Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics III / 5 / a

For a nonnegative integer-valued random variable, its probability generating function isThe convention $s^0=1$, including at $s=0$, gives $G(0)=p(0)$. The power series has absolute convergence for every real $|s|\le 1$, since ${\sum }_{j}p(j)|s{|}^j\le 1$. More generally its real domain as a finite expected value is $\{s\in \mathbb{R}:{\sum }_{j}p(j)|s{|}^j<\infty \}$, which may extend beyond $[-1,1]$. No extension beyond that guaranteed interval is automatic.

Under the extra hypothesis that $G(s)$ exists for every real $s$, its power series has infinite radius of convergence. Termwise differentiation of a power series is therefore valid to every finite order. For $k\ge 1$,At $s=1$ this is the expected value of the falling factorial of $X$:In particular $\boxed{G^{\prime }(1)=\mathbb{E}[X]}$. The terms with $j<k$ vanish because their falling factorials contain a zero factor. These factorial moments are finite under the stated hypothesis.

Since $X^2=X(X-1)+X$, the second moment is $\mathbb{E}[X^2]=G^{\prime \prime }(1)+G^{\prime }(1)$. Subtracting the squared mean gives

For the specified Poisson distribution, the exponential series gives the probability generating functionEvery derivative of $G$ is again $e^{s-1}$, so the factorial moments and variance areAlso $\mathbb{E}[X]=1$.