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Consider a Birth-and-Death Process with individual birth rates $\lambda(t)$ and individual death rates $\mu(t)$, starting at $n_0$. My question is if there is a formula for something like

$$\mathbb P_{n_0} (\text{more than one birth happens up to time $t\ge0)$}$$

or—in a similar spirit—the expected time until the first birth happens. I know that the time spend at a position $n$ is distributed exponentially with parameter $n(\lambda(t)+\mu(t))$ but I couldn't find anything concerning probabilities where we look only at e.g. births.

user568706
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  • What exactly do you mean by "individual birth rates $\lambda(t)$, $\mu(t)$"? That the process is the superposition of two birth processes with rates $\lambda(t)$ and $\mu(t)$? Recall that the superposition of Poisson processes is again Poisson. – Math1000 Jun 12 '18 at 07:04
  • @Math1000 Sorry, that was just a typo. I meant to write "with individual birth rates $\lambda(t)$ and death rates $\mu(t)$". Meaning, that for $n$-Individuals at time $t$ we have a birth rate $n\lambda(t)$ and a death rate $n\mu(t)$ – user568706 Jun 12 '18 at 07:15
  • If the birth and death rates vary with $t$, then it would be necessary to specify $\lambda(t)$ and $\mu(t)$. Or may we assume that $\lambda(t)=\lambda$ and $\mu(t)=\mu$ for all $t$? – Math1000 Jun 19 '18 at 21:15

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