probability distribution

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OR:

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 expressing it through a set o ...
and
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, a probability distribution is the mathematical function 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 hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs wh ...
. It is a mathematical description of a
random In common usage, 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. Ind ...
phenomenon in terms of its sample space and the
probabilities Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
of
events Event may refer to: Gatherings of people * Ceremony A ceremony (, ) is a unified ritualistic event with a purpose, usually consisting of a number of artistic components, performed on a special occasion. The word may be of Etruscan language, ...
(
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s 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 to be conducted, etc.

# Introduction

A probability distribution is a mathematical description of the probabilities of events, subsets of the 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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
, a set of vectors, 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 A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on randomness, random events. It is a mapping or a function from possible Outcome (pr ...
(so the sample space can be seen as a numeric set), it is common to distinguish between discrete and absolutely continuous
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
s. In the discrete case, it is sufficient to specify a
probability mass function In probability theory, probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. T ...
$p$ assigning a probability to each possible outcome: for example, when throwing a fair die, each of the six values 1 to 6 has the probability 1/6. The probability of an event 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. Absolutely continuous probability distributions can be described in several ways. The
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ...
describes the
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to 0, zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century New Latin, Modern Latin coinage ''infinitesimus'', which ori ...
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 and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
, which describes the probability that the random variable is no larger than a given value (i.e., $P\left(X < x\right)$ for some $x$). The cumulative distribution function is the area under the
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ...
from $-\infty$ to $x$, as described by the picture to the right.

# General probability 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 absolutely continuous and discrete variables, is by means of a probability function $P\colon \mathcal \to \Reals$ whose input space $\mathcal$ is related to the sample space, and gives a
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
probability as its output. 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 and . However, because of the widespread use of
random variables A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on randomness, random events. It is a mapping or a function from possible Outcome (pr ...
, which transform the sample space into a set of numbers (e.g., $\R$, $\N$), 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 $P\left(X \in E\right)$ the probability that a certain value of the 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 Russian mathematician Andrey Kolmogorov in 1933. These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probabili ...
, that is: # $P\left(X \in E\right) \ge 0 \; \forall E \in \mathcal$, so the probability is non-negative # $P\left(X \in E\right) \le 1 \; \forall E \in \mathcal$, so no probability exceeds $1$ # $P\left(X \in \bigsqcup_ E_i \right) = \sum_i P\left(X \in E_i\right)$ 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, a probability space or a probability triple (\Omega, \mathcal, P) is a space (mathematics), mathematical construct that provides a formal model of a randomness, random process or "experiment". For example, one can define a ...
$\left(X, \mathcal, P\right)$, 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 usually belong to one of two classes. A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is 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 theory, probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. T ...
. On the other hand, absolutely 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 absolutely continuous case, probabilities are described by a
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ...
, and the probability distribution is by definition the integral of the probability density function. The
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The param ...
is a commonly encountered absolutely continuous probability distribution. More complex experiments, such as those involving
stochastic processes In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a Indexed family, family of random variables. Stochastic processes are widely used as mathematical models of systems and phen ...
defined in continuous time, 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 mea ...
s. A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called univariate, while a distribution whose sample space is a
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
of dimension 2 or more is called multivariate. A univariate distribution gives the probabilities of a single
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
taking on various different values; a multivariate distribution (a
joint probability distribution Given two random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from ...
) gives the probabilities of a
random vector In probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a num ...
– 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 and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' statistical independence, independent experiment (prob ...
, the
hypergeometric distribution In probability theory and statistics, the hypergeometric distribution is a Probability distribution#Discrete probability distribution, discrete probability distribution that describes the probability of k successes (random draws for which the o ...
, and the
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The param ...
. A commonly encountered multivariate distribution is the
multivariate normal distribution In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One d ...
. Besides the probability function, the cumulative distribution function, the probability mass function and the probability density function, the moment generating function and the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the Function (mathematics), function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'' ...
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.

