Remember that in a Poisson process, the number of events that occur in a particular area (any subset of a 3D space, i.e. a subset of ) is a Poisson random variable with the parameter where is a constant and is the volume of that area
Now, let be a random variable which "measures" the distance from an event to its nearest neighbour.
So, we need to find the probability density function of the random variable (notice that is a continuous random variable).
Actually, let's first find the cumulative distribution function.
First, let's examine what event is, for some
This event means that the distance from an event to its nearest neighbour is than In fact, this means that there are no events (so, 0 events) in a sphere (around some point) of radius Let's denote that sphere with so that
Now we can analyze:
Finally, remember that a probability density function is simply a derivative of the cumulative distribution function, which means that