A stochastic birth-death process has a master equation given by

$$\frac{d p(n, t)}{d t}=\lambda[p(n-1, t)-p(n, t)]+\beta[(n+1) p(n+1, t)-n p(n, t)]$$

where $p(n, t)$ is the probability that there are $n$ individuals in the population at time $t$ for $n=0,1,2, \ldots$ and $p(n, t)=0$ for $n<0$.

Give the corresponding Fokker-Planck equation for this system.

Use this Fokker-Planck equation to find expressions for $\frac{d}{d t}\langle x\rangle$ and $\frac{d}{d t}\left\langle x^{2}\right\rangle$.

[Hint: The general form for a Fokker-Planck equation in $P(x, t)$ is

$$\frac{\partial P}{\partial t}=-\frac{\partial}{\partial x}(A P)+\frac{1}{2} \frac{\partial^{2}}{\partial x^{2}}(B P)$$

You may use this general form, stating how $A(x)$ and $B(x)$ are constructed. Alternatively, you may derive a Fokker-Plank equation directly by working from the master equation.]