Past exam of the mathematics course of the University of Cambridge / 2012 / ia / Paper 2 / 9F / ii / Solution

Finiteness of the moment-generating function on an open neighborhood of zero gives its differentiability there. Using the permitted interchange of differentiation and expectation,Hence $M_X(0)=1$, $M_X^{\prime }(0)=\mu$ and $M_X^{\prime \prime }(0)=s^2$. The second-order Taylor expansion with its Peano remainder givesHere $s^2$ is the raw second moment, not the variance: $\mathrm{Var}(X)=s^2-{\mu }^2$. This distinction matters before specializing to a centered variable.