(a) What is meant by a birth process $N=(N(t): t \geqslant 0)$ with strictly positive rates $\lambda_{0}, \lambda_{1}, \ldots ?$ Explain what is meant by saying that $N$ is non-explosive.

(b) Show that $N$ is non-explosive if and only if

$$\sum_{n=0}^{\infty} \frac{1}{\lambda_{n}}=\infty$$

(c) Suppose $N(0)=0$, and $\lambda_{n}=\alpha n+\beta$ where $\alpha, \beta>0$. Show that

$$\mathbb{E}(N(t))=\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right) .$$