Distribution of a critical linear birth-death population ($P_0=\frac{\lambda t}{1+\lambda t},P_n=\frac{(\lambda t{)}^{n-1}}{(1+\lambda t{)}^{n+1}}(n\ge 1)$)

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 $F_t=\lambda (z-1{)}^2{F}_z$ 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 $1+\lambda t$.