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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]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p0749001.png" /> is taken as a model of a stochastic phenomenon being considered. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p0749002.png" /> is a sample space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p0749003.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p0749004.png" />-algebra of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p0749005.png" /> specified in some way and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p0749006.png" /> is a measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p0749007.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p0749008.png" /> (a [[Probability measure|probability measure]]).
 
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]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p0749001.png" /> is taken as a model of a stochastic phenomenon being considered. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p0749002.png" /> is a sample space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p0749003.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p0749004.png" />-algebra of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p0749005.png" /> specified in some way and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p0749006.png" /> is a measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p0749007.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p0749008.png" /> (a [[Probability measure|probability measure]]).
  
Any such measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p0749009.png" /> is called a probability distribution (see [[#References|[1]]]). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p07490010.png" /> be "perfect" (see [[#References|[2]]]). 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 [[#References|[3]]]).
+
Any such measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p0749009.png" /> is called a probability distribution (see [[#References|[1]]]). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p07490010.png" /> be "perfect" (see [[#References|[2]]]). 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 [[#References|[3]]]).
  
 
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 [[#References|[4]]]. 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 [[#References|[5]]].
 
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 [[#References|[4]]]. 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 [[#References|[5]]].
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p07490011.png" />-distribution]], and others.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p07490011.png" />-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]]).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p07490012.png" /> 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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p07490012.png" /> 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.
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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.N. Kolmogorov,   "Foundations of the theory of probability" , Chelsea, reprint (1950) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.V. Gnedenko,   A.N. Kolmogorov,   "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> Yu.V. Prokhorov,   "The method of characteristic functionals" , ''Proc. 4-th Berkeley Symp. Math. Stat. Probab.'' , '''2''' , Univ. California Press (1961) pp. 403–419</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> W. Feller,   "An introduction to probability theory and its applications" , '''1–2''' , Wiley (1957–1971)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> 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)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> B.V. Gnedenko,   "A course of probability theory" , Moscow (1969) (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> H. Cramér,   "Mathematical methods of statistics" , Princeton Univ. Press (1946)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> J. Neveu,   "Bases mathématiques du calcul des probabilités" , Masson (1970)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1950) (Translated from Russian) {{MR|0032961}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> 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}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> 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}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> W. Feller, "An introduction to probability theory and its applications" , '''1–2''' , Wiley (1957–1971) {{MR|0779091}} {{MR|0779090}} {{MR|0270403}} {{MR|0228020}} {{MR|1534302}} {{MR|0243559}} {{MR|0242202}} {{MR|0210154}} {{MR|1570945}} {{MR|0088081}} {{MR|1528130}} {{MR|0067380}} {{MR|0038583}} {{ZBL|0598.60003}} {{ZBL|0598.60002}} {{ZBL|0219.60003}} {{ZBL|0155.23101}} {{ZBL|0158.34902}} {{ZBL|0151.22403}} {{ZBL|0138.10207}} {{ZBL|0115.35308}} {{ZBL|0077.12201}} {{ZBL|0039.13201}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> 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}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> B.V. Gnedenko, "A course of probability theory" , Moscow (1969) (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) {{MR|0016588}} {{ZBL|0063.01014}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> J. Neveu, "Bases mathématiques du calcul des probabilités" , Masson (1970) {{MR|0272004}} {{ZBL|0203.49901}} </TD></TR></table>
  
  
  
 
====Comments====
 
====Comments====
The term "probability distribution" is most frequently applied to the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p07490013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p07490014.png" /> or a space of real functions.
+
The term "probability distribution" is most frequently applied to the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p07490013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074900/p07490014.png" /> or a space of real functions.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Loève,   "Probability theory" , '''I-II''' , Springer (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P. Billingsley,   "Probability and measure" , Wiley (1979)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Loève, "Probability theory" , '''I-II''' , Springer (1977) {{MR|0651017}} {{MR|0651018}} {{ZBL|0359.60001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P. Billingsley, "Probability and measure" , Wiley (1979) {{MR|0534323}} {{ZBL|0411.60001}} </TD></TR></table>

Revision as of 10:31, 27 March 2012

One of the basic concepts in probability theory and mathematical statistics. In the modern approach, a suitable probability space is taken as a model of a stochastic phenomenon being considered. Here is a sample space, is a -algebra of subsets of specified in some way and is a measure on such that (a probability measure).

Any such measure on is called a probability distribution (see [1]). 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 be "perfect" (see [2]). 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 and also [3]).

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 [4]. They are described either by probabilities of discrete values (see Discrete distribution) or by probability densities (see Continuous distribution). There are also tables compiled in certain cases where they are necessary [5].

Among the basic probability distributions, some are connected with sequences of independent trials (see Binomial distribution; Geometric 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; Poisson distribution; Arcsine distribution). But these limit distributions may also appear directly in exact form, as in the theory of stochastic processes (see Wiener process; Poisson process), or as solutions of certain equations arising in so-called characterization theorems (see also Normal distribution; Exponential distribution). A 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, a non-central "chi-squared" distribution, a Student distribution, a Fisher -distribution, and others.

Important classes of distributions were discovered in connection with asymptotic methods in probability theory and mathematical statistics (see Limit theorems; Stable distribution; Infinitely-divisible 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 can in different ways be turned into a topological space. Weak convergence of probability distributions plays a basic role here (see Distributions, convergence of). In the one-dimensional and finite-dimensional cases the apparatus of characteristic functions (cf. 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 or a 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 (the average value), the dispersion (or variance), the median (in statistics), and the moments (cf. Moment). For numerical characteristics of multivariate probability distributions see Correlation (in statistics); Regression.

The statistical analogue of a probability distribution is an 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). For ways to measure how well an empirical distribution fits a hypothetical one see Statistical hypotheses, verification of; Non-parametric methods in statistics.

References

[1] A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1950) (Translated from Russian) MR0032961
[2] B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian) MR0062975 Zbl 0056.36001
[3] Yu.V. Prokhorov, "The method of characteristic functionals" , Proc. 4-th Berkeley Symp. Math. Stat. Probab. , 2 , Univ. California Press (1961) pp. 403–419 Zbl 0158.36502
[4] W. Feller, "An introduction to probability theory and its applications" , 1–2 , Wiley (1957–1971) MR0779091 MR0779090 MR0270403 MR0228020 MR1534302 MR0243559 MR0242202 MR0210154 MR1570945 MR0088081 MR1528130 MR0067380 MR0038583 Zbl 0598.60003 Zbl 0598.60002 Zbl 0219.60003 Zbl 0155.23101 Zbl 0158.34902 Zbl 0151.22403 Zbl 0138.10207 Zbl 0115.35308 Zbl 0077.12201 Zbl 0039.13201
[5] 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) Zbl 0529.62099
[6] B.V. Gnedenko, "A course of probability theory" , Moscow (1969) (In Russian)
[7] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) MR0016588 Zbl 0063.01014
[8] J. Neveu, "Bases mathématiques du calcul des probabilités" , Masson (1970) MR0272004 Zbl 0203.49901


Comments

The term "probability distribution" is most frequently applied to the case when , or a space of real functions.

References

[a1] M. Loève, "Probability theory" , I-II , Springer (1977) MR0651017 MR0651018 Zbl 0359.60001
[a2] P. Billingsley, "Probability and measure" , Wiley (1979) MR0534323 Zbl 0411.60001
How to Cite This Entry:
Probability distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Probability_distribution&oldid=13959
This article was adapted from an original article by Yu.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article