Bernoulli and Binomial Offspring Models

Two Basic Offspring Models

In a Galton–Watson process, each individual produces a random number of children, independently of the other individuals and of reproduction in other generations. Two useful models arise when reproduction is described by a fixed number of opportunities: an individual may have at most one child, or may have at most \(r\) children. The first gives a Bernoulli offspring law; the second gives a binomial offspring law. Their simple formulas make them useful for checking calculations and for representing limited reproduction opportunities.

Work on a probability space \((\Omega,\mathcal{F},\mathbb{P})\). The generation sizes \((Z_n)_{n\ge0}\) take values in \(S=\{0,1,2,\ldots\}\), with discrete generation index \(n\). Write \(\xi_{n,i}\) for the number of children of the \(i\)th individual in generation \(n\). In the Galton–Watson model these offspring counts are independent and identically distributed. The population update is \(Z_{n+1}=\sum_{i=1}^{Z_n}\xi_{n,i}\), where a sum over no individuals is zero. Here the focus is the offspring law and the distribution of \(Z_1\) for a fixed initial size \(Z_0=z\), not the behavior of later generations.

For a nonnegative integer-valued offspring count \(\xi\), its probability generating function is \(G(s)=\mathbb{E}[s^\xi]\), for \(0\le s\le1\). Thus \(G(s)=\sum_k\mathbb{P}(\xi=k)s^k\). The earlier material on offspring distributions and probability generating functions establishes this encoding: the coefficient of \(s^k\) is the probability of \(k\) offspring. For both models here, the sum is finite, so \(G\) is a polynomial and its derivatives at \(1\) are well-defined.

Generating Functions and Moments

Proof. Write \(a_k=\mathbb{P}(\xi=k)\), so \(G(s)=\sum_k a_k s^k\), with only finitely many nonzero terms. Differentiating the polynomial gives \(G'(1)=\sum_k k a_k=\mathbb{E}[\xi]\), and \(G''(1)=\sum_k k(k-1)a_k=\mathbb{E}[\xi(\xi-1)]\). Since \(\xi^2=\xi(\xi-1)+\xi\), it follows that \(\mathbb{E}[\xi^2]=G''(1)+G'(1)\). Subtracting \((\mathbb{E}[\xi])^2\) from \(\mathbb{E}[\xi^2]\) proves the variance formula. \(\square\)

For Bernoulli reproduction, the only possible values are \(0\) and \(1\). Directly from the probabilities,

Its derivatives are \(G'(s)=p\) and \(G''(s)=0\). The theorem therefore gives mean \(p\) and variance \(p-p^2=p(1-p)\). These expressions include the boundary cases: if \(p=0\), the individual has no children with probability one; if \(p=1\), the individual has exactly one child with probability one. In either case the variance is zero.

For binomial reproduction, the binomial theorem gives

For \(r\ge1\), differentiation yields \(G'(s)=rp((1-p)+ps)^{r-1}\), so \(G'(1)=rp\), and \(G''(1)=r(r-1)p^2\). The variance formula gives \(r(r-1)p^2+rp-r^2p^2=rp(1-p)\). If \(r=0\), the offspring count is identically zero, and the same mean and variance formulas give zero. At \(p=0\) the count is also identically zero; at \(p=1\) it is identically \(r\). Both boundary cases have variance zero.

Several Initial Individuals

When \(Z_0=z\) is fixed, the first generation pools the offspring of \(z\) independent individuals. For Bernoulli reproduction, each of those individuals has one opportunity, so \(Z_1\) is binomial with \(z\) opportunities and success probability \(p\). For binomial offspring with \(r\) opportunities per individual, there are \(rz\) independent opportunities in total. This aggregation gives a direct way to compute the first-generation distribution.

Proof. Each of the \(z\) individuals has \(r\) independent opportunities, and every opportunity succeeds with probability \(p\). Independence of offspring across individuals means all \(rz\) opportunities are independent. The total \(Z_1\) is the number of successful opportunities. For \(0\le k\le rz\), any specified set of \(k\) successful opportunities and \(rz-k\) failures has probability \(p^k(1-p)^{rz-k}\), and there are \(\binom{rz}{k}\) such sets. Summing these disjoint possibilities gives the stated probability. If \(rz=0\), there are no opportunities, so the total is zero. When \(p=0\), all opportunities fail, and when \(p=1\), all succeed; these give \(Z_1=0\) and \(Z_1=rz\), respectively. \(\square\)

Equivalently, for \(rz>0\), the generating function of \(Z_1\) is \(((1-p)+ps)^{rz}\). This follows by multiplying the generating function \((1-p)+ps\) for each independent opportunity. The coefficients of the resulting polynomial are exactly the first-generation probabilities in the theorem. This is a statement about \(Z_1\) for fixed \(Z_0\); it does not require a formula for later generation sizes.

Interpreting Parameters and Avoiding a Common Error

The parameter \(p\) is the success probability of one reproduction opportunity, not the expected number of children by itself in the binomial model. The parameter \(r\) counts the available opportunities and sets the largest possible offspring count. Thus the binomial mean is \(rp\), while the Bernoulli mean is \(p\) because Bernoulli reproduction has just one opportunity. The restriction \(r\in\{0,1,2,\ldots\}\) is essential: a non-integer number of opportunities does not define this binomial offspring model. The restriction \(0\le p\le1\) ensures that the stated probabilities are valid.

A common mistake is to treat the second derivative \(G''(1)\) as the variance. It is instead the second factorial moment \(\mathbb{E}[\xi(\xi-1)]\); the mean and its square must also be used. For example, with \(r=3\) and \(p=\tfrac12\), \(G''(1)=3\cdot2\cdot(\tfrac12)^2=\tfrac32\), but the variance is \(\tfrac32+\tfrac32-(\tfrac32)^2=\tfrac34\), not \(\tfrac32\). The correction is to apply \(\operatorname{Var}(\xi)=G''(1)+G'(1)-(G'(1))^2\) after computing both derivatives.

These models also differ in their possible offspring counts. Bernoulli reproduction permits only \(0\) or \(1\), so one individual cannot produce multiple children. Binomial reproduction permits every count from \(0\) through \(r\) when \(0<p<1\) and \(r>0\). At the boundary \(p=0\) it permits only zero children; at \(p=1\) it permits exactly \(r\). A formula for a generating function or moment remains useful at these boundaries, but the resulting law is deterministic rather than variable.