View source for Probability distribution

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One of the basic concepts in [[Probability theory|probability theory]] and [[Mathematical statistics|mathematical statistics]]. In the modern approach, a suitable [[Probability space|probability space]] $\{\Omega,S,\operatorname P\}$ is taken as a model of a stochastic phenomenon being considered. Here $\Omega$ is a sample space, $S$ is a $\sigma$-algebra of subsets of $\Omega$ specified in some way and $\operatorname P$ is a measure on $S$ such that $\operatorname P(\Omega)=1$ (a [[Probability measure|probability measure]]).

Any such measure on $\{\Omega,S\}$ is called a probability distribution (see {{Cite|K}}). But this definition, basic in the axiomatics introduced by A.N. Kolmogorov in 1933, proved to be too general in the course of the further development of the theory and was replaced by more restrictive ones in order to exclude some "pathological" cases. An example was the requirement that the measure $\operatorname P$ be "[[Perfect measure|perfect]]" (see {{Cite|GK}}). Probability distributions in function spaces are usually required to satisfy some regularity property, usually formulated as separability but also admitting a characterization in different terms (see [[Separable process|Separable process]] and also {{Cite|P}}).

Many of the probability distributions that appear in the specific problems in probability theory and mathematical statistics have been known for a long time and are connected with the basic probability schemes {{Cite|F}}. They are described either by probabilities of discrete values (see [[Discrete distribution|Discrete distribution]]) or by probability densities (see [[Continuous distribution|Continuous distribution]]). There are also tables compiled in certain cases where they are necessary {{Cite|BS}}.

Among the basic probability distributions, some are connected with sequences of independent trials (see [[Binomial distribution|Binomial distribution]]; [[Geometric distribution|Geometric distribution]]; [[Multinomial distribution|Multinomial distribution]]) and others with the limit laws corresponding to such a probability scheme when the number of trials increases indefinitely (see [[Normal distribution|Normal distribution]]; [[Poisson distribution|Poisson distribution]]; [[Arcsine distribution|Arcsine distribution]]). But these limit distributions may also appear directly in exact form, as in the theory of stochastic processes (see [[Wiener process|Wiener process]]; [[Poisson process|Poisson process]]), or as solutions of certain equations arising in so-called [[Characterization theorems|characterization theorems]] (see also [[Normal distribution|Normal distribution]]; [[Exponential distribution|Exponential distribution]]). A [[Uniform distribution|uniform distribution]], usually considered as a mathematical way of expressing that outcomes of an experiment are equally possible, can also be obtained as a limit distribution (say, by considering sums of large numbers of random variables or some other random variables with sufficiently smooth and "spread out" distributions modulo 1). More probability distributions can be obtained from those mentioned above by means of functional transformations of the corresponding random variables. For example, in mathematical statistics random variables with a normal distribution are used to obtain variables with a [[Chi-squared distribution| "chi-squared" distribution]], a [[Non-central chi-squared distribution|non-central "chi-squared" distribution]], a [[Student distribution|Student distribution]], a [[Fisher-F-distribution|Fisher $F$-distribution]], and others.

Important classes of distributions were discovered in connection with asymptotic methods in probability theory and mathematical statistics (see [[Limit theorems|Limit theorems]]; [[Stable distribution|Stable distribution]]; [[Infinitely-divisible distribution|Infinitely-divisible distribution]]; [[Omega-squared distribution| "Omega-squared" distribution]]).

It is important, both for the theory and in applications, to be able to define a concept of proximity of distributions. The collection of all probability distributions on $\{\Omega,S\}$ can in different ways be turned into a topological space. Weak convergence of probability distributions plays a basic role here (see [[Distributions, convergence of|Distributions, convergence of]]). In the one-dimensional and finite-dimensional cases the apparatus of characteristic functions (cf. [[Characteristic function|Characteristic function]]) is a principal instrument for studying convergence of probability distributions.

A complete description of a probability distribution (say, by means of the [[Density of a probability distribution|density of a probability distribution]] or a [[Distribution function|distribution function]]) is often replaced by a limited collection of characteristics. The most widely used of these in the one-dimensional case are the [[Mathematical expectation|mathematical expectation]] (the average value), the [[Dispersion|dispersion]] (or variance), the [[Median (in statistics)|median (in statistics)]], and the moments (cf. [[Moment|Moment]]). For numerical characteristics of multivariate probability distributions see [[Correlation (in statistics)|Correlation (in statistics)]]; [[Regression|Regression]].

The statistical analogue of a probability distribution is an [[Empirical distribution|empirical distribution]]. An empirical distribution and its characteristics can be used for the approximate representation of a theoretical distribution and its characteristics (see [[Statistical estimator|Statistical estimator]]). For ways to measure how well an empirical distribution fits a hypothetical one see [[Statistical hypotheses, verification of|Statistical hypotheses, verification of]]; [[Non-parametric methods in statistics|Non-parametric methods in statistics]].

====References====
{|
|valign="top"|{{Ref|K}}|| A.N. Kolmogorov, "Foundations of the theory of probability", Chelsea, reprint (1950) (Translated from Russian) {{MR|0032961}} {{ZBL|}}
|-
|valign="top"|{{Ref|GK}}|| B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables", Addison-Wesley (1954) (Translated from Russian) {{MR|0062975}} {{ZBL|0056.36001}}
|-
|valign="top"|{{Ref|P}}|| Yu.V. Prokhorov, "The method of characteristic functionals", ''Proc. 4-th Berkeley Symp. Math. Stat. Probab.'', '''2''', Univ. California Press (1961) pp. 403–419 {{MR|}} {{ZBL|0158.36502}}
|-
|valign="top"|{{Ref|F}}|| W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''1–2''', Wiley (1957–1971)
|-
|valign="top"|{{Ref|BS}}|| L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics", ''Libr. math. tables'', '''46''', Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova) {{MR|}} {{ZBL|0529.62099}}
|-
|valign="top"|{{Ref|G}}|| B.V. Gnedenko, [[Gnedenko, "A course in the theory of probability"|"A course of probability theory"]], Moscow (1969) (In Russian)
|-
|valign="top"|{{Ref|C}}|| H. Cramér, "Mathematical methods of statistics", Princeton Univ. Press (1946) {{MR|0016588}} {{ZBL|0063.01014}}
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|valign="top"|{{Ref|N}}|| J. Neveu, "Bases mathématiques du calcul des probabilités", Masson (1970) {{MR|0272004}} {{ZBL|0203.49901}}
|}

====Comments====
The term "probability distribution" is most frequently applied to the case when $S=\mathbf R$, $\mathbf R^n$ or a space of real functions.

====References====
{|
|valign="top"|{{Ref|L}}|| M. Loève, "Probability theory" , '''I-II''' , Springer (1977) {{MR|0651017}} {{MR|0651018}} {{ZBL|0359.60001}}
|-
|valign="top"|{{Ref|B}}|| P. Billingsley, "Probability and measure" , Wiley (1979) {{MR|0534323}} {{ZBL|0411.60001}}
|}