In here, we obtained the Poisson distribution by successively solving a differential-difference equation. While getting successive solutions of the Poisson distribution was easy and simple, the same thing can’t be said in general. In this note, we discuss another method of obtaining probability distributions. Let $P(x,t)=\sum_{n=0}^\infty p_n(t)x^n$. $P(x,t)$ is called a probability generating function for the probability distribution $p_n(t)$. Recall the differential-difference equation (2) in here: $$\frac{dp_n(t)}{dt}=\lambda\{p_{n-1}(t)-p_n(t)\},\ n\geq 0$$ with $p_{-1}(t):=0$ and $p_0(0)=1$. Multiplying the equation by $x^n$ and then summing over $n$ from $n=1$ to $\infty$, we obtain the differential equation \begin{equation}\label{eq:genfn}\frac{\partial P(x,t)}{\partial t}=\lambda(x-1)P(x,t)\end{equation} with $P(x,0)=1$. \eqref{eq:genfn} only contains derivative with respect to $t$, so it is a separable equation. Its solution is given by $$P(x,t)=e^{\lambda t(x-1)}$$ From this we easily obtain the Poisson distribution $$p_n(t)=\frac{e^{-\lambda t}(\lambda t)^n}{n!}$$

References:

1. Norman T. J. Bailey, The Elements of Stochastic Processes with Applications to the Natural Sciences, John Wiley & Sons, Inc., 1964.