Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 27K / a

Table of contents
- Solution a

Solution

For every Borel set

A \subseteq R^{d}

, let

N (A) = # (Π \cap A)

be its random point count. A Non-homogeneous Poisson point process with intensity function of a point process

λ

is characterized by:

$N (A)$ is finite almost surely for every bounded Borel set $A$ ;
for every bounded Borel set $A$ ,

N (A) \sim Poisson (Λ (A)), Λ (A) = \int_{A} λ (x) d x;

the counts $N (A_{1}), \dots, N (A_{m})$ are independent random variables whenever $A_{1}, \dots, A_{m}$ are pairwise disjoint bounded Borel sets.

Because

Λ

has a density with respect to Lebesgue measure, it is atomless, so the process is simple: almost surely no location occurs more than once.

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version