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# Introduction; Review of Random Variables

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Probability   ::   Set Theory   ::   Definition   ::   Conditional   ::   Random   ::   Distribution   ::   R.V.S.

## Pure types of r.v.s

1. Discrete r.v.s -- an r.v. whose possible values can be enumerated.
2. Continuous r.v.s -- an r.v. whose distribution function can be written as the (regular) integral of another function
3. Singular, but not discrete -- Any other r.v.

## Discrete r.v.s

A random variable that can take on at most a countable number of possible values is said to be a discrete r.v.:

Properties of pmfs:
1. Nonnegativity:

2. Total probability:

(These two properties completely characterize the class of all pdfs on a given set .)
3. Relation to cdf:

(To the pdf and the cdf contain the same information. Note that for a discrete r.v., the cdf is piecewise constant. Draw picture)

## Continuous r.v.s

A r.v. is said to be continuous is there is a function such that

for all . In this case, is an absolutely continuous function. 1

Properties:

1. .
These two properties completely characterize the s.

called the probability density function (pdf) of .

1. .
2. for all .
3. .

## Properties of Gaussian Random Variables

Let .

1. If then
2. .
3. .
4. A Gaussian r.v. is completely characterized by its first two moments (i.e., the mean and variance).
We will also see, and make use of, many other properties of Gaussian random processes, such as the fact that an uncorrelated Gaussian r.p. is also independent.

## Bibliography

1. P. Billingsley, Probability and Measure . New York: Wiley, 1986.

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 31). Introduction; Review of Random Variables. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lec1_7.html. This work is licensed under a Creative Commons License