## Basic terms

*''
Random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
'': takes values from a sample space; probabilities describe which values and set of values are taken more likely. *'' Event'': set of possible values (outcomes) of a random variable that occurs with a certain probability. *'' Probability function'' or ''probability measure'': describes the probability $P\left(X \in E\right)$ that the event $E,$ occurs.Chapters 1 and 2 of *''
Cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
'': function evaluating the
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
that $X$ will take a value less than or equal to $x$ for a random variable (only for real-valued random variables). *'' Quantile function'': the inverse of the cumulative distribution function. Gives $x$ such that, with probability $q$, $X$ will not exceed $x$.

## Discrete probability distributions

*Discrete probability distribution: for many random variables with finitely or countably infinitely many values. *''
Probability mass function In probability theory, probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. T ...
'' (''pmf''): function that gives the probability that a discrete random variable is equal to some value. *'' Frequency distribution'': a table that displays the frequency of various outcomes . *''
Relative frequency In statistics, the frequency (or absolute frequency) of an Event (probability theory), event i is the number n_i of times the observation has occurred/recorded in an experiment or study. These frequencies are often depicted graphically or in tabu ...
distribution'': a frequency distribution where each value has been divided (normalized) by a number of outcomes in a sample (i.e. sample size). *''
Categorical distribution In probability theory and statistics, a categorical distribution (also called a generalized Bernoulli distribution, multinoulli distribution) is a discrete probability distribution that describes the possible results of a random variable that can ...
'': for discrete random variables with a finite set of values.

## Absolutely continuous probability distributions

*Absolutely continuous probability distribution: for many random variables with uncountably many values. *''
Probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ...
'' (''pdf'') or ''probability density'': function whose value at any given sample (or point) in the 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.

## Related terms

* ''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, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded: that is, they have heavier tails than the exponential distribution. In many applications it is the right tail of the distrib ...
, Long-tailed distribution, fat-tailed distribution
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, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
'' or ''mean'': the weighted average of the possible values, using their probabilities as their weights; or the continuous analog thereof. *''
Median In statistics and probability theory, the median is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be th ...
'': 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. * ''Mode'': for a discrete random variable, the value with highest probability; for an absolutely continuous random variable, a location at which the probability density function has a local peak. *''
Quantile In statistics and probability, quantiles are cut points dividing the Range (statistics), range of a probability distribution into continuous intervals with equal probabilities, or dividing the Observation (statistics), observations in a Sample (s ...
'': the q-quantile is the value $x$ such that $P\left(X < x\right) = q$. *''
Variance In probability theory and statistics, variance is the expected value, expectation of the squared Deviation (statistics), deviation of a random variable from its population mean or sample mean. Variance is a measure of statistical dispersion, di ...
'': the second moment of the pmf or pdf about the mean; an important measure of the dispersion of the distribution. *''
Standard deviation In statistics, the standard Deviation (statistics), 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 (al ...
'': the square root of the variance, and hence another measure of dispersion. * ''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 and statistics, skewness is a measure of the asymmetry of the probability distribution of a real number, real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined. For ...
'': a measure of the extent to which a pmf or pdf "leans" to one side of its mean. The third
standardized moment In probability theory and statistics, a standardized moment of a probability distribution is a moment (often a higher degree central moment) that is normalized, typically by a power of the standard deviation, rendering the moment scale invariant. ...
of the distribution. *''
Kurtosis In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real number, real-valued random variable. Like skew ...
'': a measure of the "fatness" of the tails of a pmf or pdf. The fourth standardized moment of the distribution.

# Cumulative distribution function

In the special case of a real-valued random variable, the probability distribution can equivalently be represented by a cumulative distribution function instead of a probability measure. The cumulative distribution function of a random variable $X$ with regard to a probability distribution $p$ is defined as $F(x) = P(X \leq x).$ The cumulative distribution function of any real-valued random variable has the properties: *
• $F\left(x\right)$ is non-decreasing;
• *
• $F\left(x\right)$ is right-continuous;
• *
• $0 \le F\left(x\right) \le 1$;
• *
• $\lim_ F\left(x\right) = 0$ and $\lim_ F\left(x\right) = 1$; and
• *
• $\Pr\left(a < X \le b\right) = F\left(b\right) - F\left(a\right)$.
• Conversely, any function $F:\mathbb\to\mathbb$ that satisfies the first four of the properties above is the cumulative distribution function of some probability distribution on the real numbers. Any probability distribution can be decomposed as the sum of a discrete, an absolutely continuous and a singular continuous distribution, and thus any cumulative distribution function admits a decomposition as the sum of the three according cumulative distribution functions.

