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In
probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...
and
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...

statistics
, a probability distribution is the mathematical
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
that gives the probabilities of occurrence of different possible outcomes for an
experiment An experiment is a procedure carried out to support or refute a , or determine the or of something previously untried. Experiments provide insight into by demonstrating what outcome occurs when a particular factor is manipulated. Experime ...
. It is a mathematical description of a
random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wikt:order, order and does not follow an intelligible pattern or combination. I ...
phenomenon in terms of its
sample space In probability theory Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these axiom ...

sample space
and the
probabilities Probability is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathe ...

probabilities
of
events Event may refer to: Gatherings of people * Ceremony, an event of ritual significance, performed on a special occasion * Convention (meeting), a gathering of individuals engaged in some common interest * Event management, the organization of event ...
(subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that the coin is fair). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a
survey Survey may refer to: Statistics and human research * Statistical survey Survey methodology is "the study of survey methods". As a field of applied statistics concentrating on Survey (human research), human-research surveys, survey methodology s ...
to be conducted, etc.


Introduction

A probability distribution is a mathematical description of the probabilities of events, subsets of the
sample space In probability theory Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these axiom ...

sample space
. The sample space, often denoted by \Omega, is the set of all possible outcomes of a random phenomenon being observed; it may be any set: a set of
real numbers Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...

real numbers
, a set of
vectors Vector may refer to: Biology *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector *Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...
, a set of arbitrary non-numerical values, etc. For example, the sample space of a coin flip would be . To define probability distributions for the specific case of
random variables In probability and statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventio ...
(so the sample space can be seen as a numeric set), it is common to distinguish between discrete and continuous
random variable A random variable is a variable whose values depend on outcomes of a random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...
s. In the discrete case, it is sufficient to specify a
probability mass function In probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quant ...
p assigning a probability to each possible outcome: for example, when throwing a fair
die Die, as a verb, refers to death, the cessation of life. Die may also refer to: Games * Die, singular of dice, small throwable objects used for producing random numbers Manufacturing * Die (integrated circuit), a rectangular piece of a semiconduct ...

die
, each of the six values 1 to 6 has the probability 1/6. The probability of an
event Event may refer to: Gatherings of people * Ceremony A ceremony (, ) is a unified ritual A ritual is a sequence of activities involving gestures, words, actions, or objects, performed in a sequestered place and according to a set sequence. Rit ...
is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event "the die rolls an even value" is :p(2) + p(4) + p(6) = 1/6+1/6+1/6=1/2. In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, consider measuring the weight of a piece of ham in the supermarket, and assume the scale has many digits of precision. The probability that it weighs ''exactly'' 500 g is zero, as it will most likely have some non-zero decimal digits. Nevertheless, one might demand, in quality control, that a package of "500 g" of ham must weigh between 490 g and 510 g with at least 98% probability, and this demand is less sensitive to the accuracy of measurement instruments. Continuous probability distributions can be described in several ways. The
probability density function and probability density function of a normal distribution . Image:visualisation_mode_median_mean.svg, 150px, Geometric visualisation of the mode (statistics), mode, median (statistics), median and mean (statistics), mean of an arbitrary probabilit ...
describes the
infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...
probability of any given value, and the probability that the outcome lies in a given interval can be computed by integrating the probability density function over that interval. An alternative description of the distribution is by means of the
cumulative distribution function In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by exp ...
, which describes the probability that the random variable is no larger than a given value (i.e., for some ''x''). The cumulative distribution function is the area under the
probability density function and probability density function of a normal distribution . Image:visualisation_mode_median_mean.svg, 150px, Geometric visualisation of the mode (statistics), mode, median (statistics), median and mean (statistics), mean of an arbitrary probabilit ...
from -\infty to ''x'', as described by the picture to the right.


