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About: Random Variables and Probabilities

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

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Name: Random Variables and Probabilities
ID: m23260
Language: English (en)
Summary: Often, each outcome of an experiment is characterized by a number. If the outcome is observed as a physical quantity, the size of that quantity (in prescribed units) is the entity actually observed. In many nonnumerical cases, it is convenient to assign a number to each outcome. For example, in a coin flipping experiment, a “head” may be represented by a 1 and a “tail” by a 0. In a Bernoulli trial, a success may be represented by a 1 and a failure by a 0. In a sequence of trials, we may be interested in the number of successes in a sequence of n component trials. One could assign a distinct number to each card in a deck of playing cards. Observations of the result of selecting a card could be recorded in terms of individual numbers. In each case, the associated number becomes a property of the outcome. The fundamental idea of a real random variable is the assignment of a real number to each elementary outcome ω in the basic space Ω. Such an assignment amounts to determining a function X, whose domain is Ω and whose range is a subset of the real line R. Each ω is mapped into exactly one value t, although several ω may have the same image point. Except in special cases, we cannot write a formula for a random variable X. However, random variables share some important general properties of functions which play an essential role in determining their usefulness. Associated with a function X as a mapping are the inverse mapping and the inverse images it produces. By the inverse image of a set of real numbers M under the mapping X, we mean the set of all those ω∈Ω which are mapped into M by X. If X does not take a value in M, the inverse image is the empty set (impossible event). If M includes the range of X, (the set of all possible values of X), the inverse image is the entire basic space Ω. The class of inverse images of the Borel sets on the real line play an essential role in probability analysis.
Subject: Mathematics and Statistics
Keywords: applied probability, mapping, probability, random variables,
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: 1.9 (history)
First publication date: Apr 15, 2009 3:18 pm +0000
Last revision to module: Sep 18, 2009 1:48 pm +0000


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Version History

Version: 1.9 Sep 18, 2009 1:48 pm +0000 by Daniel Williamson
added google analytics tracking code

Version: 1.8 Jul 28, 2009 10:00 am +0000 by Paul E Pfeiffer
summary added

Version: 1.7 Jul 27, 2009 12:22 pm +0000 by Daniel Williamson
disjoint union fix

Version: 1.6 Jun 19, 2009 11:28 am +0000 by Daniel Williamson
m-file fixes, supplemental links, metadata updated

Version: 1.5 Jun 9, 2009 5:36 pm +0000 by Daniel Williamson
updated links, images and accessibility

Version: 1.4 May 26, 2009 4:44 pm +0000 by Daniel Williamson
fixed module names, and updated content structure

Version: 1.3 May 14, 2009 2:50 pm +0000 by Paul E Pfeiffer

Version: 1.2 May 13, 2009 4:49 pm +0000 by Paul E Pfeiffer

Version: 1.1 May 13, 2009 12:17 pm +0000 by Paul E Pfeiffer

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