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# Introduction; Review of Random Variables

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

## Distribution functions

The cumulative distribution function (cdf) of an r.v. is defined for each as

Properties of cdf:
1. is non-decreasing: If then .
2. is right-continuous: .
Draw "typical'' picture.

These four properties completely characterize the family of cdfs on the real line. Any function which satisfies these has a corresponding probability distribution.

1. For : .
2. . Thus, if is continuous at , .
From these properties, we can assign probabilities to all intervals from knowledge of the cdf. Thus we can extend this to all Borel sets.

Thus determines a unique probability distribution on , so and are uniquely related.

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_6.html. This work is licensed under a Creative Commons License