General definition

A probability distribution can be described in various forms, such as by a probability mass function or a cumulative distribution function. One of the most general descriptions, which applies for continuous and discrete variables, is by means of a probability function P\colon \mathcal \rightarrow \mathbb whose input space \mathcal is related to the
sample space In probability theory Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these axiom ...

sample space
, and gives a
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
probability as its output.Chapters 1 and 2 of The probability function ''P'' can take as argument subsets of the sample space itself, as in the coin toss example, where the function ''P'' was defined so that ''P'' and ''P''. However, because of the widespread use of
random variables In probability and statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventio ...
, which transform the sample space into a set of numbers (e.g., \mathbb, \mathbb), it is more common to study probability distributions whose argument are subsets of these particular kinds of sets (number sets), and all probability distributions discussed in this article are of this type. It is common to denote as the probability that a certain variable ''X'' belongs to a certain event ''E''. The above probability function only characterizes a probability distribution if it satisfies all the
Kolmogorov axioms The Kolmogorov axioms are the foundations of probability theory introduced by Andrey Kolmogorov in 1933. These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probability cases. An alternativ ...
, that is: # P(X \in E) \ge 0 \; \forall E \in \mathcal, so the probability is non-negative # P(X \in E) \le 1 \; \forall E \in \mathcal, so no probability exceeds 1 # P(X \in \bigsqcup_ E_i ) = \sum_i P(X \in E_i) for any disjoint family of sets \ The concept of probability function is made more rigorous by defining it as the element of a
probability space In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expres ...
(X, \mathcal, P), where X is the set of possible outcomes, \mathcal is the set of all subsets E \subset X whose probability can be measured, and P is the probability function, or probability measure, that assigns a probability to each of these measurable subsets E \in \mathcal. Probability distributions are generally divided into two classes. A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is
discrete Discrete in science is the opposite of :wikt:continuous, continuous: something that is separate; distinct; individual. Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic c ...

discrete
(e.g. a coin toss, a roll of a die) and the probabilities are encoded by a discrete list of the probabilities of the outcomes; in this case the discrete probability distribution is known as
probability mass function In probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quant ...
. On the other hand, continuous probability distributions are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In the case of real numbers, the continuous probability distribution is the
cumulative distribution function In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by exp ...
. In general, in the continuous case, probabilities are described by a
probability density function and probability density function of a normal distribution . Image:visualisation_mode_median_mean.svg, 150px, Geometric visualisation of the mode (statistics), mode, median (statistics), median and mean (statistics), mean of an arbitrary probabilit ...
, and the probability distribution is by definition the integral of the probability density function. The
normal distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by ex ...

normal distribution
is a commonly encountered continuous probability distribution. More complex experiments, such as those involving
stochastic processes In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
defined in
continuous timeIn mathematical dynamics, discrete time and continuous time are two alternative frameworks within which to model variables that evolve over time. Discrete time Discrete sampled signal Discrete time views values of variables as occurring at disti ...
, may demand the use of more general
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability meas ...
s. A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called
univariate In mathematics, a univariate object is an expression, equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geom ...
, while a distribution whose sample space is a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of dimension 2 or more is called multivariate. A univariate distribution gives the probabilities of a single
random variable A random variable is a variable whose values depend on outcomes of a random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...
taking on various different values; a multivariate distribution (a
joint probability distribution Given random variables X,Y,\ldots, that are defined on the same probability space, the joint probability distribution for X,Y,\ldots is a probability distribution that gives the probability that each of X,Y,\ldots falls in any particular range o ...
) gives the probabilities of a
random vector In probability Probability is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculu ...
– a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the
binomial distribution In probability theory Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically thes ...

binomial distribution
, the
hypergeometric distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, and the
normal distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by ex ...

normal distribution
. A commonly encountered multivariate distribution is the
multivariate normal distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
. Besides the probability function, the cumulative distribution function, the probability mass function and the probability density function, the
moment generating function In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compared ...
and the
characteristic functionIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
also serve to identify a probability distribution, as they uniquely determine an underlying cumulative distribution function.


Terminology

Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below.


Functions for discrete variables

*Probability function: describes the probability P(X \in E) that the event E, from the sample space occurs. * Probability mass function (pmf): function that gives the probability that a discrete random variable is equal to some value. *
Frequency distributionIn statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a ...

Frequency distribution
: a table that displays the frequency of various outcomes in a sample. *Relative frequency distribution: a
frequency distributionIn statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a ...

frequency distribution
where each value has been divided (normalized) by a number of outcomes in a sample (i.e. sample size). *Discrete probability distribution function: general term to indicate the way the total probability of 1 is distributed over all various possible outcomes (i.e. over entire population) for discrete random variable. *
Cumulative distribution function In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by exp ...
: function evaluating the
probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

probability
that X will take a value less than or equal to x for a discrete random variable. *
Categorical distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expres ...
: for discrete random variables with a finite set of values.