# 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 (
almost surely 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 means that the probability of any event $E$ can be expressed as a (finite or
countably infinite In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
) sum: $P(X\in E) = \sum_ P(X = \omega),$ where $A$ is a countable set. Thus the discrete random variables are exactly those with a
probability mass function In probability theory, probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. T ...
$p\left(x\right) = P\left(X=x\right)$. 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 $p\left(n\right) = \tfrac$ for $n = 1, 2, ...$, the sum of probabilities would be $1/2 + 1/4 + 1/8 + \dots = 1$. A discrete random variable is a random variable whose probability distribution is discrete. Well-known discrete probability distributions used in statistical modeling include the
Poisson distribution In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...
, the
Bernoulli distribution In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,James Victor Uspensky: ''Introduction to Mathematical Probability'', McGraw-Hill, New York 1937, page 45 is the discrete probabili ...
, the
binomial distribution In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' statistical independence, independent experiment (prob ...
, the
geometric distribution In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions: * The probability distribution of the number ''X'' of Bernoulli trials needed to get one success, supported on the set \; * ...
, the
negative binomial distribution In probability theory and statistics, the negative binomial distribution is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non-r ...
and
categorical distribution In probability theory and statistics, a categorical distribution (also called a generalized Bernoulli distribution, multinoulli distribution) is a discrete probability distribution that describes the possible results of a random variable that can ...
. When a sample (a set of observations) is drawn from a larger population, the sample points have an empirical distribution that is discrete, and which provides information about the population distribution. Additionally, the
discrete uniform distribution In probability theory and statistics, the discrete uniform distribution is a Symmetric distribution, symmetric discrete probability distribution, probability distribution wherein a finite number of values are equally likely to be observed; ever ...
is commonly used in computer programs that make equal-probability random selections between a number of choices.

## Cumulative distribution function

A real-valued discrete random variable can equivalently be defined as a random variable whose cumulative distribution function increases only by jump discontinuities—that is, its cdf increases only where it "jumps" to a higher value, and is constant in intervals without jumps. The points where jumps occur are precisely the values which the random variable may take. Thus the cumulative distribution function has the form $F(x) = P(X \leq x) = \sum_ p(\omega).$ Note that the points where the cdf jumps always form a countable set; this may be any countable set and thus may even be dense in the real numbers.

## Dirac delta representation

A discrete probability distribution is often represented with Dirac measures, the probability distributions of deterministic random variables. For any outcome $\omega$, let $\delta_\omega$ be the Dirac measure concentrated at $\omega$. Given a discrete probability distribution, there is a countable set $A$ with $P\left(X \in A\right) = 1$ and a probability mass function $p$. If $E$ is any event, then $P(X \in E) = \sum_ p(\omega) \delta_\omega(E),$ or in short, $P_X = \sum_ p(\omega) \delta_\omega.$ Similarly, discrete distributions can be represented with the
Dirac delta function In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
as a generalized
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ...
$f$, where $f(x) = \sum_ p(\omega) \delta(x - \omega),$ which means $P(X \in E) = \int_E f(x) \, dx = \sum_ p(\omega) \int_E \delta(x - \omega) = \sum_ p(\omega)$ for any event $E.$

## Indicator-function representation

For a discrete random variable $X$, let $u_0, u_1, \dots$ be the values it can take with non-zero probability. Denote $\Omega_i=X^(u_i)= \,\, i=0, 1, 2, \dots$ These are disjoint sets, 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 $u_0, u_1, \dots$ 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. Expressed formally, the random variable $X$ has a one-point distribution if it has a possible outcome $x$ such that $P\left(Xx\right)=1.$ All other possible outcomes then have probability 0. Its cumulative distribution function jumps immediately from 0 to 1.

