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