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Basic Concepts of Random Processes

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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:

1. The number of events occurring in non-overlapping intervals of time are independent.
2. The probability of one event exactly in any interval of length is equal to for sufficiently small.

(That is, is the generic term for terms of order higher than .) Also, the probability of more than one event occurring during an interval of length .

Now define a r.p. by as the number of events occurring in the interval . Then has the following properties:

1. is Poisson with parameter :

2. and are independent r.v.s for all nonoverlapping intervals and .
The parameter is called the rate of . Property 2 follows from the first assumption. We say that such a process has independent increments .

Such a process is called a Poisson counting process (PCP) with rate . These two properties complete determine a PCP. All finite-dimensional 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

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, June 07). Basic Concepts of Random Processes. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lecture6_5.htm. This work is licensed under a Creative Commons License