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Module by: Paul E Pfeiffer. E-mail the author

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Name: Distribution and Density Functions
ID: m23267
Language: English (en)
Summary: In the unit on Random Variables and Probability, we introduce real random variables as mappings from the basic space to the real line. The mapping induces a transfer of the probability mass on the basic space to subsets of the real line in such a way that the probability that X takes a value in a set M is exactly the mass assigned to that set by the transfer. To perform probability calculations, we need to describe analytically the distribution on the line. For simple random variables, we have at each possible value of X a point mass equal to the probability X takes that value. For more general cases, we need a more useful description: the distribution function which at each t has the value of the probability mass at or to the left of t. If the probability mass in the induced distribution is spread smoothly along the real line, with no point mass concentrations, there is a probability density function such that the probability mass in any interval is the area under the curve over that interval.
Subject: Mathematics and Statistics
Keywords: Beta, Binomial, Common discrete distributions, Distribution function, Exponential, Gamma, Geometric, Negative binomial, Normal or Gaussian, Poisson, Simple random variable, Symmetric triangular, Uniform, Weibull
License: Creative Commons Attribution License CC-BY 3.0

Authors: Paul E Pfeiffer (
Copyright Holders: Paul E Pfeiffer (
Maintainers: Paul E Pfeiffer (, Daniel Williamson (, C. Sidney Burrus (

Latest version: 1.7 (history)
First publication date: Apr 15, 2009 3:56 pm -0500
Last revision to module: Sep 18, 2009 12:32 pm -0500


PDF: m23267_1.7.pdf PDF file, for viewing content offline and printing. Learn more.
XML: m23267_1.7.cnxml XML that defines the structure and contents of the module, minus any included media files. Can be reimported in the editing interface. Learn more.

Version History

Version: 1.7 Sep 18, 2009 12:32 pm -0500 by Daniel Williamson
added google analytics tracking code

Version: 1.6 Jul 27, 2009 11:52 am -0500 by Daniel Williamson
disjoint union fix

Version: 1.5 Jun 19, 2009 11:03 am -0500 by Daniel Williamson
m-file fixes, supplemental links, metadata updated

Version: 1.4 Jun 9, 2009 5:05 pm -0500 by Daniel Williamson
updated links, images, and accessibility

Version: 1.3 May 26, 2009 4:22 pm -0500 by Daniel Williamson
fixed module names, and updated content structure

Version: 1.2 May 14, 2009 2:52 pm -0500 by Paul E Pfeiffer

Version: 1.1 May 13, 2009 5:18 pm -0500 by Paul E Pfeiffer

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