Functions for continuous variables

*
Probability density function and probability density function of a normal distribution . Image:visualisation_mode_median_mean.svg, 150px, Geometric visualisation of the mode (statistics), mode, median (statistics), median and mean (statistics), mean of an arbitrary probabilit ...
(pdf): function whose value at any given sample (or point) in the
sample space In probability theory Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these axiom ...

sample space
(the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would equal that sample. * Continuous probability distribution function: most often reserved for continuous random variables. *
Cumulative distribution function In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by exp ...
: function evaluating the
probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

probability
that X will take a value less than or equal to x for a continuous variable. *
Quantile function In probability and statistics, the quantile function, associated with a probability distribution of a random variable, specifies the value of the random variable such that the probability of the variable being less than or equal to that value equa ...
: the inverse of the cumulative distribution function. Gives x such that, with probability q, X will not exceed x.


Basic terms

* Mode: for a discrete random variable, the value with highest probability; for a continuous random variable, a location at which the probability density function has a local peak. * Support: set of values that can be assumed with non-zero probability by the random variable. For a random variable X, it is sometimes denoted as R_X. * Tail:More information and examples can be found in the articles
Heavy-tailed distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressi ...
,
Long-tailed distribution In statistics and business, a long tail of some probability distribution, distributions of numbers is the portion of the distribution having many occurrences far from the "head" or central part of the distribution. The distribution could involve ...
,
fat-tailed distribution A fat-tailed distribution is a probability distribution In probability theory and statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statist ...
the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form X > a, X < b or a union thereof. *Head: the region where the pmf or pdf is relatively high. Usually has the form a < X < b. *
Expected value In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by exp ...
or mean: the
weighted average The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of the possible values, using their probabilities as their weights; or the continuous analog thereof. *
Median In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin wi ...

Median
: the value such that the set of values less than the median, and the set greater than the median, each have probabilities no greater than one-half. *
Variance In probability theory Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these ax ...

Variance
: the second moment of the pmf or pdf about the mean; an important measure of the
dispersion Dispersion may refer to: Economics and finance *Dispersion (finance), a measure for the statistical distribution of portfolio returns *Price dispersion, a variation in prices across sellers of the same item *Wage dispersion, the amount of variation ...
of the distribution. *
Standard deviation In statistics, the standard deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected v ...

Standard deviation
: the square root of the variance, and hence another measure of dispersion. *
Quantile In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin wi ...

Quantile
: the q-quantile is the value x such that P(X < x) = q. * Symmetry: a property of some distributions in which the portion of the distribution to the left of a specific value (usually the median) is a mirror image of the portion to its right. *
Skewness In probability theory Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these ...

Skewness
: a measure of the extent to which a pmf or pdf "leans" to one side of its mean. The third standardized moment of the distribution. *
Kurtosis In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expres ...
: a measure of the "fatness" of the tails of a pmf or pdf. The fourth standardized moment of the distribution. * Continuity: a property of some distributions whose values do not change abruptly.


Discrete probability distribution

A discrete probability distribution is the probability distribution of a random variable that can take on only a countable number of values. In the case where the range of values is countably infinite, these values have to decline to zero fast enough for the probabilities to add up to 1. For example, if \operatorname(X=n) = \tfrac for ''n'' = 1, 2, ..., the sum of probabilities would be 1/2 + 1/4 + 1/8 + ... = 1. Well-known discrete probability distributions used in statistical modeling include the
Poisson distribution In probability theory and statistics, the Poisson distribution (; ), named after France, French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a f ...
, the
Bernoulli distribution In probability theory Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these ax ...

Bernoulli distribution
, the
binomial distribution In probability theory Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically thes ...

binomial distribution
, the
geometric distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expres ...
, and the
negative binomial distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
. Additionally, the
discrete uniform distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
is commonly used in computer programs that make equal-probability random selections between a number of choices. When a sample (a set of observations) is drawn from a larger population, the sample points have an that is discrete, and which provides information about the population distribution.


Cumulative distribution function

Equivalently to the above, a discrete random variable can be defined as a random variable whose
cumulative distribution function In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by exp ...
(cdf) increases only by jump discontinuities—that is, its cdf increases only where it "jumps" to a higher value, and is constant between those jumps. Note however that the points where the cdf jumps may form a dense set of the real numbers. The points where jumps occur are precisely the values which the random variable may take.


