From: horst.kraemer@snafu.de (Horst Kraemer)
Subject: Re: A Question about Poisson
Date: 1 Oct 1999 04:30:08 -0500
Newsgroups: sci.math.research
Keywords: Probability an event occurs (or not) in a Poisson process

On 30 Sep 1999 17:30:07 -0500, ian <ianl@dl.tc.faa.gov> wrote:

> The following applies to a Poisson process:
 
> I am wondering how to calculate the probability that there are zero
> events in an interval given that one event occurs _in the interval
> before_.  That is, Given the probability of an event occuring once in a
> random interval, what is the probability of there being zero events in
> the NEXT interval?

It's a _defining_ property of a POISSON process that the probability
of events occurring in disjoint intervals are independent.

If you focus on one specific interval the probability doesn't "see"
what happens outside this interval, or in terms of time: A POISSON
process doesn't "remember" what happened before.

Thus your probability is a plain

	exp(-lambda(t2-t1))

if lambda is the average number of events in a unit time interval.


You may test this property easily in every programming language. If a
function random() returns a random number in the range 0<=r<1 and
r1,...r_n is a sequence of n random numbers then let

	l_i = -ln(1-r_i)

The sequence 

s1 = l1
s2 = s1+l2
s3 = s2+l3
..
s_n = s_{n-1}+l_n


will simulate a POISSON process with density 1.


Regards
Horst