I don't know how to show it.

Answer 1

1. This is normally called the characteristic function.

2. It wants you to show that $E(\exp{(itX)})$ is finite for any probability distribution. This amounts to $$ \lvert E(e^{itX}) \rvert = \left| \int_{\Omega} e^{itx} dP(x) \right| \leq \int_{\Omega} \lvert e^{itx} \rvert \, dP(x) \leq \int_{\Omega} 1 \, dP(x) = 1. $$