Delta-function representation

Consequently, a discrete probability distribution is often represented as a generalized
probability density function and probability density function of a normal distribution . Image:visualisation_mode_median_mean.svg, 150px, Geometric visualisation of the mode (statistics), mode, median (statistics), median and mean (statistics), mean of an arbitrary probabilit ...
involving
Dirac delta function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is n ...
s, which substantially unifies the treatment of continuous and discrete distributions. This is especially useful when dealing with probability distributions involving both a continuous and a discrete part.


Indicator-function representation

For a discrete random variable ''X'', let ''u''0, ''u''1, ... be the values it can take with non-zero probability. Denote :\Omega_i=X^(u_i)= \,\, i=0, 1, 2, \dots These are
disjoint set Image:Disjunkte Mengen.svg, Two disjoint sets. In mathematics, two Set (mathematics), sets are said to be disjoint sets if they have no element (mathematics), element in common. Equivalently, two disjoint sets are sets whose intersection (set theor ...
s, and for such sets :P\left(\bigcup_i \Omega_i\right)=\sum_i P(\Omega_i)=\sum_i P(X=u_i)=1. It follows that the probability that ''X'' takes any value except for ''u''0, ''u''1, ... is zero, and thus one can write ''X'' as :X(\omega)=\sum_i u_i 1_(\omega) except on a set of probability zero, where 1_A is the indicator function of ''A''. This may serve as an alternative definition of discrete random variables.


One-point distribution

A special case is the discrete distribution of a random variable that can take on only one fixed value; in other words, it is a
deterministic distribution In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
. Expressed formally, the random variable X has a one-point distribution if it has a possible outcome x such that P(Xx)=1. All other possible outcomes then have probability 0. Its cumulative distribution function jumps immediately from 0 to 1.


Continuous probability distribution

A continuous probability distribution is a probability distribution whose support is an uncountable set, such as an interval in the real line. They are uniquely characterized by a
cumulative distribution function In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by exp ...
that can be used to calculate the probability for each subset of the support. There are many examples of continuous probability distributions: ,
uniform A uniform is a variety of clothing A kanga, worn throughout the African Great Lakes region Clothing (also known as clothes, apparel, and attire) are items worn on the body. Typically, clothing is made of fabrics or textiles, but over ti ...
, chi-squared, and others. A random variable X has a continuous probability distribution if there is a function f: \mathbb \rightarrow , \infty/math> such that for each interval I \subset \mathbb the probability of X belonging to I is given by the integral of f over I. For example, if I =
, b The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
/math>, then we would have: : \operatorname\left \le X \le b\right= \int_a^b f(x) \, dx In particular, the probability for X to take any single value a (that is, a \le X \le a) is zero, because an
integral In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

integral
with coinciding upper and lower limits is always equal to zero. A variable that satisfies the above is called continuous random variable. Its cumulative density function is defined as : F(x) = \operatorname\left \infty < X \le x\right= \int_^x f(x) \, dx which, by this definition, has the properties:
  • F(x) is non-decreasing;
  • 0 \le F(x) \le 1;
  • \lim_ F(x) = 0 and \lim_ F(x) = 1;
  • \operatorname P(a < X \le b) = F(b) - F(a); and
  • F(x) is continuous due to the
    Riemann integral The partition does not need to be regular, as shown here. The approximation works as long as the width of each subdivision tends to zero. In the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...
    properties.
It is also possible to think in the opposite direction, which allows more flexibility: if F(x) is a function that satisfies all but the last of the properties above, then F represents the cumulative density function for some random variable: a discrete random variable if F is a step function, and a continuous random variable otherwise. This allows for continuous distributions that have a cumulative density function, but not a probability density function, such as the
Cantor distribution The Cantor distribution is the probability distribution In probability theory and statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statis ...
. It is often necessary to generalize the above definition for more arbitrary subsets of the real line. In these contexts, a continuous probability distribution is defined as a probability distribution with a cumulative distribution function that is
absolutely continuousIn calculus, absolute continuity is a smoothness property of function (mathematics), functions that is stronger than continuous function, continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the ...
. Equivalently, it is a probability distribution on the
real numbers Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...

