Basic Concepts of Random Processes
Definitions :: Ergodicity :: Autocorrelations :: Sinusoidal :: Poisson :: Gaussian :: Properties :: Spectra :: Cases :: Random :: Independent
The Homogeneous Poisson Counting Process
Let . Suppose events occur randomly in time in the following fashion:
 The number of events occurring in nonoverlapping intervals of time are independent.

The probability of one event exactly in any interval of length
is equal to
for
sufficiently small.
Now define a r.p. by as the number of events occurring in the interval . Then has the following properties:

is Poisson with parameter
:
 and are independent r.v.s for all nonoverlapping intervals and .
Such a process is called a Poisson counting process (PCP) with rate . These two properties complete determine a PCP. All finitedimensional distributions (fdds) of the process can be determined from these two properties.
How do we show the Poisson distribution property? Pick
.
Let
for . Then
By assumption 2,
Now
So
Now . When we get
so
We have another boundary condition: , giving
Now we could proceed solve the set of equations for For example, when :
This could be solved, e.g., using Laplace transforms. In general we would find
As stated, the properties allow us to find all finite dimensional
distributions. For example, suppose we want to find the joint
distribution of
and
for
.
where the factorization occurs because of independent increments.
Draw a typical sample path...
The process is called homogeneous because the rate at which the events occur does not depend on .
Let us work out the mean and autocorrelation functions.
(Poisson).
Assume
:
and if , .
This process is not WSS! The mean is not constant, and the autocorrelation is not a function of the time difference.
Now create a function , for some fixed . The random process is WSS. The increase in the number of counts (over some fixed interval) does not depend on the time.
We could create an
inhomogeneous
Poisson if the probability of
an occurrence in the interval
is
. Then