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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, 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 order and does not follow an intelligible pattern or combination. Individual rando ...
phenomenon in terms of its
sample space In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually de ...
and the
probabilities Probability is the branch of mathematics concerning numerical descriptions of how likely an 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 1, where, roughly speakin ...
of events (
subset In mathematics, set ''A'' is a subset of a set ''B'' if all 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 unequal, then ''A'' is a proper subset o ...
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 In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually de ...
. 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, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
, 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 (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 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. The probability mass ...
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) c ...
describes the infinitesimal 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. Ev ...
, which describes the probability that the random variable is no larger than a given value (i.e., P(X < x) 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) c ...
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 In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually de ...
, 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, 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(X \in E) 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, 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, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
(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 usually belong to one of two classes. A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a ...
(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 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. The probability mass ...
. 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) c ...
, 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-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu i ...
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 family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that a ...
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 properties such as ''countable additivity''. The difference between a probability measure and the more g ...
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, function or polynomial involving only one variable. Objects involving more than one variable are multivariate. In some cases the distinction between the univariate and multivariat ...
, while a distribution whose sample space is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
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 variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considere ...
) gives the probabilities of a
random vector In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its valu ...
– 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'' independent experiments, each asking a yes–no qu ...
, the
hypergeometric distribution In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of k successes (random draws for which the object drawn has a specified feature) in n draws, ''without' ...
, and the
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu i ...
. 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 ...
. 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 ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at point ...
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(X \in E) 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. Ev ...
'': 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 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 e ...
'': 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 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. The probability mass ...
'' (''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 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 ca ...
'': 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) c ...
'' (''pdf'') or ''probability density'': function whose value at any given sample (or point) in the
sample space In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually de ...
(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 ...
'' 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'': 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'': the q-quantile is the value x such that P(X < x) = q. *''
Variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...
'': the second moment of the pmf or pdf about the mean; an important measure of the dispersion of the distribution. *'' Standard deviation'': 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-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined. For a unimo ...
'': 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 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-valued random variable. Like skewness, kur ...
'': 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(x) is non-decreasing;
  • *
  • F(x) is
    right-continuous In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
    ;
  • *
  • 0 \le F(x) \le 1;
  • *
  • \lim_ F(x) = 0 and \lim_ F(x) = 1; and
  • *
  • \Pr(a < X \le b) = F(b) - F(a).
  • 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 Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a ...
    , 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, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0 ...
    ) which means that the probability of any event E can be expressed as a (finite or
    countably infinite In mathematics, a set is countable if either it is 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 from it into the natural numbers ...
    ) 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 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. The probability mass ...
    p(x) = P(X=x). 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(n) = \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 ...
    , 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 probab ...
    , 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'' independent experiments, each asking a yes–no qu ...
    , 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 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 expr ...
    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 ca ...
    . 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 probability distribution wherein a finite number of values are equally likely to be observed; every one of ''n'' values has equal probability 1/''n''. Ano ...
    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 measure In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element ''x'' or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields ...
    s, 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(X \in A) = 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, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
    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) c ...
    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 In mathematics, a degenerate distribution is, according to some, a probability distribution in a space with support only on a manifold of lower dimension, and according to others a distribution with support only at a single point. By the latter d ...
    . 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.


    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 f: \Reals \to , \infty/math> such that for each interval ,b\subset \mathbb the probability of X belonging to ,b/math> is given by the integral of f over I: P\left(a \le X \le b \right) = \int_a^b f(x) \, dx . This is the definition of 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) c ...
    , 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 functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
    with coinciding upper and lower limits is always equal to zero. If the interval ,b/math> is replaced by any measurable set A, the according equality still holds: P(X \in A) = \int_A f(x) \, dx . An absolutely continuous random variable is a random variable whose probability distribution is absolutely continuous. There are many examples of absolutely continuous probability distributions: normal,
    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 The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. This distribution has neither a probability density function nor a probability mass function, since although its cumulative ...
    . 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 measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is i ...
    X from a
    probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
    (\Omega, \mathcal, \mathbb) to a
    measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. Definition Consider a set X and a σ-algebra \mathcal A on X. Then ...
    (\mathcal,\mathcal). 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 properties such as ''countable additivity''. The difference between a probability measure and the more g ...
    on (\mathcal,\mathcal) 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 or balls. 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 mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
    \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 (commonly known as the
    Rabinovich–Fabrikant equations The Rabinovich–Fabrikant equations are a set of three coupled ordinary differential equations exhibiting Chaos theory, chaotic behaviour for certain values of the parameters. They are named after Mikhail Rabinovich and Anatoly Fabrikant, who de ...
    ) that can be used to model the behaviour of Langmuir waves in 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 describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
    . 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 Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...
    . 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 numbers. The PRNG-generate ...
    that produces numbers X that are uniformly distributed in the half-open interval . These
    random variate In probability and statistics, a random variate or simply variate is a particular outcome of a ''random variable'': the random variates which are other outcomes of the same random variable might have different values ( random numbers). A random ...
    s 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 X = \begin 1,& \text U so that \Pr(X=1) = \Pr(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 an absolutely continuous random variable, an absolutely 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 \\ pt&\Leftrightarrow e^ = 1-u \\ pt&\Leftrightarrow -\lambda x = \ln(1-u) \\ pt&\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 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 deter ...
    ) 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 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 quantum chemistry, qu ...
    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, anti ...
    . 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 ...
    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 In probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection ...
    .


