= Distribution of a critical linear birth-death population
{title2=$P_0=\frac{\lambda t}{1+\lambda t},\quad P_n=\frac{(\lambda t)^{n-1}}{(1+\lambda t)^{n+1}}\ (n\geq1)$}
For equal individual birth and death rates $\lambda$ and one initial individual, the <probability generating function> is $F(z,t)=1-(1-z)/[1+\lambda t(1-z)]$. It solves $F_t=\lambda(z-1)^2F_z$ with $F(z,0)=z$, and its geometric expansion gives the displayed <probabilities>. The <mean> remains one and the <variance> is $2\lambda t$, although extinction occurs with <probability> tending to one. The surviving fraction $1/(1+\lambda t)$ has conditional <mean> population $1+\lambda t$.
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