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probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
and
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, a probability distribution is a function that gives the probabilities of occurrence of possible events 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 definite pattern or predictability in information. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. ...
phenomenon in terms of its sample space and the
probabilities Probability is a branch of mathematics and statistics concerning Event (probability theory), events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probab ...
of events (
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
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). More commonly, probability distributions are used to compare the relative occurrence of many different random values. Probability distributions can be defined in different ways and for discrete or for continuous variables. Distributions with special properties or for especially important applications are given specific names.


Introduction

A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often represented in notation 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. The sample space may be any set: a set of
real numbers In mathematics, a real number is a number that can be used to measurement, measure a continuous variable, continuous one-dimensional quantity such as a time, duration or temperature. Here, ''continuous'' means that pairs of values can have arbi ...
, a set of descriptive labels, a set of vectors, a set of arbitrary non-numerical values, etc. For example, the sample space of a coin flip could be To define probability distributions for the specific case of
random variables A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' in its mathematical definition refers ...
(so the sample space can be seen as a numeric set), it is common to distinguish between discrete and continuous
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
s. In the discrete case, it is sufficient to specify a
probability mass function In probability and statistics, a probability mass function (sometimes called ''probability function'' or ''frequency function'') is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes i ...
p assigning a probability to each possible outcome (e.g. when throwing a fair die, each of the six digits to , corresponding to the number of dots on the die, has probability \tfrac). The probability of an event is then defined to be the sum of the probabilities of all outcomes that satisfy the event; for example, the probability of the event "the die rolls an even value" is p(\text2\text) + p(\text4\text) + p(\text6\text) = \frac + \frac + \frac = \frac. In contrast, when a random variable takes values from a continuum then by convention, any individual outcome is assigned probability zero. For such continuous random variables, only events that include infinitely many outcomes such as intervals have probability greater than 0. For example, consider measuring the weight of a piece of ham in the supermarket, and assume the scale can provide arbitrarily many digits of precision. Then, the probability that it weighs ''exactly'' 500  g must be zero because no matter how high the level of precision chosen, it cannot be assumed that there are no non-zero decimal digits in the remaining omitted digits ignored by the precision level. However, for the same use case, it is possible to meet quality control requirements such as that a package of "500 g" of ham must weigh between 490 g and 510 g with at least 98% probability. This is possible because this measurement does not require as much precision from the underlying equipment. Continuous probability distributions can be described by means of the
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
, which describes the probability that the random variable is no larger than a given value (i.e., for some . The cumulative distribution function is the area under the
probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
from to , as shown in figure 1. Most continuous probability distributions encountered in practice are not only continuous but also absolutely continuous. Such distributions can be described by their
probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
. Informally, the probability density f of a random variable X describes the
infinitesimal In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...
probability that X takes any value x — that is P(x \leq X < x + \Delta x) \approx f(x) \, \Delta x as \Delta x > 0 becomes is arbitrarily small. The probability that X lies in a given interval can be computed rigorously by integrating the probability density function over that interval.


General probability definition

Let (\Omega, \mathcal, P) be 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 ...
, (E, \mathcal) be a measurable space, and X : \Omega \to E be a (E, \mathcal) -valued random variable. Then the probability distribution of X is the pushforward measure of the probability measure P onto (E, \mathcal) induced by X. Explicitly, this pushforward measure on (E, \mathcal) is given by X_ (P) (B) = P \left( X^ (B) \right) for B \in \mathcal. Any probability distribution is a
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
on (E, \mathcal) (in general different from P, unless X happens to be the identity map). 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 a σ-algebra, 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
probability as its output, particularly, a number in ,1\subseteq \Reals. The probability function P can take as argument subsets of the sample space itself, as in the coin toss example, where the function P was defined so that and . However, because of the widespread use of
random variables A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' in its mathematical definition refers ...
, 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 \bigcup_ E_i ) = \sum_i P(X \in E_i) for any countable 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 ...
(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, ...
(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 (sometimes called ''probability function'' or ''frequency function'') is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes i ...
. 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), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
, and the probability distribution is by definition the integral of the probability density function. The
normal distribution In probability theory and 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 ...
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 measure In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
s. A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called univariate, while a distribution whose sample space is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
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 Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
taking on various different values; a multivariate distribution (a
joint probability distribution A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
) gives the probabilities of a
random vector In probability, and statistics, a multivariate random variable or random vector is a list or vector of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge ...
– 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 and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
, the hypergeometric distribution, and the
normal distribution In probability theory and 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 ...
. 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 A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
'': takes values from a sample space; probabilities describe which values and set of values are more likely taken. *'' 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. Ever ...
'': function evaluating the
probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
that X will take a value less than or equal to x for a random variable (only for real-valued random variables). *'' Quantile function'': the inverse of the cumulative distribution function. Gives x such that, with probability q, X will not exceed x.


