(a) Let $\left(N_{t}\right)_{t \geqslant 0}$ be a Poisson process of rate $\lambda>0$. Let $p$ be a number between 0 and 1 and suppose that each jump in $\left(N_{t}\right)$ is counted as type one with probability $p$ and type two with probability $1-p$, independently for different jumps and independently of the Poisson process. Let $M_{t}^{(1)}$ be the number of type-one jumps and $M_{t}^{(2)}=N_{t}-M_{t}^{(1)}$ the number of type-two jumps by time $t$. What can you say about the pair of processes $\left(M_{t}^{(1)}\right)_{t \geqslant 0}$ and $\left(M_{t}^{(2)}\right)_{t \geqslant 0}$ ? What if we fix probabilities $p_{1}, \ldots, p_{m}$ with $p_{1}+\ldots+p_{m}=1$ and consider $m$ types instead of two?

(b) A person collects coupons one at a time, at jump times of a Poisson process $\left(N_{t}\right)_{t \geqslant 0}$ of rate $\lambda$. There are $m$ types of coupons, and each time a coupon of type $j$ is obtained with probability $p_{j}$, independently of the previously collected coupons and independently of the Poisson process. Let $T$ be the first time when a complete set of coupon types is collected. Show that

$$\mathbb{P}(T<t)=\prod_{j=1}^{m}\left(1-e^{-p_{j} \lambda t}\right)$$

Let $L=N_{T}$ be the total number of coupons collected by the time the complete set of coupon types is obtained. Show that $\lambda \mathbb{E} T=\mathbb{E} L$. Hence, or otherwise, deduce that $\mathbb{E} L$ does not depend on $\lambda$.