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.

Solution

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

 Solution (source code)