Discrete probability distributions

*Discrete probability distribution: for many random variables with finitely or countably infinitely many values. *''
Probability mass function In probability and statistics, a probability mass function (sometimes called ''probability function'' or ''frequency function'') is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes i ...
'' (''pmf''): function that gives the probability that a discrete random variable is equal to some value. *''
Frequency distribution In statistics, the frequency or absolute frequency of an Event (probability theory), event i is the number n_i of times the observation has occurred/been recorded in an experiment or study. These frequencies are often depicted graphically or tabu ...
'': a table that displays the frequency of various outcomes . *'' Relative frequency distribution'': a
frequency distribution In statistics, the frequency or absolute frequency of an Event (probability theory), event i is the number n_i of times the observation has occurred/been recorded in an experiment or study. These frequencies are often depicted graphically or tabu ...
where each value has been divided (normalized) by a number of outcomes in a sample (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), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
'' (''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 (or probability density in the case of a continuous distribution) 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, Long-tailed distribution,
fat-tailed distribution A fat-tailed distribution is a probability distribution that exhibits a large skewness or kurtosis, relative to that of either a normal distribution or an exponential distribution. In common usage, the terms fat-tailed and Heavy-tailed distribut ...
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, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
'' or ''mean'': the weighted average of the possible values, using their probabilities as their weights; or the continuous analog thereof. *''
Median The median of a set of numbers is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be thought of as the “ ...
'': 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 t ...
'': the q-quantile is the value x such that P(X < x) = q. *''
Variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
'': the second moment of the pmf or pdf about the mean; an important measure of the dispersion of the distribution. *''
Standard deviation In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
'': 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
    mixture In chemistry, a mixture is a material made up of two or more different chemical substances which can be separated by physical method. It is an impure substance made up of 2 or more elements or compounds mechanically mixed together in any proporti ...
    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, ...
    , an absolutely continuous and a singular continuous distribution, and thus any cumulative distribution function admits a decomposition as the convex 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) 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 numbe ...
    ) sum: P(X\in E) = \sum_ P(X = \omega), where A is a countable set with P(X \in A) = 1. Thus the discrete random variables (i.e. random variables whose probability distribution is discrete) are exactly those with a
    probability mass function In probability and statistics, a probability mass function (sometimes called ''probability function'' or ''frequency function'') is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes i ...
    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. 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 if these events occur with a known const ...
    , the Bernoulli distribution, the
    binomial distribution In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
    , 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 \mathbb = \; * T ...
    , the negative binomial distribution and categorical distribution. 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 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). 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, also called one-point distributions (see below), 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 mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
    as a generalized
    probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
    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, a Dirac measure. 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 before x to 1 at x. It is closely related to a deterministic distribution, which cannot take on any other value, while a one-point distribution can take other values, though only with probability 0. For most practical purposes the two notions are equivalent.


    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 I = ,b\subset \mathbb the probability of X belonging to I 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), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
    , 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 is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
    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 costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
    , 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 inverse is not true, there exist singular distributions, which are neither absolutely continuous nor discrete nor a mixture of those, and do not have a density. An example is given by the Cantor distribution. Some authors however use the term "continuous distribution" to denote all distributions whose cumulative distribution function is absolutely continuous, i.e. refer to absolutely continuous distributions as continuous distributions. For a more general definition of density functions and the equivalent absolutely continuous measures see absolutely continuous measure.