# Absolutely continuous probability distribution

An absolutely continuous probability distribution is a probability distribution on the real numbers with uncountably many possible values, such as a whole interval in the real line, and where the probability of any event can be expressed as an integral. More precisely, a real random variable $X$ has an absolutely continuous probability distribution if there is a function
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ...
, so that absolutely continuous probability distributions are exactly those with a probability density function. 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, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
with coinciding upper and lower limits is always equal to zero. If the interval
uniform A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...
, chi-squared, and others.

## Cumulative distribution function

Absolutely continuous probability distributions as defined above are precisely those with an absolutely continuous cumulative distribution function. In this case, the cumulative distribution function $F$ has the form $F(x) = P(X \leq x) = \int_^x f(t)\,dt$ where $f$ is a density of the random variable $X$ with regard to the distribution $P$. ''Note on terminology:'' Absolutely continuous distributions ought to be distinguished from continuous distributions, which are those having a continuous cumulative distribution function. Every absolutely continuous distribution is a continuous distribution but the converse is not true, there exist singular distributions, 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. Some authors however use the term "continuous distribution" to denote all distributions whose cumulative distribution function is absolutely continuous, i.e. refer to absolutely continuous distributions as continuous distributions. For a more general definition of density functions and the equivalent absolutely continuous measures see absolutely continuous measure.

# Kolmogorov definition

In the measure-theoretic formalization of
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 expressing it through a set o ...
, a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
is defined as a
measurable function In mathematics and in particular Mathematical analysis#Measure_theory, measure theory, a measurable function is a function between the underlying sets of two measurable space, measurable spaces that preserves the structure of the spaces: the prei ...
$X$ from a
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a space (mathematics), mathematical construct that provides a formal model of a randomness, random process or "experiment". For example, one can define a ...
$\left(\Omega, \mathcal, \mathbb\right)$ to a
measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a Set (mathematics), set and a Sigma-algebra, σ-algebra, which defines the subsets that will be measured. Definition Consider a set X and a Si ...
$\left(\mathcal,\mathcal\right)$. Given that probabilities of events of the form $\$ satisfy Kolmogorov's probability axioms, the probability distribution of $X$ is the image measure $X_*\mathbb$ 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 mea ...
on $\left(\mathcal,\mathcal\right)$ satisfying $X_*\mathbb = \mathbbX^$.

# Other kinds of distributions

Absolutely 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, a hypercube is an N-dimensional space, ''n''-dimensional analogue of a Square (geometry), square () and a cube (). It is a Closed set, closed, Compact space, compact, Convex polytope, convex figure whose 1-N-skeleton, skeleton consis ...
or balls. However, this is not always the case, and there exist phenomena with supports that are actually complicated curves 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 equations. Such a system can be either linear differential equations, linear or non-linear differential equations, non-linear. Also, such a system can be either a sy ...
(commonly known as the Rabinovich–Fabrikant equations) that can be used to model the behaviour of Langmuir waves in
plasma Plasma or plasm may refer to: Science * Plasma (physics) Plasma () 1, where \nu_ is the electron gyrofrequency and \nu_ is the electron collision rate. It is often the case that the electrons are magnetized while the ions are not. Magnetized ...
. 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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
. 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
ergodic theory Ergodic theory (Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assum ...
. Note that even in these cases, the probability distribution, if it exists, might still be termed "absolutely 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 A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random number generation, random n ...
that produces numbers $X$ that are uniformly distributed in the half-open interval . 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
Monte Carlo method Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determini ...
) is the generation of 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 properties of gases to the
quantum mechanical Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including qua ...
description of
fundamental particles In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions (quarks, leptons, antiqu ...
. For these and many other reasons, simple
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
s 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, absolutely 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 In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The param ...
(Gaussian distribution), for a single such quantity; the most commonly used absolutely continuous distribution

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

*
Log-normal distribution In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed, ...
, for a single such quantity whose log is normally distributed *
Pareto distribution The Pareto distribution, named after the Italian civil engineer, economist An economist is a professional and practitioner in the social sciences, social science discipline of economics. The individual may also study, develop, and apply th ...
, for a single such quantity whose log is exponentially distributed; the prototypical
power law In statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, a ...
distribution

