Past exam of the probability course of the University of Oxford / 2015 / 1 / b

The moment-generating function of a real random variable $X$ isfor real $t$ for which this expected value is finite. Because the integrand is nonnegative, the extended expected value is always defined, but it may equal $+\infty$. We always have $M_X(0)=1$; the usual uniqueness and convergence theorems require finiteness on an open interval containing zero. When this condition holds, derivatives at zero recover the moments, $M_X^{(r)}(0)=\mathbb{E}[X^r]$.

Using the probability mass function of the Poisson distribution and the power series of the exponential function,This moment-generating function of a Poisson distribution is finite for every real $t$. It also covers $\mu =0$, when $X=0$ with probability one and the moment-generating function is identically one.

Complete the square in the integral against the standard normal density:The remaining integral is one because its integrand is the probability density function of the normal distribution $N(t,1)$. Hence the moment-generating function of a standard normal variable is finite on the whole real line.