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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 ...
and
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, 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 whe ...
. 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 ran ...
phenomenon in terms of its sample space and the probabilities 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 of ...
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 Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of all possible outcomes of a random phenomenon being observed; it may be any set: a set of real numbers, 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 variables. In the discrete case, it is sufficient to specify a probability mass function p assigning a probability to each possible outcome: for example, when throwing a fair
die Die, as a verb, refers to death, the cessation of life. Die may also refer to: Games * Die, singular of dice, small throwable objects used for producing random numbers Manufacturing * Die (integrated circuit), a rectangular piece of a semicondu ...
, each of the six values 1 to 6 has the probability 1/6. The probability of an
event Event may refer to: Gatherings of people * Ceremony, an event of ritual significance, performed on a special occasion * Convention (meeting), a gathering of individuals engaged in some common interest * Event management, the organization of ev ...
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) ca ...
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, 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) ca ...
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 measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
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 (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 (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. 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) ca ...
, 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 ...
is a commonly encountered absolutely continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures. 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 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 can ...
of dimension 2 or more is called
multivariate Multivariate may refer to: In mathematics * Multivariable calculus * Multivariate function * Multivariate polynomial In computing * Multivariate cryptography * Multivariate division algorithm * Multivariate interpolation * Multivariate optical c ...
. A univariate distribution gives the probabilities of a single random variable taking on various different values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector – 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, the hypergeometric distribution, 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 ...
. A commonly encountered multivariate distribution is the multivariate normal distribution. 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 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'': takes values from a sample space; probabilities describe which values and set of values are taken more likely. *''
Event Event may refer to: Gatherings of people * Ceremony, an event of ritual significance, performed on a special occasion * Convention (meeting), a gathering of individuals engaged in some common interest * Event management, the organization of ev ...
'': 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'': function evaluating the
probability 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 speaking, ...
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'' (''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 Sample or samples may refer to: Base meaning * Sample (statistics), a subset of a population – complete data set * Sample (signal), a digital discrete sample of a continuous analog signal * Sample (material), a specimen or small quantity of ...
(i.e. sample size). *'' Categorical distribution'': 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) ca ...
'' (''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'' or ''mean'': the weighted average 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 In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one fewer quantile th ...
'': 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 numbe ...
'': the second moment of the pmf or pdf about the mean; an important measure of the
dispersion Dispersion may refer to: Economics and finance *Dispersion (finance), a measure for the statistical distribution of portfolio returns *Price dispersion, a variation in prices across sellers of the same item *Wage dispersion, the amount of variatio ...
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'': 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'': 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;
  • *
  • 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, 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 number ...
    ) 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 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, 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 probabi ...
    , the binomial distribution, the geometric distribution, the negative binomial distribution and categorical distribution. When a
    sample Sample or samples may refer to: Base meaning * Sample (statistics), a subset of a population – complete data set * Sample (signal), a digital discrete sample of a continuous analog signal * Sample (material), a specimen or small quantity of ...
    (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 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(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 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) ca ...
    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(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) ca ...
    , 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, ...
    , 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 cumulat ...
    . 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 ...
    , a random variable is defined as a measurable function X from a probability space (\Omega, \mathcal, \mathbb) to a measurable space (\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 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\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. It is not simple to establish that the system has a probability measure, and the main problem is the following. Let t_1 \ll t_2 \ll t_3 be instants in time and O a subset of the support; if the probability measure exists for the system, one would expect the frequency of observing states inside set O would be equal in interval _1,t_2/math> and _2,t_3/math>, which might not happen; for example, it could oscillate similar to a sine, \sin(t), whose limit when t \rightarrow \infty does not converge. Formally, the measure exists only if the limit of the relative frequency converges when the system is observed into the infinite future. The branch of dynamical systems that studies the existence of a probability measure is ergodic theory. Note that even in these cases, the probability distribution, if it exists, might still be termed "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 that produces numbers X that are uniformly distributed in the
    half-open interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Othe ...
    . 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 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 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 number ...
    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 Many probability distributions that are important in theory or applications have been given specific names. Discrete distributions With finite support * The Bernoulli distribution, which takes value 1 with probability ''p'' and value 0 with pr ...
    , 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 collectio ...
    .


