Past exam of the Probability, statistics and options II course of the University of Oxford / 2011 / M / M4 / a / i / Solution

The moment-generating function is $M_X(t)=\mathbb{E}[e^{tX}]$ for real $t$ where this expectation is finite. Usually one requires finiteness on a neighbourhood of zero when using it to characterize a distribution. The characteristic function isIt always exists because the integrand has absolute value one. In both definitions the expectation is taken with respect to the distribution of $X$.