Spatial Poisson Processes

Abstract

Spatial Poisson process are typically used to model the random scattering of configuration points within a plane or a three-dimensional space X. In case $X = {\mathord {\mathbb R}}_+$ is the real half line, these random points can be identified with the jump times $(T_k)_{k \ge 1}$ of the standard Poisson process $(N_t)_{t\in {\mathord {\mathbb R}}_+}$ introduced in Sect. 9.1. However, in contrast with the previous chapter, no time ordering is a priori imposed here on the index set X.

Exercises

Exercise 11.1

Consider a standard Poisson process $(N_t)_{t\in {\mathord {\mathbb R}}_+}$ on ${\mathord {\mathbb R}}_+$ with intensity $\lambda = 2$ and jump times $(T_k)_{k\ge 1}$. Compute

$$ {\mathrm{I}}\!{\mathrm{E}}[ T_1 + T_2 + T_3 \mid N_2 = 2 ]. $$

Exercise 11.2

Consider a spatial Poisson process on ${\mathord {\mathbb R}}^2$ with intensity $\lambda = 0.5$ per square meter. What is the probability that there are 10 events within a circle of radius 3 m.

Exercise 11.3

Some living organisms are distributed in space according to a Poisson process of intensity $\theta = 0.6$ units per mm$^3$. Compute the probability that more than two living organisms are found within a 10 mm$^3$ volume.

Exercise 11.4

Defects are present over a piece of fabric according to a Poisson process with intensity of one defect per piece of fabric. Both halves of the piece is checked separately. What is the probability that both inspections record at least one defect?

Exercise 11.5

Let $\lambda > 0$ and suppose that N points are independently and uniformly distributed over the interval [0, N]. Determine the probability distribution for the number of points in the interval $[0,\lambda ]$ as $N \rightarrow \infty$.

Exercise 11.6

Suppose that X(A) is a spatial Poisson process of discrete items scattered on the plane ${\mathbb R}^2$ with intensity $\lambda = 0.5$ per square meter. We let

$$ D((x,y),r) = \{ (u, v) \in {\mathbb R}^2 \ : \ (x-u)^2+(y-v)^2 \le r^2 \} $$

denote the disc with radius r centered at (x, y) in ${\mathbb R}^2$. No evaluation of numerical expressions is required in this exercise.

• (a) What is the probability that 10 items are found within the disk D((0, 0), 3) with radius 3 meters centered at the origin?

• (b) What is the probability that 5 items are found within the disk D((0, 0), 3) and 3 items are found within the disk D((x, y), 3) with $(x, y)=(7,0)$?

• (c) What is the probability that 8 items are found anywhere within $D((0,0), 3) \bigcup D((x,y), 3)$ with $(x, y)=(7,0)$?

• (d) Given that 5 items are found within the disk D((0, 0), 1), what is the probability that 3 of them are located within the disk D((1 / 2, 0), 1 / 2) centered at (1 / 2, 0) with radius 1 / 2?