(i) The fire alarm in Mill Lane is set off at random times. The probability of an alarm during the time-interval $(u, u+h)$ is $\lambda(u) h+o(h)$ where the 'intensity function' $\lambda(u)$ may vary with the time $u$. Let $N(t)$ be the number of alarms by time $t$, and set $N(0)=0$. Show, subject to reasonable extra assumptions to be stated clearly, that $p_{i}(t)=P(N(t)=i)$ satisfies

$$p_{0}^{\prime}(t)=-\lambda(t) p_{0}(t), \quad p_{i}^{\prime}(t)=\lambda(t)\left\{p_{i-1}(t)-p_{i}(t)\right\}, \quad i \geqslant 1 .$$

Deduce that $N(t)$ has the Poisson distribution with parameter $\Lambda(t)=\int_{0}^{t} \lambda(u) d u$.

(ii) The fire alarm in Clarkson Road is different. The number $M(t)$ of alarms by time $t$ is such that

$$P(M(t+h)=m+1 \mid M(t)=m)=\lambda_{m} h+o(h),$$

where $\lambda_{m}=\alpha m+\beta, m \geqslant 0$, and $\alpha, \beta>0$. Show, subject to suitable extra conditions, that $p_{m}(t)=P(M(t)=m)$ satisfies a set of differential-difference equations to be specified. Deduce without solving these equations in their entirety that $M(t)$ has mean $\beta\left(e^{\alpha t}-1\right) / \alpha$, and find the variance of $M(t)$.