(i) Define an inhomogeneous Poisson process with rate function $\lambda(u)$.

(ii) Show that the number of arrivals in an inhomogeneous Poisson process during the interval $(0, t)$ has the Poisson distribution with mean

$$\int_{0}^{t} \lambda(u) \mathrm{d} u$$

(iii) Suppose that $\Lambda=\{\Lambda(t), t \geqslant 0\}$ is a non-negative real-valued random process. Conditional on $\Lambda$, let $N=\{N(t), t \geqslant 0\}$ be an inhomogeneous Poisson process with rate function $\Lambda(u)$. Such a process $N$ is called a doubly-stochastic Poisson process. Show that the variance of $N(t)$ cannot be less than its mean.

(iv) Now consider the process $M(t)$ obtained by deleting every odd-numbered point in an ordinary Poisson process of rate $\lambda$. Check that

$$\mathbb{E} M(t)=\frac{2 \lambda t+e^{-2 \lambda t}-1}{4}, \quad \operatorname{Var} M(t)=\frac{4 \lambda t-8 \lambda t e^{-2 \lambda t}-e^{-4 \lambda t}+1}{16}$$

Deduce that $M(t)$ is not a doubly-stochastic Poisson process.