Factorial Moments from PGF Derivatives

Why Differentiate a Probability-Generating Function?

For a nonnegative integer-valued random variable, the probability-generating function (PGF) packages all point probabilities into one function. Its derivatives reveal weighted sums of those probabilities. The weights are not the ordinary powers of the count: differentiating \(s^k\) \(r\) times produces the falling factorial \(k(k-1)\cdots(k-r+1)\). This makes PGF derivatives a direct tool for calculating factorial moments.

There is one important boundary issue. A PGF is always defined at \(s=1\), but it need not be differentiable there with a finite derivative. We will first differentiate at \(0<s<1\), where power-series differentiation is valid, and then justify the limit to \(1\) under a precise finiteness condition.

Factorial Moments and PGF Derivatives

The convention \(0^0=1\) is used in the PGF term for \(X=0\), so \(G(0)=P(X=0)\). For \(0<s<1\), the series for \(G\) is a power series that converges absolutely. It can therefore be differentiated term by term at interior points. The boundary \(s=1\), however, requires a separate argument.

Why the Boundary Identity Holds

The formula inside the unit interval follows by differentiating each power \(s^k\). To reach the endpoint, take \(s\) upwards to \(1\). Every summand in the derivative series is nonnegative and increases to \((k)_rP(X=k)\). The Monotone Convergence Theorem therefore gives the extended-valued limit. When the resulting sum is finite, the successive derivatives also have finite limits, which ensures that these limits are genuine one-sided derivatives at the endpoint.

Here is the endpoint argument in more detail. Write \(p_k=P(X=k)\), and for \(0<s<1\) let \(H_j(s)=\sum_{k=j}^{\infty}(k)_j p_k s^{k-j}\) for \(1\leq j\leq r\), with \(H_0(s)=G(s)\). If \(E[(X)_r]<\infty\), then each \(E[(X)_j]\) for \(j<r\) is finite: for \(k\geq r\), \((k)_j\leq(k)_r\), and the remaining values of \(k\) form a finite set. Monotone Convergence gives a finite limit \(H_j(1)=\sum_{k=j}^{\infty}(k)_j p_k\) for every \(j\leq r\). Thus each \(H_j\) extends continuously to \(1\).

On any interval inside \((0,1)\), termwise differentiation gives \(H_j'=H_{j+1}\) for \(j<r\). The Fundamental Theorem of Calculus, followed by taking the upper endpoint to \(1\), now gives \[ H_j(1)-H_j(s)=\int_s^1 H_{j+1}(t)\,dt. \] The integral is well-defined because \(H_{j+1}\) has a finite continuous extension to \(1\). Dividing by \(1-s\) and letting \(s\uparrow1\), continuity implies that the average of \(H_{j+1}\) over \([s,1]\) tends to \(H_{j+1}(1)\). Hence the left derivative of \(H_j\) at \(1\) is \(H_{j+1}(1)\). Applying this successively identifies \(G^{(r)}(1-)\) with \(H_r(1)=E[(X)_r]\). \(\square\)

If the factorial moment is infinite, the Monotone Convergence argument still applies, but it says the interior derivatives tend to \(+\infty\); it does not give a finite derivative at \(1\). This distinction is why the finiteness condition belongs in the boundary identity.

Worked Applications

Common Pitfall: Differentiating at One Without Checking Finiteness

A PGF’s value at \(1\) is always \(G(1)=1\), but that fact does not imply that its derivative at \(1\) is finite. For a concrete example, let \(P(X=k)=1/[k(k+1)]\) for \(k\geq1\). These probabilities sum to one because \[ \sum_{k=1}^{N}\frac{1}{k(k+1)} =\sum_{k=1}^{N}\left(\frac{1}{k}-\frac{1}{k+1}\right) =1-\frac{1}{N+1}\longrightarrow1. \] Yet \[ E[X]=\sum_{k=1}^{\infty}k\frac{1}{k(k+1)} =\sum_{k=1}^{\infty}\frac{1}{k+1}=\infty. \] Here \(G'(s)=\sum_{k=1}^{\infty}s^{k-1}/(k+1)\) for \(0<s<1\), and \(G'(s)\) tends to infinity as \(s\uparrow1\).

The mistaken reasoning is: “The PGF is defined at \(1\), so I can differentiate its series there and obtain a finite answer.” Being able to evaluate a series at an endpoint does not justify differentiating it there. The correction is to differentiate at an interior point first, then use the theorem: the boundary limit equals the factorial moment, and it is finite only when that factorial moment is finite.

Why This Identity Is Useful

The derivative formula converts a distributional calculation into a calculation with a function. Once a PGF has a manageable algebraic or exponential form, differentiating it can be much simpler than summing a probability series term by term. In applications involving counts, derivatives at one provide expected falling-factorial counts, with the order \(r\) specifying how many successive factors appear.

The identity also provides a useful check on calculations. A proposed \(r\)th factorial moment must be nonnegative, and it must be zero for a distribution supported only on values below \(r\). Most importantly, the endpoint condition makes clear what a derivative calculation has established: an interior derivative formula is valid for \(s<1\), while its value at one requires a finite factorial moment if a finite boundary derivative is claimed.