    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 ...
    (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 norma ...
    , for a single such quantity whose log is normally distributed * Pareto distribution, for a single such quantity whose log is exponentially distributed; the prototypical power law distribution


    Uniformly distributed quantities

    * Discrete uniform distribution, for a finite set of values (e.g. the outcome of a fair die) *
    Continuous uniform distribution In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies bet ...
    , 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 probabi ...
    , for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no) ** Binomial distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed total number of
    independent 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, for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs ** Geometric distribution, for binomial-type observations but where the quantity of interest is the number of failures before the first success; a special case of the negative binomial distribution * Related to sampling schemes over a finite population: ** Hypergeometric distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, using sampling without replacement **
    Beta-binomial distribution In probability theory and statistics, the beta-binomial distribution is a family of discrete probability distributions on a finite support of non-negative integers arising when the probability of success in each of a fixed or known number of Ber ...
    , 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 statistics, a Pólya urn model (also known as a Pólya urn scheme or simply as Pólya's urn), named after George Pólya, is a type of statistical model used as an idealized mental exercise framework, unifying many treatments. In an urn model, ...
    (in some sense, the "opposite" of sampling without replacement)


    Categorical outcomes (events with possible outcomes)

    * Categorical distribution, 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 probabi ...
    * Multinomial distribution, for the number of each type of categorical outcome, given a fixed number of total outcomes; a generalization of the binomial distribution * Multivariate hypergeometric distribution, similar to the multinomial distribution, but using sampling without replacement; a generalization of the hypergeometric distribution


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

    * Poisson distribution, for the number of occurrences of a Poisson-type event in a given period of time * Exponential distribution, for the time before the next Poisson-type event occurs *
    Gamma distribution 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 di ...
    , for the time before the next k Poisson-type events occur


    Absolute values of vectors with normally distributed components

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


    Normally distributed quantities operated with sum of squares

    *
    Chi-squared distribution 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 numbe ...
    of normally distributed samples (see
    chi-squared test A chi-squared test (also chi-square or test) is a statistical hypothesis test used in the analysis of contingency tables when the sample sizes are large. In simpler terms, this test is primarily used to examine whether two categorical variables ...
    ) * 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 '' ar ...
    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, the distribution of the ratio of two scaled chi squared variables; useful e.g. for inferences that involve comparing variances or involving
    R-squared In statistics, the coefficient of determination, denoted ''R''2 or ''r''2 and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s). It is a statistic used ...
    (the squared correlation coefficient)


    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 parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...
    , 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 probabi ...
    and binomial distribution *
    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 di ...
    , for a non-negative scaling parameter; conjugate to the rate parameter of a Poisson distribution or exponential distribution, 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 numbe ...
    ) 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 ...
    , 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 \bold ...
    , for a vector of probabilities that must sum to 1; conjugate to the categorical distribution and multinomial distribution; 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 parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...
    *
    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 defi ...
    , for a symmetric non-negative definite matrix; conjugate to the inverse of the covariance matrix of a multivariate normal distribution; 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 di ...


    Some specialized applications of probability distributions

    * The cache language models and other statistical language models used in natural language processing to assign probabilities to the occurrence of particular words and word sequences do so by means of probability distributions. * In quantum mechanics, the probability density of finding the particle at a given point is proportional to the square of the magnitude of the particle's wavefunction at that point (see Born rule). Therefore, the probability distribution function of the position of a particle is described by P_ (t) = \int_a^b d x\,, \Psi(x,t), ^2 , probability that the particle's position will be in the interval in dimension one, and a similar triple integral in dimension three. This is a key principle of quantum mechanics. * Probabilistic load flow in power-flow study explains the uncertainties of input variables as probability distribution and provides the power flow calculation also in term of probability distribution. * Prediction of natural phenomena occurrences based on previous frequency 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 ...
    * Joint probability distribution *
    Quasiprobability distribution A quasiprobability distribution is a mathematical object similar to a probability distribution but which relaxes some of Kolmogorov's axioms of probability theory. Quasiprobabilities share several of general features with ordinary probabilities, ...
    * Empirical probability distribution * Histogram * Riemann–Stieltjes integral application to probability theory


    Lists

    *
    List of probability distributions Many probability distributions that are important in theory or applications have been given specific names. Discrete distributions With finite support * The Bernoulli distribution, which takes value 1 with probability ''p'' and value 0 with pr ...
    * 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