Birth-death master equation

For birth rates $b_n$, death rates $d_n$, and state probabilities $P_n$,

Since births and deaths change the state by $+1$ and $-1$,

For per-capita birth and death rates $B$ and $D$, the rates are $b_n=Bn$ and $d_n=Dn$. If $\mu =\mathbb{E}{N}_t$ and ${\sigma }^2=\mathrm{var}{N}_t$, thenThe expected value sees only the net growth rate $B-D$, whereas the variance also sees the total event rate $B+D$.

The $r$th falling factorial moment is $\mathbb{E}[(N{)}_r]$, where $(N{)}_r=N(N-1)\cdots (N-r+1)$.

If $N$ is Poisson with mean $\lambda$, then $\mathbb{E}[(N{)}_r]={\lambda }^r$; in particular, $\mathbb{E}N=\lambda$ and $\mathbb{E}[N(N-1)]={\lambda }^2$.