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# Basic Concepts of Random Processes

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Definitions  ::  Ergodicity  ::  Autocorrelations  ::  Sinusoidal  ::  Poisson  ::  Gaussian  ::  Properties  ::  Spectra  ::  Cases  ::  Random  ::  Independent

## Basic definitions and concepts

• If is a singleton (one element) then is a r.v.
• If , then is a bivariate r.v.
• If consists of a finite number of elements, then is a random vector.
• If is countable, then is a random sequence.
For most applications we think of as time.'' In some cases, is multidimensional. Then is called a random field.

## Three interpretations of a r.p.

1. A collection of waveforms that occur randomly. That is, it is defined on some probability space. For each there is a corresponding waveform as a function of with fixed.

Think of having a big bag of waveforms. We reach into the bag and pick out a waveform -- a function of -- at random.

2. A collection of random variables. In this case, that is, for each fixed , we have a random variable .
3. A real-valued function of two variables .

A realization is just a function. It does not exhibit the randomness.

We will assume this set completely characterizes the statistical distribution of the process.

Strict stationarity is a fairly strong condition, and we don't necessarily need it always.

Copyright 2008, Todd Moon. 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_1.htm. This work is licensed under a Creative Commons License