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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space
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
process. In particular, they imply that the time between consecutive events (called interarrival
times) are
independent
random variables. For the homogeneous Poisson process, these inter
arrival times are
exponentially
distributed with parameter λ. Also, the memorylessness property
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 Fall '07
 SLHendricks
 Probability theory, probability density function, CDF

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