Offspring distribution of a branching process ($p_k=\mathbb{P}(\xi =k),F(s)={\sum }_{k\ge 0}{p}_k{s}^k$)

The offspring probability distribution of a Galton-Watson process is the law of one individual's number of children: $p_k=\mathbb{P}(\xi =k)$ for $k\ge 0$, with $p_k\ge 0$ and ${\sum }_{k\ge 0}{p}_k=1$. Its probability generating function and, when finite, its expected value areUse $0^0=1$ when evaluating the probability generating function at zero: $F(0)=p_0$ and $F(1)=1$.

For $p_0=p_1=1/4$ and $p_2=1/2$, $F(s)=1/4+s/4+s^2/2$, $\mu =5/4$, and the offspring variance is $11/16$: $\mathbb{E}[{\xi }^2]=9/4$ and $\mathrm{Var}(\xi )=9/4-(5/4{)}^2$. The generation probability mass functions pictured under Galton-Watson process come from repeatedly composing this probability generating function. They describe the whole generation, whereas $(p_k)$ describes one individual's reproduction. ^[ref]

The whole offspring distribution matters for extinction. Another law, $p_0=3/8$ and $p_2=5/8$, has the same offspring mean $5/4$ but branching extinction probability $3/5$, since $3/8+5s^2/8=s$ has roots of a polynomial $3/5$ and $1$. Consequently the two laws have the same expected generation size in a Galton-Watson process at each generation but different survival chances. A binomial branching process and a Poisson branching process likewise specify different offspring distributions; a common mean does not make their probability generating functions identical.

For $0<\alpha ,\beta <1$, this probability generating function has probabilities $p_0=1-\alpha$ and $p_k=\alpha \beta (1-\beta )\cdots (k-1-\beta )/k!$ for $k\ge 1$, where the product after $\beta$ is empty for $k=1$. All probabilities are positive and sum to one, while the expected value is infinite. In a Galton-Watson process starting with one individual, the generation-$n$ probability generating function is $G_n(s)=1-{\alpha }^{(1-{\beta }^n)/(1-\beta )}(1-s{)}^{{\beta }^n}$. The branching extinction probability is $1-{\alpha }^{1/(1-\beta )}$, obtained by taking the limit of $G_n(0)$ and using that extinction by generation $n$ is an increasing sequence of events.