Interpoint distances for two colours of points
Suppose red points form a one-dimensional Poisson spatial process
with density λr. Suppose blue points are distributed
similarly and independently with density λb.
In the figure, λr=1 and λb=1.5.
What is the probability that two neighbouring points are the same or
Let r,b be the interpoint distance between red and blue points
respectively. These RVs have pdf
The difference d=r-b thus has pdf (here  is the indicator function)
Integrating this, we get the simple final result: if I am a red point,
my nearest neighbour on the right is blue if d>0, and
Prob[two neighbours are of different colours]
=Prob[red]Prob[neighbour blue]+Prob[blue]Prob[neighbour red]