## Uniformly distributed quantities

*
Discrete uniform distribution In probability theory and statistics, the discrete uniform distribution is a Symmetric distribution, symmetric discrete probability distribution, probability distribution wherein a finite number of values are equally likely to be observed; ever ...
, for a finite set of values (e.g. the outcome of a fair die) *
Continuous uniform distribution In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of Symmetric distribution, symmetric probability distributions. The distribution describes an experiment where there is an arbitrary ...
, for absolutely continuously distributed values

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

* Basic distributions: **
Bernoulli distribution In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,James Victor Uspensky: ''Introduction to Mathematical Probability'', McGraw-Hill, New York 1937, page 45 is the discrete probabili ...
, for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no) **
Binomial distribution In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' statistical independence, independent experiment (prob ...
, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed total number of
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independen ...
occurrences **
Negative binomial distribution In probability theory and statistics, the negative binomial distribution is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non-r ...
, for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs **
Geometric distribution In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions: * The probability distribution of the number ''X'' of Bernoulli trials needed to get one success, supported on the set \; * ...
, 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 and statistics, the negative binomial distribution is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non-r ...
* Related to sampling schemes over a finite population: **
Hypergeometric distribution In probability theory and statistics, the hypergeometric distribution is a Probability distribution#Discrete probability distribution, discrete probability distribution that describes the probability of k successes (random draws for which the o ...
, 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 possible outcomes)

*
Categorical distribution In probability theory and statistics, a categorical distribution (also called a generalized Bernoulli distribution, multinoulli distribution) is a discrete probability distribution that describes the possible results of a random variable that can ...
, for a single categorical outcome (e.g. yes/no/maybe in a survey); a generalization of the
Bernoulli distribution In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,James Victor Uspensky: ''Introduction to Mathematical Probability'', McGraw-Hill, New York 1937, page 45 is the discrete probabili ...
*
Multinomial distribution In probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for each side of a ''k''-sided dice rolled ''n'' times. For ''n'' statistical independence, ind ...
, 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 and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' statistical independence, independent experiment (prob ...
* Multivariate hypergeometric distribution, similar to the
multinomial distribution In probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for each side of a ''k''-sided dice rolled ''n'' times. For ''n'' statistical independence, ind ...
, but using sampling without replacement; a generalization of the
hypergeometric distribution In probability theory and statistics, the hypergeometric distribution is a Probability distribution#Discrete probability distribution, discrete probability distribution that describes the probability of k successes (random draws for which the o ...

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

*
Poisson distribution In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...
, for the number of occurrences of a Poisson-type event in a given period of time *
Exponential distribution In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
, for the time before the next Poisson-type event occurs *
Gamma distribution In probability theory and statistics, the gamma distribution is a two-statistical parameter, parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special ca ...
, for the time before the next k Poisson-type events occur

## Absolute values of vectors with normally distributed components

*
Rayleigh distribution In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom. The distributi ...
, 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 Rice is the seed of the Poaceae, grass species ''Oryza sativa'' (Asian rice) or less commonly ''Oryza glaberrima'' (African rice). The name wild rice is usually used for species of the genera ''Zizania (genus), Zizania'' and ''Porteresia'', bo ...
, 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 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 ...
, the distribution of a sum of squared
standard normal In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The param ...
variables; useful e.g. for inference regarding the
sample variance In probability theory and statistics, variance is the expected value, expectation of the squared Deviation (statistics), deviation of a random variable from its population mean or sample mean. Variance is a measure of statistical dispersion, di ...
of normally distributed samples (see
chi-squared test A chi-squared test (also chi-square or test) is a Statistical hypothesis testing, statistical hypothesis test used in the analysis of contingency table, contingency tables when the sample sizes are large. In simpler terms, this test is primaril ...
) * Student's t distribution, the distribution of the ratio of a
standard normal In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The param ...
variable and the square root of a scaled chi squared variable; useful for inference regarding the
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude (mathematics), magnitude and sign (mathematics), sign) of a gi ...
of normally distributed samples with unknown variance (see
Student's t-test A ''t''-test is any statistical hypothesis test in which the test statistic A test statistic is a statistic (a quantity derived from the Sample (statistics), sample) used in statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Sta ...
) *
F-distribution In probability theory and statistics, the ''F''-distribution or F-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor) is a continuous probability distribution th ...
, the distribution of the ratio of two scaled chi squared variables; useful e.g. for inferences that involve comparing variances or involving R-squared (the squared
correlation coefficient A correlation coefficient is a numerical measure of some type of correlation and dependence, correlation, meaning a statistical relationship between two variable (mathematics), variables. The variables may be two column (database), columns of a giv ...
)

