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.

1. 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.

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

2. 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.

3. 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 $\text{P}(R>1)$.