The intensity of a Poisson point process
an event in 
}. We can
simulate a Poisson point process with given 
on an interval
I as follows:
Simulate the number of events N from 
. Form 
, and generate N i.i.d
random numbers 
. Then 
are the points of a Poisson process with intensity
.
Ex 2. Prove this.
Suppose the left ends of clones are a Poisson point process with
intensity 
. And the clone lengths 
are
i.i.d. 
with mean L. Let 
{ t is uncovered }.
Thin the original Poisson process back to 
by removing
clones that do not overlap t. Thus a clone with left end at s is retained
with probability 
, and the thinned process has intensity
So
Corollary.
Note that this result is the same as the exponential approximation of
the exact calculation of 
in the
case of constant clone length. It goes to 0 as 
.