## As conjugate prior distributions in Bayesian inference

*
Beta distribution In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive Statistical parameter, parameters, denoted by ''alpha'' (''α'') and ''beta'' ...
, for a single probability (real number between 0 and 1); conjugate to the
Bernoulli distribution In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,James Victor Uspensky: ''Introduction to Mathematical Probability'', McGraw-Hill, New York 1937, page 45 is the discrete probabili ...
and
binomial distribution In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' statistical independence, independent experiment (prob ...
*
Gamma distribution In probability theory and statistics, the gamma distribution is a two-statistical parameter, parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special ca ...
, for a non-negative scaling parameter; conjugate to the rate parameter of a
Poisson distribution In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...
or
exponential distribution In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
, the precision (inverse
variance In probability theory and statistics, variance is the expected value, expectation of the squared Deviation (statistics), deviation of a random variable from its population mean or sample mean. Variance is a measure of statistical dispersion, di ...
) of a
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The param ...
, etc. *
Dirichlet distribution In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted \operatorname(\boldsymbol\alpha), is a family of Continuous probability distribution, continuous multivariate random variable, multiva ...
, for a vector of probabilities that must sum to 1; conjugate to the
categorical distribution In probability theory and statistics, a categorical distribution (also called a generalized Bernoulli distribution, multinoulli distribution) is a discrete probability distribution that describes the possible results of a random variable that can ...
and
multinomial distribution In probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for each side of a ''k''-sided dice rolled ''n'' times. For ''n'' statistical independence, ind ...
; generalization of the
beta distribution In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive Statistical parameter, parameters, denoted by ''alpha'' (''α'') and ''beta'' ...
* Wishart distribution, for a symmetric non-negative definite matrix; conjugate to the inverse of the
covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square Matrix (mathematics), matrix giving the covariance between ea ...
of a
multivariate normal distribution In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One d ...
; generalization of the
gamma distribution In probability theory and statistics, the gamma distribution is a two-statistical parameter, parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special ca ...

## Some specialized applications of probability distributions

* The cache language models and other statistical language models used in
natural language processing Natural language processing (NLP) is an interdisciplinary subfield of linguistics, computer science, and artificial intelligence concerned with the interactions between computers and human language, in particular how to program computers to proc ...
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 A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex number, complex-valued probability amplitude, and the probabilities for the possible results of ...
at that point (see
Born rule The Born rule (also called Born's rule) is a key postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result. In its simplest form, it states that the probability density of fin ...
). 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 distributions such as
tropical cyclone A tropical cyclone is a rapidly rotating storm, storm system characterized by a Low-pressure area, low-pressure center, a closed low-level atmospheric circulation, Beaufort scale, strong winds, and a spiral arrangement of thunderstorms tha ...
s, hail, time in between events, etc.

# Fitting

*
Conditional probability distribution In probability theory and statistics, given two joint probability distribution, jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particu ...
*
Joint probability distribution Given two random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from ...
* Quasiprobability distribution * Empirical probability distribution *
Histogram A histogram is an approximate representation of the frequency distribution, distribution of numerical data. The term was first introduced by Karl Pearson. To construct a histogram, the first step is to "Data binning, bin" (or "Data binning, buck ...
* Riemann–Stieltjes integral application to probability theory

## Lists

* List of probability distributions * List of statistical topics

* *