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# Introduction; Review of Random Variables

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

## Random variables

Up to this point, the outcomes in could be anything: they could be elephants, computers, or mitochondria, since is simple expressed in terms of sets. But we frequently deal with numbers , and want to describe events associated with sets of numbers. This leads to the idea of a random variable.

A function such that , that is, such that the events involved are in , is said to be measurable with respect to . That is, is divided into sufficiently small pieces that the events in it can describe all of the sets associated with .

We observe that a random variable cannot generate partitions of the underlying sample space which are not events in the -field .

Another way of saying that is measurable: For any ,

Recalling the idea of Borel sets associated with the real line, we see that a random variable is a measurable function from to :

We will abbreviate the term random variable as r.v. .

We will use a notational shorthand for random variables.

By this definition, we can identify a new probability space. Using as the pre-probability space, we use the measure

So we get the probability space .

As a matter of practicality, if the sample space is , with the Borel field, most mappings to will be random variables.

To summarize:

where

for .

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_5.html. This work is licensed under a Creative Commons License