Let $M$ be a Poisson random measure of intensity $\lambda$ on the plane $\mathbb{R}^{2}$. Denote by $C(r)$ the circle $\left\{x \in \mathbb{R}^{2}:\|x\|<r\right\}$ of radius $r$ in $\mathbb{R}^{2}$ centred at the origin and let $R_{k}$ be the largest radius such that $C\left(R_{k}\right)$ contains precisely $k$ points of $M$. [Thus $C\left(R_{0}\right)$ is the largest circle about the origin containing no points of $M, C\left(R_{1}\right)$ is the largest circle about the origin containing a single point of $M$, and so on.] Calculate $\mathbb{E} R_{0}, \mathbb{E} R_{1}$ and $\mathbb{E} R_{2}$.

Now let $N$ be a Poisson random measure of intensity $\lambda$ on the line $\mathbb{R}^{1}$. Let $L_{k}$ be the length of the largest open interval that covers the origin and contains precisely $k$ points of $N$. [Thus $L_{0}$ gives the length of the largest interval containing the origin but no points of $N, L_{1}$ gives the length of the largest interval containing the origin and a single point of $N$, and so on.] Calculate $\mathbb{E} L_{0}, \mathbb{E} L_{1}$ and $\mathbb{E} L_{2}$.