real numbers
that is
absolutely continuousIn calculus, absolute continuity is a smoothness property of function (mathematics), functions that is stronger than continuous function, continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the ...
with respect to the
Lebesgue measure In Measure (mathematics), measure theory, a branch of mathematics, the Lebesgue measure, named after france, French mathematician Henri Lebesgue, is the standard way of assigning a measure (mathematics), measure to subsets of ''n''-dimensional Eucli ...
. Such distributions can be represented by their
probability density function and probability density function of a normal distribution . Image:visualisation_mode_median_mean.svg, 150px, Geometric visualisation of the mode (statistics), mode, median (statistics), median and mean (statistics), mean of an arbitrary probabilit ...
s. If X is such an absolutely continuous random variable, then it has a
probability density function and probability density function of a normal distribution . Image:visualisation_mode_median_mean.svg, 150px, Geometric visualisation of the mode (statistics), mode, median (statistics), median and mean (statistics), mean of an arbitrary probabilit ...
f(x), and its probability of falling into a Lebesgue-measurable set A \subset \mathbb is: : \operatorname\left \in A\right= \int_A f(x) \, d\mu where \mu is the Lebesgue measure. Note on terminology: some authors use the term "continuous distribution" to denote distributions whose cumulative distribution functions are
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
, rather than
absolutely continuousIn calculus, absolute continuity is a smoothness property of function (mathematics), functions that is stronger than continuous function, continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the ...
. These distributions are the ones \mu such that \mu\\,=\,0 for all \,x. This definition includes the (absolutely) continuous distributions defined above, but it also includes
singular distributionIn probability, a singular distribution is a probability distribution concentrated on a set of Lebesgue measure zero, where the probability of each point in that set is zero. Other names These distributions are sometimes called singular continuou ...
s, which are neither absolutely continuous nor discrete nor a mixture of those, and do not have a density. An example is given by the
Cantor distribution The Cantor distribution is the probability distribution In probability theory and statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statis ...
.


Kolmogorov definition

In the measure-theoretic formalization of
probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...
, a
random variable A random variable is a variable whose values depend on outcomes of a random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...
is defined as a
measurable function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
X from a
probability space In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expres ...
(\Omega, \mathcal, \mathbb) to a
measurable space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

measurable space
(\mathcal,\mathcal). Given that probabilities of events of the form \ satisfy Kolmogorov's probability axioms, the probability distribution of ''X'' is the
pushforward measure In measure theory, a discipline within mathematics, a pushforward measure (also push forward, push-forward or image measure) is obtained by transferring ("pushing forward") a measure (mathematics), measure from one measurable space to another using ...
X_*\mathbb of X , which is a
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability meas ...
on (\mathcal,\mathcal) satisfying X_*\mathbb = \mathbbX^.


Other kinds of distributions

Continuous and discrete distributions with support on \mathbb^k or \mathbb^k are extremely useful to model a myriad of phenomena, since most practical distributions are supported on relatively simple subsets, such as
hypercubes In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
or
balls A ball A ball is a round object (usually spherical, but can sometimes be ovoid An oval (from Latin ''ovum'', "egg") is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas ( p ...
. However, this is not always the case, and there exist phenomena with supports that are actually complicated curves \gamma:
, b The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
\rightarrow \mathbb^n within some space \mathbb^n or similar. In these cases, the probability distribution is supported on the image of such curve, and is likely to be determined empirically, rather than finding a closed formula for it. One example is shown in the figure to the right, which displays the evolution of a
system of differential equations In mathematics, a system of differential equations is a finite set of differential equation In mathematics, a differential equation is an equation that relates one or more function (mathematics), functions and their derivatives. In applications, t ...
(commonly known as the Rabinovich–Fabrikant equations) that can be used to model the behaviour of
Langmuir wavesPlasma oscillations, also known as Langmuir waves (after Irving Langmuir), are rapid oscillations of the electron density in conducting media such as Plasma (physics), plasmas or metals in the ultraviolet region. The oscillations can be described as ...
in
plasma Plasma or plasm may refer to: Science * Plasma (physics), one of the four fundamental states of matter * Plasma (mineral) or heliotrope, a mineral aggregate * Quark–gluon plasma, a state of matter in quantum chromodynamics Biology * Blood plasma ...
. When this phenomenon is studied, the observed states from the subset are as indicated in red. So one could ask what is the probability of observing a state in a certain position of the red subset; if such a probability exists, it is called the probability measure of the system. This kind of complicated support appears quite frequently in
dynamical systems In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in a Manifold, geometrical space. Examples include the mathematical models that describe the ...
. It is not simple to establish that the system has a probability measure, and the main problem is the following. Let t_1 \ll t_2 \ll t_3 be instants in time and O a subset of the support; if the probability measure exists for the system, one would expect the frequency of observing states inside set O would be equal in interval _1,t_2/math> and _2,t_3/math>, which might not happen; for example, it could oscillate similar to a sine, \sin(t), whose limit when t \rightarrow \infty does not converge. Formally, the measure exists only if the limit of the relative frequency converges when the system is observed into the infinite future. The branch of dynamical systems that studies the existence of a probability measure is ergodic theory. Note that even in these cases, the probability distribution, if it exists, might still be termed "continuous" or "discrete" depending on whether the support is uncountable or countable, respectively.


