Moment equations of a linear birth-death process
= Moment equations of a linear birth-death process
For per-capita birth and death rates $B$ and $D$, the rates are $b_n=Bn$ and $d_n=Dn$. If $\mu=\mathbb EN_t$ and $\sigma^2=\operatorname{var}N_t$, then
$$
\mu'=(B-D)\mu,
\qquad
(\sigma^2)'=2(B-D)\sigma^2+(B+D)\mu.
$$
The <expected value> sees only the net growth rate $B-D$, whereas the <variance> also sees the total event rate $B+D$.