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Types of Poisson Processes
[
edit
] Homogeneous Poisson process
A
homogeneous
Poisson process is characterized by a rate parameter λ such that the number of
events in time interval (
t
,
t
+ τ] follows a
Poisson distribution
with associated parameter λτ. This
relation is given as
where
N
(
t
+ τ) −
N
(
t
) describes the number of events in time interval (
t
,
t
+ τ].
Just as a Poisson random variable is characterized by its scalar parameter λ, a homogeneous
Poisson process is characterized by its rate parameter λ, which is the
expected
number of
"events" or "arrivals" that occur per unit time.
N
(
t
) is a sample homogeneous Poisson process, not to be confused with a density or distribution
function.
[
edit
] NonHomogeneous Poisson process
In general, the rate parameter may change over time. In this case, the generalized rate function
is given as λ(
t
). Now the expected number of events between time
a
and time
b
is
Thus, the number of arrivals in the time interval (
a
,
b
], given as
N
(
b
) −
N
(
a
), follows a
Poisson
distribution
with associated parameter λ
a
,
b
A homogeneous Poisson process may be viewed as a special case when λ(
t
) = λ, a constant rate.
[
edit
] Spatial Poisson process
A further variation on the Poisson process, called the spatial Poisson process, introduces a
spatial dependence on the rate function and is given as
where
for some
vector
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V
(e.g.
R
2
or
R
3
). For any set
(e.g. a spatial region) with finite
measure
, the
number of events occurring inside this region can be modelled as a Poisson process with
associated rate function λ
S
(
t
) such that
In the special case that this generalized rate function is a separable function of time and space,
we have:
for some function
. Without loss of generality, let
else we may scale
and λ(
t
) appropriately. Now,
represents the spatial
probability
density function
of these random events in the following sense. The act of sampling this spatial
Poisson process is equivalent to sampling a Poisson process with rate function λ(
t
), and
associating with each event a random vector
sampled from the probability density function
. A similar result can be shown for the general (nonseparable) case.
[
edit
] General characteristics of the Poisson process
In its most general form, the only two conditions for a
stochastic process
to be a Poisson
process are:
•
Orderliness
: which roughly means
which implies that arrivals don't occur simultaneously (but this is actually a
mathematicallystronger statement).
•
Memorylessness
(also called evolution without aftereffects): the number of arrivals
occurring in any bounded interval of time after time
t
is
independent
of the number of
arrivals occurring before time
t
.
These seemingly unrestrictive conditions actually impose a great deal of structure in the Poisson
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 Fall '07
 SLHendricks

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