Past exam of the probability course of the University of Oxford / 2017 / 2 / a

The moment-generating function of a real random variable $X$ isfor real $t$ where the expectation is finite. One may regard it as an extended nonnegative function elsewhere. In particular $M_X(0)=1$; finiteness on a neighborhood of zero is the usual hypothesis for recovering moments and using its local expansion.

Differentiating the moment-generating function at zero givesThe second derivative is the second raw moment, so the variance is $M_X^{\prime \prime }(0)-(M_X^{\prime }(0){)}^2$. The assumed exponential integrability justifies these derivatives.

Integrate against the standard normal density by completing the square:Thus the moment-generating function of a standard normal variable is

Let $Y_1,Y_2,\dots$ be real independent and identically distributed random variables with finite expected value $\mu$ and variance $0<{\sigma }^2<\infty$. The central limit theorem statesEquivalently, the cumulative distribution functions of these normalized sums converge at every real point to the standard normal distribution function. The theorem itself does not require a finite moment-generating function.