    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-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu i ...
    (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 normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...
    , for a single such quantity whose log is normally distributed *
    Pareto distribution The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto ( ), is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actu ...
    , for a single such quantity whose log is exponentially distributed; the prototypical
    power law In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one qua ...
    distribution


    Uniformly distributed quantities

    *
    Discrete uniform distribution In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein a finite number of values are equally likely to be observed; every one of ''n'' values has equal probability 1/''n''. Ano ...
    , for a finite set of values (e.g. the outcome of a fair die) * Continuous uniform distribution, 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 probab ...
    , 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'' independent experiments, each asking a yes–no qu ...
    , 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 * Independe ...
    occurrences **
    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 expr ...
    , 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 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 expr ...
    * Related to sampling schemes over a finite population: **
    Hypergeometric distribution In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of k successes (random draws for which the object drawn has a specified feature) in n draws, ''without' ...
    , 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 ca ...
    , 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 probab ...
    *
    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'' independent trials each of w ...
    , 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'' independent experiments, each asking a yes–no qu ...
    * 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'' independent trials each of w ...
    , but using sampling without replacement; a generalization of the
    hypergeometric distribution In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of k successes (random draws for which the object drawn has a specified feature) in n draws, ''without' ...


    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 ...
    , 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 averag ...
    , for the time before the next Poisson-type event occurs *
    Gamma distribution In probability theory and statistics, the gamma distribution is a two- parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma dis ...
    , 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 distribu ...
    , 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 In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squar ...
    , the distribution of a sum of squared standard normal variables; useful e.g. for inference regarding the
    sample variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...
    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 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 and sign) of a given data set. For a data set, the '' ari ...
    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 follows a Student's ''t''-distribution under the null hypothesis. It is most commonly applied when the test statistic would follow a normal distribution if the value of ...
    ) *
    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 ...
    , 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, meaning a statistical relationship between two variables. The variables may be two columns of a given data set of observations, often called a sample, or two componen ...
    )


    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 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 probab ...
    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'' independent experiments, each asking a yes–no qu ...
    *
    Gamma distribution In probability theory and statistics, the gamma distribution is a two- parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma dis ...
    , 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 ...
    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 averag ...
    , the precision (inverse
    variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...
    ) of a
    normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu i ...
    , 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 multivariate probability distributions parameterized by a vector \bolds ...
    , 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 ca ...
    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'' independent trials each of w ...
    ; generalization of the beta distribution *
    Wishart distribution In statistics, the Wishart distribution is a generalization to multiple dimensions of the gamma distribution. It is named in honor of John Wishart, who first formulated the distribution in 1928. It is a family of probability distributions defin ...
    , for a symmetric non-negative definite matrix; conjugate to the inverse of the covariance matrix 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 ...
    ; generalization of the
    gamma distribution In probability theory and statistics, the gamma distribution is a two- parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma dis ...


    Some specialized applications of probability distributions

    * The
    cache language model A cache language model is a type of statistical language model. These occur in the natural language processing subfield of computer science and assign probabilities to given sequences of words by means of a probability distribution. Statistical lan ...
    s 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-valued probability amplitude, and the probabilities for the possible results of measurements ma ...
    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 distributions such as
    tropical cyclone A tropical cyclone is a rapidly rotating storm system characterized by a low-pressure center, a closed low-level atmospheric circulation, strong winds, and a spiral arrangement of thunderstorms that produce heavy rain and squalls. Dep ...
    s, hail, time in between events, etc.


    Fitting


    See also

    *
    Conditional probability distribution In probability theory and statistics, given two 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 particular value; in some cases the c ...
    *
    Joint probability distribution Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considere ...
    * Quasiprobability distribution * Empirical probability distribution *
    Histogram A histogram is an approximate representation of the distribution of numerical data. The term was first introduced by Karl Pearson. To construct a histogram, the first step is to " bin" (or " bucket") the range of values—that is, divide the ent ...
    * 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 Mathematical and quantitative methods (economics) it:Variabile casuale#Distribuzione di probabilità