Geometric Offspring Laws and Their Generating Functions

Two Geometric Laws for Offspring Counts

A geometric distribution can describe the number of failures before the first success, or the number of trials up to and including the first success. These differ by one, so using the word “geometric” without specifying the support can change an offspring model substantially. Here we derive both conventions and compare their generating functions, moments, and probabilities.

Work on a probability space \((\Omega,\mathcal{F},\mathbb{P})\). In a Galton–Watson process, an individual’s offspring count is a random variable \(\xi\) taking values in \(\mathbb{N}_0=\{0,1,2,\ldots\}\); the population process \((Z_n)_{n\ge0}\) is indexed by discrete generation \(n\) and has state space \(\mathbb{N}_0\). This tutorial concerns the possible laws for \(\xi\). We use a parameter \(p\) with \(0<p\le1\), and write \(q=1-p\). The endpoint \(p=1\) is included because it gives a valid degenerate law; \(p=0\) is excluded because the proposed probabilities would not sum to one.

These are probability distributions: for \(0<p<1\), the geometric-series identity \(\sum_{k=0}^{\infty}q^k=1/(1-q)=1/p\) gives total mass one in the first convention, and \(\sum_{k=1}^{\infty}q^{k-1}=1/p\) gives total mass one in the second. When \(p=1\), \(q=0\): the first law puts probability one at zero, while the second puts probability one at one.

Generating Functions and Moments

For the nonnegative-integer convention, summing the probability mass function gives \(G(s)=p\sum_{k=0}^{\infty}(qs)^k\). For the positive-integer convention the corresponding sum starts at \(k=1\), so one factor of \(s\) is taken out. These series converge for \(0\le s\le1\), since \(q<1\).

The second formula is \(s\) times the first. This reflects the shift in support: if \(Y\) has the nonnegative-integer geometric law, then \(1+Y\) has the positive-integer geometric law. Indeed, for every \(k\ge1\), \(\mathbb{P}(1+Y=k)=\mathbb{P}(Y=k-1)=p q^{k-1}\).

Proof. For \(0<p<1\), differentiate the displayed PGFs at \(s<1\), where their geometric series and derivatives converge. The nonnegative-integer PGF has derivatives \(G'(s)=pq/(1-qs)^2\) and \(G''(s)=2pq^2/(1-qs)^3\). The positive-integer PGF has derivatives \(G'(s)=p/(1-qs)^2\) and \(G''(s)=2pq/(1-qs)^3\). Since \(1-q=p\), evaluating at \(s=1\) gives the means and second factorial moments stated in the theorem. For any nonnegative integer-valued random variable with finite second moment, \(\mathbb{E}[\xi^2]=\mathbb{E}[\xi(\xi-1)]+\mathbb{E}[\xi]\). Thus the nonnegative-integer variance is \(2q^2/p^2+q/p-(q/p)^2=q/p^2\). The positive-integer variance is \(2q/p^2+1/p-(1/p)^2=q/p^2\), using \(q=1-p\). When \(p=1\), the two laws are constant, respectively zero and one, so their means, variances, and factorial moments are \(0,0,0\) and \(1,0,0\), as the formulas give. \(\square\)

For general PGFs, the derivative identities require care at \(s=1\). For \(0\le s<1\), termwise differentiation gives \(G'(s)=\sum_{k\ge1}k\mathbb{P}(\xi=k)s^{k-1}\) and \(G''(s)=\sum_{k\ge2}k(k-1)\mathbb{P}(\xi=k)s^{k-2}\). As \(s\) increases to one, the nonnegative terms increase, so their limits are the corresponding moments, possibly infinite. In particular, \(G'(1)=\mathbb{E}[\xi]\) when the mean is finite, and \(G''(1)=\mathbb{E}[\xi(\xi-1)]\) when the second factorial moment is finite. Differentiability once at \(1\) does not by itself imply a finite second factorial moment. For the two geometric PGFs above, the rational formulas have finite derivatives of every order near \(1\), so the moment calculations are valid.

Worked Calculations

Parameter Identification and a Common Mistake

Proof. In the first convention \(m=q/p=(1-p)/p=1/p-1\), so \(m+1=1/p\) and \(p=1/(m+1)\). Since \(0<p\le1\), this gives exactly \(m\ge0\). In the second convention \(m=1/p\), so \(p=1/m\); the condition \(0<p\le1\) is equivalent to \(m\ge1\). Conversely, for every mean in the stated range, the displayed parameter belongs to \((0,1]\) and substitution into the relevant mean formula returns \(m\). Thus the parameter is both necessary and sufficient. \(\square\)

Common mistake. For a positive-integer geometric law with \(p=\tfrac12\), using \(p q^k\) at \(k=0,1,\ldots\) incorrectly assigns probability \(\tfrac12\) to zero offspring. That contradicts the positive support. The correct formula is \(p q^{k-1}\) for \(k\ge1\): it gives \(\mathbb{P}(\xi=1)=\tfrac12\), \(\mathbb{P}(\xi=2)=\tfrac14\), and, in general, zero probability at \(k=0\). Its total mass is \(\sum_{k=1}^{\infty}\tfrac12(\tfrac12)^{k-1}=1\).

When specifying a geometric offspring model, state its support along with its parameter. The nonnegative version permits no offspring and has mean \(q/p\); the positive version forces at least one offspring and has mean \(1/p\). Their PGFs make the shift explicit, and their probability formulas allow individual offspring probabilities to be computed without ambiguity.