Random number generation

Most algorithms are based on a pseudorandom number generator that produces numbers ''X'' that are uniformly distributed in the half-open interval [0,1). These random variates ''X'' are then transformed via some algorithm to create a new random variate having the required probability distribution. With this source of uniform pseudo-randomness, realizations of any random variable can be generated. For example, suppose U has a uniform distribution between 0 and 1. To construct a random Bernoulli variable for some 0 < p < 1, we define so that \textrm(X=1) = \textrm(U This random variable X has a Bernoulli distribution with parameter p. Note that this is a transformation of discrete random variable. For a distribution function F of a continuous random variable, a continuous random variable must be constructed. F^, an inverse function of F, relates to the uniform variable U: = . For example, suppose a random variable that has an exponential distribution F(x) = 1 - e^ must be constructed. \begin F(x) = u &\Leftrightarrow 1-e^ = u \\ &\Leftrightarrow e^ = 1-u \\&\Leftrightarrow -\lambda x = \ln(1-u) \\ &\Leftrightarrow x = \frac\ln(1-u) \end so F^(u) = \frac\ln(1-u) and if U has a U(0,1) distribution, then the random variable X is defined by X = F^(U) = \frac \ln(1-U). This has an exponential distribution of \lambda. A frequent problem in statistical simulations (the Monte Carlo method) is the generation of Pseudorandomness, pseudo-random numbers that are distributed in a given way.


Common probability distributions and their applications

The concept of the probability distribution and the random variables which they describe underlies the mathematical discipline of probability theory, and the science of statistics. There is spread or variability in almost any value that can be measured in a population (e.g. height of people, durability of a metal, sales growth, traffic flow, etc.); almost all measurements are made with some intrinsic error; in physics, many processes are described probabilistically, from the Kinetic theory of gases, kinetic properties of gases to the quantum mechanical description of fundamental particles. For these and many other reasons, simple numbers are often inadequate for describing a quantity, while probability distributions are often more appropriate. The following is a list of some of the most common probability distributions, grouped by the type of process that they are related to. For a more complete list, see list of probability distributions, which groups by the nature of the outcome being considered (discrete, continuous, multivariate, etc.) All of the univariate distributions below are singly peaked; that is, it is assumed that the values cluster around a single point. In practice, actually observed quantities may cluster around multiple values. Such quantities can be modeled using a mixture distribution.


Linear growth (e.g. errors, offsets)

* Normal distribution (Gaussian distribution), for a single such quantity; the most commonly used continuous distribution


Exponential growth (e.g. prices, incomes, populations)

* Log-normal distribution, for a single such quantity whose log is Normal distribution, normally distributed * Pareto distribution, for a single such quantity whose log is Exponential distribution, exponentially distributed; the prototypical power law distribution


Uniformly distributed quantities

* Discrete uniform distribution, for a finite set of values (e.g. the outcome of a fair die) * Continuous uniform distribution, for continuously distributed values


Bernoulli trials (yes/no events, with a given probability)

* Basic distributions: **
Bernoulli distribution In probability theory Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these ax ...

Bernoulli distribution
, for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no) ** Binomial distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed total number of Independent (statistics), independent occurrences ** Negative binomial distribution, for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs ** Geometric distribution, for binomial-type observations but where the quantity of interest is the number of failures before the first success; a special case of the
negative binomial distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
* Related to sampling schemes over a finite population: ** Hypergeometric distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, using sampling without replacement ** Beta-binomial distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, sampling using a Pólya urn model (in some sense, the "opposite" of sampling without replacement)


Categorical outcomes (events with ''K'' possible outcomes)

*
Categorical distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expres ...
, for a single categorical outcome (e.g. yes/no/maybe in a survey); a generalization of the
Bernoulli distribution In probability theory Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these ax ...

