Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability / 8 / a

The probability generating function of a nonnegative integer-valued random variable isUse $s^0=1$, including at $s=0$. The power series converges absolutely for every real $s\in [-1,1]$, since ${\sum }_{j}p(j)|s{|}^j\le 1$. In particular it is always defined on the usual probability-generating interval $[0,1]$. It may also be defined for $|s|>1$, depending on the radius of convergence and convergence at its endpoints; the precise real domain of finite expectation is $\{s\in \mathbb{R}:{\sum }_{j}p(j)|s{|}^j<\infty \}$.

By hypothesis the power series converges at every positive real argument, so its radius of convergence is infinite. Termwise differentiation of a power series is therefore valid in a neighborhood of $s=1$. For $k\ge 1$,At $s=1$, the right side is the expectation of the falling factorial of $X$:The special case $k=1$ givesFor $k=0$, take the empty product as $1$; the equality becomes $G(1)=1$. These are the factorial moments.

Since $X^2=X(X-1)+X$, the probability generating function identities giveSubtracting the squared mean yields the variance

For the Poisson distribution with parameter $1$, the exponential series givesEvery derivative is $G^{(k)}(s)=e^{s-1}$, so the factorial moments areIn particular $G^{\prime }(1)=G^{\prime \prime }(1)=1$, and the variance is