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A Galton-Watson process whose offspring law is a binomial distribution $\mathrm{Bin}(n,p)$ has mean $np$ and the displayed offspring probability generating function. Its branching extinction probability is the smallest root of $q=f(q)$ in $[0,1]$. It dominates a breadth-first exploration of a binomial random graph by allowing each explored vertex up to $n$ independent possible children, including previously exposed or already discovered vertices as phantom children.

Let $q\in (0,1]$ be the branching extinction probability of a Galton-Watson process with offspring probability generating function $f$. Conditional on extinction, the probability of $j$ children is multiplied by $q^j/q$: every child's descendants must also become extinct. Hence the new probability generating function is $f_q$ and its mean is $f^{\prime }(q)$. If $f^{\prime }(q)<1$, the total progeny of a branching process has conditional expected value $1/(1-f^{\prime }(q))$, and unconditionally $\mathbb{E}[T;T<\infty ]=q/(1-f^{\prime }(q))$. The Markov inequality gives a finite-total-progeny tail bound without discarding the survival mass.

For a positive-integer geometric distribution $P(N=j)=p{q}^{j-1}$ with $q=1-p$, the capped time has masses $p{q}^{j-1}$ for $1\le j<k$ and $q^{k-1}$ at $k$. The last mass includes all later arrivals, not only $N=k$. Its probability generating function and expected value areThe expectation follows either by differentiating the PGF or summing the tail probabilities up to $k$.

The probability distribution of the nonnegative integer number of claims over a specified accounting period. Its probability generating function determines the count probabilities. A Poisson distribution gives equal count expected value and variance; a Poisson mixture allows heterogeneity in the underlying risk intensity.

For a Galton-Watson process with one ancestor this is the probability that some generation is empty. It is the smallest fixed point of the offspring probability generating function in $[0,1]$; the complementary survival probability is $1-q$. A branching process conditioned on extinction reweights reproduction using this fixed point.

For the simple symmetric random walk on $\mathbb{Z}$, the renewal equation relates first-return and return probability generating functions. The central-binomial return series is $U(s)=(1-s^2{)}^{-1/2}$, giving $F(s)=1-U(s{)}^{-1}$. Its value tends to one at $s=1$, but its derivative diverges there: the walk has null recurrent states.

A power series encoding a sequence through its coefficients. It turns combinatorial splitting and concatenation rules into algebraic identities. A counting generating function is not necessarily a probability generating function: its coefficients need not sum to one.

In a Galton-Watson process, each individual independently has $j$ children with probability $p_j$. This probability distribution determines its offspring probability generating function, its mean $f^{\prime }(1)$ when finite, and the branching extinction probability. A binomial branching process and a Poisson branching process have different offspring distributions even when their means agree.

The nonnegative integer laws whose probabilities satisfy this recurrence for all $n\ge 1$. Their probability generating function solves $(1-az)G^{\prime }(z)=(a+b)G(z)$. This is the $(a,b,0)$ count class used in the Panjer aggregate algorithm; the count recurrence itself should be distinguished from that aggregate recursion. It includes the Poisson distribution, binomial distribution, and negative binomial distribution with suitable parameters.