# Spatial Poisson Processes, and Relatives of Poisson Processes

*You should solve these problems with as few calculations as possible, relying on properties of Poisson processes as much as possible.*

- In a 2-d spatial Poisson process with intensity \(\lambda\), let \(X\) represent the
*nearest neighbor distance*, that is, the distance between an arbitrary point and the point of the process closest to it.

- Find an expression for \(\text{P}(X > x)\), for \(x>0\).
- Find an expression for the probability density function (pdf) of \(X\).
- Find an expression for \(\text{E}(X)\).

Starting at 9 a.m., customers arrive at a store according to a nonhomogeneous Poisson process with intensity function \(\lambda(t) = t^2\), for \(t>0\), where the time is measured in hours. Find the probability mass function of the number of customers who enter the store by noon.

Suppose points are distributed in a 2-d region centered at the origin according to a nonhomogeneous, spatial Poisson process \(\{N_A\}\) with intensity function \[ \lambda(x, y) = e^{-(x^2 + y^2)} \] Let \(R\) be the distance from the origin to the nearest point. Compute \(\text{P}(R > 1)\).