# Poisson process in 3D

#### Kiuhnm

##### New Member
I haven't the slightest idea of how to tackle the following problem.

Find the probability density for the distance from an event to its nearest neighbor for a Poisson process in three-dimensional space.

All my book says about "Poisson processes" is
"The Poisson distribution often arises from a model called a Poisson process for the distribution of random events in a set $$S$$, which is typically one-, two-, or three-dimensional, corresponding to time, a plane, or a volume of space. Basically, this model states the if $$S_1,S_2,\ldots,S_n$$ are disjoint subsets of $$S$$, then the numbers of events in these subsets, $$N_1,N_2,\ldots,N_n$$, are independent random variables that follow Poisson distributions with parameters $$\lambda|S_1|,\lambda|S_2|,\ldots,\lambda|S_n|$$, where $$|S_i|$$ denotes the measure of $$S_i$$."

#### Kiuhnm

##### New Member
Problem solved. Here's how.
Let R be a random variable of the distance from a point P to its nearest neighbor.
Then
$$P(R>r)=P(\text{there are no points in Sphere(P,r)})=\exp(-4\lambda\pi r^3/3)$$
Thus
$$P(R\leq r) = 1-\exp(-4\lambda\pi r^3/3)$$
Now we must take the derivative in order to find the density function:
$$P(R=r)=4\lambda\pi r^2\exp(-4\lambda\pi r^3/3)$$
This result agrees with the solutions so I'm pretty positive it's correct.