Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / M / M4 / a

The moment generating function is $\boxed{M_X(\theta )=\mathbb{E}[e^{\theta X}]}$ for real $\theta$ where this expectation is finite. Existence on an open interval about zero determines the distribution and permits differentiation there to obtain moments, when the usual local integrability conditions hold.