Continuous distribution
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
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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
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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
[1] | W. Feller, "An introduction to probability theory and its applications" , 2 , Wiley (1971) |
[2] | M. Loève, "Probability theory" , Princeton Univ. Press (1963) |
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
.
Continuous distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Continuous_distribution&oldid=20849