# Solutions to a stochastic birth-death-immigration process

A population is undergoing a birth-death-immigration process. That is, the population size can increase by virtue of birth and immigration, and can decrease by virtue of death. The birth rate is $\lambda$, the death rate is $\mu$, and the immigration rate is $\alpha$. If we let $X(t)$ be a random variable representing the population size at any time $t\ge 0$ and let $n_0$ be the size of the population at time $t = 0$, the function $P_n(t)$ is defined as:

$P[X(t) = n | X(0) = n_0]$.

Using the probability generating function,

$\phi(s,t) = \sum_{n=0}^\infty P_n(t)s^n$

we are given that when $\lambda \ne \mu$,

$\phi(s,t) = [{\lambda-\mu\over \lambda s+\lambda(1-s)exp(\lambda-\mu)-\mu}]^{\alpha\over\lambda}$.

With all that being said, my question is this: how do we find $P_0(t)$ and $P_1(t)$ by using the above solution to the probability generation function?

• Using an internet search, I found www.csupomona.edu/~rjswift/BDI.pdf‎, which is probably the source of the question? You might want to edit the question accordingly, especially the missing exponent $\alpha/\lambda$. – Thomas Rippl Apr 24 '14 at 7:48
• Corrected, thank you. – Weston Apr 24 '14 at 23:52

## 1 Answer

$$P_0(t)=\phi(0,t) \qquad P_1(t)=\frac{\partial\phi}{\partial s}(0,t)$$ $$P_n(t)=\frac1{n!}\,\frac{\partial^n\phi}{\partial s^n}(0,t)$$