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and here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025620/c02562022.png" /> almost-everywhere (with respect to Lebesgue measure). A distribution is absolutely continuous with respect to Lebesgue measure if and only if the corresponding distribution function is absolutely continuous (as a function of a real variable). In addition to absolutely-continuous distributions there are continuous distributions that are concentrated on sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025620/c02562023.png" />-measure zero. Such distributions are called singular (cf. [[Singular distribution|Singular distribution]]) with respect to a certain measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025620/c02562024.png" />. By Lebesgue's decomposition theorem, every continuous distribution is a mixture of two distributions, one of which is absolutely continuous and the other is singular with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025620/c02562025.png" />.
 
and here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025620/c02562022.png" /> almost-everywhere (with respect to Lebesgue measure). A distribution is absolutely continuous with respect to Lebesgue measure if and only if the corresponding distribution function is absolutely continuous (as a function of a real variable). In addition to absolutely-continuous distributions there are continuous distributions that are concentrated on sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025620/c02562023.png" />-measure zero. Such distributions are called singular (cf. [[Singular distribution|Singular distribution]]) with respect to a certain measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025620/c02562024.png" />. By Lebesgue's decomposition theorem, every continuous distribution is a mixture of two distributions, one of which is absolutely continuous and the other is singular with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025620/c02562025.png" />.
  
Some of the most important (absolutely-) continuous distributions are: the [[Arcsine distribution|arcsine distribution]]; the [[Beta-distribution|beta-distribution]], the [[Gamma-distribution|gamma-distribution]], the [[Cauchy distribution|Cauchy distribution]], the [[Normal distribution|normal distribution]], the [[Uniform distribution|uniform distribution]], the [[Exponential distribution|exponential distribution]], the [[Student distribution|Student distribution]], and the [["Chi-squared" distribution| "chi-squared" distribution]].
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Some of the most important (absolutely-) continuous distributions are: the [[Arcsine distribution|arcsine distribution]]; the [[Beta-distribution|beta-distribution]], the [[Gamma-distribution|gamma-distribution]], the [[Cauchy distribution|Cauchy distribution]], the [[Normal distribution|normal distribution]], the [[Uniform distribution|uniform distribution]], the [[Exponential distribution|exponential distribution]], the [[Student distribution|Student distribution]], and the [[Chi-squared distribution| "chi-squared" distribution]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> W. Feller, "An introduction to probability theory and its applications" , '''2''' , Wiley (1971) {{MR|0270403}} {{ZBL|0219.60003}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M. Loève, "Probability theory" , Princeton Univ. Press (1963) {{MR|0203748}} {{ZBL|0108.14202}} </TD></TR></table>
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|valign="top"|{{Ref|F}}|| W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its applications"]], '''2''', Wiley (1971)
 
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|valign="top"|{{Ref|L}}|| M. Loève, "Probability theory", Princeton Univ. Press (1963) {{MR|0203748}} {{ZBL|0108.14202}}
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====Comments====
 
====Comments====

Latest revision as of 11:58, 20 October 2012

2020 Mathematics Subject Classification: Primary: 60E05 [MSN][ZBL]

A probability distribution without atoms. Thus, a continuous distribution is the opposite of a discrete distribution (see also Atomic distribution). Discrete and continuous distributions together from the basic types of distributions. By a theorem of C. Jordan, every probability distribution is a mixture of a discrete and a continuous distribution. For example, let be the distribution function corresponding to a certain distribution on the real line. Then , where and are distribution functions of the discrete and the continuous type, respectively, is such a mixture. The distribution function of a continuous distribution is a continuous function. The absolutely-continuous distributions occupy a special position among the continuous distributions. This class of distributions on a measurable space is defined, relative to a reference measure , by the fact that can be represented in the form

Here is in and is a measurable function on with . The function is called the density of relative to (usually, is Lebesgue measure and ). On the line, the corresponding distribution function then has the representation

and here almost-everywhere (with respect to Lebesgue measure). A distribution is absolutely continuous with respect to Lebesgue measure if and only if the corresponding distribution function is absolutely continuous (as a function of a real variable). In addition to absolutely-continuous distributions there are continuous distributions that are concentrated on sets of -measure zero. Such distributions are called singular (cf. Singular distribution) with respect to a certain measure . By Lebesgue's decomposition theorem, every continuous distribution is a mixture of two distributions, one of which is absolutely continuous and the other is singular with respect to .

Some of the most important (absolutely-) continuous distributions are: the arcsine distribution; the beta-distribution, the gamma-distribution, the Cauchy distribution, the normal distribution, the uniform distribution, the exponential distribution, the Student distribution, and the "chi-squared" distribution.

References

[F] W. Feller, "An introduction to probability theory and its applications", 2, Wiley (1971)
[L] M. Loève, "Probability theory", Princeton Univ. Press (1963) MR0203748 Zbl 0108.14202

Comments

Atoms are those points of the sample space that have positive probability. A discrete distribution is a distribution in which all probability is concentrated in the atoms.

An absolutely-continuous distribution as defined above is also called absolutely continuous with respect to .

How to Cite This Entry:
Continuous distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Continuous_distribution&oldid=23595
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article