Continuous distribution

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 .