(a) Let $\lambda: \mathbb{R}^{d} \rightarrow[0, \infty)$ be such that $\Lambda(A):=\int_{A} \lambda(\mathbf{x}) d \mathbf{x}$ is finite for any bounded measurable set $A \subseteq \mathbb{R}^{d}$. State the properties which define a (non-homogeneous) Poisson process $\Pi$ on $\mathbb{R}^{d}$ with intensity function $\lambda$.

(b) Let $\Pi$ be a Poisson process on $\mathbb{R}^{d}$ with intensity function $\lambda$, and let $f: \mathbb{R}^{d} \rightarrow \mathbb{R}^{s}$ be a given function. Give a clear statement of the necessary conditions on the pair $\Lambda, f$ subject to which $f(\Pi)$ is a Poisson process on $\mathbb{R}^{s}$. When these conditions hold, express the mean measure of $f(\Pi)$ in terms of $\Lambda$ and $f$.

(c) Let $\Pi$ be a homogeneous Poisson process on $\mathbb{R}^{2}$ with constant intensity 1 , and let $f: \mathbb{R}^{2} \rightarrow[0, \infty)$ be given by $f\left(x_{1}, x_{2}\right)=x_{1}^{2}+x_{2}^{2}$. Show that $f(\Pi)$ is a homogeneous Poisson process on $[0, \infty)$ with constant intensity $\pi$.

Let $R_{1}, R_{2}, \ldots$ be an increasing sequence of positive random variables such that the points of $f(\Pi)$ are $R_{1}^{2}, R_{2}^{2}, \ldots$ Show that $R_{k}$ has density function

$$h_{k}(r)=\frac{1}{(k-1) !} 2 \pi r\left(\pi r^{2}\right)^{k-1} e^{-\pi r^{2}}, \quad r>0$$