Bernoulli distribution
* Multinomial distribution, for the number of each type of categorical outcome, given a fixed number of total outcomes; a generalization of the
binomial distribution In probability theory Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically thes ...

binomial distribution
* Multivariate hypergeometric distribution, similar to the multinomial distribution, but using sampling without replacement; a generalization of the
hypergeometric distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...


Poisson process (events that occur independently with a given rate)

*
Poisson distribution In probability theory and statistics, the Poisson distribution (; ), named after France, French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a f ...
, for the number of occurrences of a Poisson-type event in a given period of time * Exponential distribution, for the time before the next Poisson-type event occurs * Gamma distribution, for the time before the next k Poisson-type events occur


Absolute values of vectors with normally distributed components

* Rayleigh distribution, for the distribution of vector magnitudes with Gaussian distributed orthogonal components. Rayleigh distributions are found in RF signals with Gaussian real and imaginary components. * Rice distribution, a generalization of the Rayleigh distributions for where there is a stationary background signal component. Found in Rician fading of radio signals due to multipath propagation and in MR images with noise corruption on non-zero NMR signals.


Normally distributed quantities operated with sum of squares

* Chi-squared distribution, the distribution of a sum of squared standard normal variables; useful e.g. for inference regarding the sample variance of normally distributed samples (see chi-squared test) * Student's t distribution, the distribution of the ratio of a standard normal variable and the square root of a scaled Chi squared distribution, chi squared variable; useful for inference regarding the mean of normally distributed samples with unknown variance (see Student's t-test) * F-distribution, the distribution of the ratio of two scaled Chi squared distribution, chi squared variables; useful e.g. for inferences that involve comparing variances or involving R-squared (the squared Pearson product-moment correlation coefficient, correlation coefficient)


As conjugate prior distributions in Bayesian inference

* Beta distribution, for a single probability (real number between 0 and 1); conjugate to the
Bernoulli distribution In probability theory Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these ax ...

Bernoulli distribution
and
binomial distribution In probability theory Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically thes ...

binomial distribution
* Gamma distribution, for a non-negative scaling parameter; conjugate to the rate parameter of a
Poisson distribution In probability theory and statistics, the Poisson distribution (; ), named after France, French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a f ...
or exponential distribution, the Precision (statistics), precision (inverse variance) of a
normal distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by ex ...

normal distribution
, etc. * Dirichlet distribution, for a vector of probabilities that must sum to 1; conjugate to the categorical distribution and multinomial distribution; generalization of the beta distribution *Wishart distribution, for a symmetric non-negative definite matrix; conjugate to the inverse of the covariance matrix of a
multivariate normal distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
; generalization of the gamma distribution


Some specialized applications of probability distributions

* The cache language models and other Statistical Language Model, statistical language models used in natural language processing to assign probabilities to the occurrence of particular words and word sequences do so by means of probability distributions. * In quantum mechanics, the probability density of finding the particle at a given point is proportional to the square of the magnitude of the particle's wavefunction at that point (see Born rule). Therefore, the probability distribution function of the position of a particle is described by P_ (t) = \int_a^b d x\,, \Psi(x,t), ^2 , probability that the particle's position will be in the interval in dimension one, and a similar triple integral in dimension three. This is a key principle of quantum mechanics. * Probabilistic load flow in power-flow study explains the uncertainties of input variables as probability distribution and provides the power flow calculation also in term of probability distribution. * Prediction of natural phenomena occurrences based on previous
frequency distributionIn statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a ...

frequency distribution
s such as tropical cyclones, hail, time in between events, etc.


See also

* Conditional probability distribution * Joint probability distribution * Quasiprobability distribution * Empirical probability, Empirical probability distribution * Histogram * Riemann–Stieltjes integral#Application to probability theory, Riemann–Stieltjes integral application to probability theory


Lists

* List of probability distributions * List of statistical topics


References


Citations


Sources

* *


External links

*
Field Guide to Continuous Probability Distributions
Gavin E. Crooks. {{DEFAULTSORT:Probability Distribution Probability distributions, Mathematical and quantitative methods (economics) it:Variabile casuale#Distribuzione di probabilità