    Kolmogorov definition

    In the measure-theoretic formalization of
    probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
    , a
    random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
    is defined as a measurable function 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 ...
    (\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 In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
    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 In geometry, a hypercube is an N-dimensional space, ''n''-dimensional analogue of a Square (geometry), square (two-dimensional, ) and a cube (Three-dimensional, ); the special case for Four-dimensional space, is known as a ''tesseract''. It is ...
    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) 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 (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
    . 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 A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random number generation, random n ...
    that produces numbers X that are uniformly distributed in the half-open interval . These random variates X are then transformed via some algorithm to create a new random variate having the required probability distribution. With this source of uniform pseudo-randomness, realizations of any random variable can be generated. For example, suppose has a uniform distribution between 0 and 1. To construct a random Bernoulli variable for some , define X = \begin 1& \text U We thus have P(X=1) = P(U Therefore, the random variable has a Bernoulli distribution with parameter . This method can be adapted to generate real-valued random variables with any distribution: for be any cumulative distribution function , let be the generalized left inverse of F, also known in this context as the '' quantile function'' or ''inverse distribution function'': F^(p) = \inf \. Then, if and only if . As a result, if is uniformly distributed on , then the cumulative distribution function of is . For example, suppose we want to generate a random variable having an exponential distribution with parameter \lambda — that is, with cumulative distribution function F : x \mapsto 1 - e^. \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) = -\tfrac\ln(1-u), and if has a uniform distribution on then X = -\tfrac\ln(1-U) has an exponential distribution with parameter \lambda. Although from a theoretical point of view this method always works, in practice the inverse distribution function is unknown and/or cannot be computed efficiently. In this case, other methods (such as 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 ...
    ) are used.


    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. For these and many other reasons, simple
    number A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
    s are often inadequate for describing a quantity, while probability distributions are often more appropriate. The following is a list of some of the most common probability distributions, grouped by the type of process that they are related to. For a more complete list, see list of probability distributions, which groups by the nature of the outcome being considered (discrete, absolutely continuous, multivariate, etc.) All of the univariate distributions below are singly peaked; that is, it is assumed that the values cluster around a single point. In practice, actually observed quantities may cluster around multiple values. Such quantities can be modeled using a mixture distribution.


    Linear growth (e.g. errors, offsets)

    *
    Normal distribution In probability theory and 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 ...
    (Gaussian distribution), for a single such quantity; the most commonly used absolutely continuous distribution


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

    * Log-normal distribution, 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, actuarial scien ...
    , for a single such quantity whose log is exponentially distributed; the prototypical
    power law In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a relative change in the other quantity proportional to the ...
    distribution


    Uniformly distributed quantities

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


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

    * Basic distributions: ** Bernoulli distribution, 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 and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
    , for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed total number of independent occurrences ** Negative binomial distribution, for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs **
    Geometric distribution 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 \mathbb = \; * T ...
    , 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, 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, for a single categorical outcome (e.g. yes/no/maybe in a survey); a generalization of the Bernoulli distribution * Multinomial distribution, for the number of each type of categorical outcome, given a fixed number of total outcomes; a generalization of the
    binomial distribution In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
    * 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 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 if these events occur with a known const ...
    , 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 or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuousl ...
    , for the time before the next Poisson-type event occurs * Gamma distribution, for the time before the next k Poisson-type events occur


    Absolute values of vectors with normally distributed components

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


    Normally distributed quantities operated with sum of squares

    * Chi-squared distribution, the distribution of a sum of squared standard normal variables; useful e.g. for inference regarding the
    sample variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, ...
    of normally distributed samples (see
    chi-squared test A chi-squared test (also chi-square or test) is a Statistical hypothesis testing, statistical hypothesis test used in the analysis of contingency tables when the sample sizes are large. In simpler terms, this test is primarily used to examine w ...
    ) * 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 A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
    of normally distributed samples with unknown variance (see
    Student's t-test Student's ''t''-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's ''t''- ...
    ) * 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 (the squared
    correlation coefficient A correlation coefficient is a numerical measure of some type of linear 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 c ...
    )


    As conjugate prior distributions in Bayesian inference

    * Beta distribution, for a single probability (real number between 0 and 1); conjugate to the Bernoulli distribution and
    binomial distribution In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
    * Gamma distribution, for a non-negative scaling parameter; conjugate to the rate parameter of a
    Poisson distribution In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...
    or
    exponential distribution In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuousl ...
    , the precision (inverse
    variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
    ) of a
    normal distribution In probability theory and 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 ...
    , etc. * Dirichlet distribution, for a vector of probabilities that must sum to 1; conjugate to the categorical distribution and multinomial distribution; generalization of the beta distribution * Wishart distribution, for a symmetric non-negative definite matrix; conjugate to the inverse of the covariance matrix of a multivariate normal distribution; generalization of the gamma distribution


    Some specialized applications of probability distributions

    * The cache language models and other statistical language models used in
    natural language processing Natural language processing (NLP) is a subfield of computer science and especially artificial intelligence. It is primarily concerned with providing computers with the ability to process data encoded in natural language and is thus closely related ...
    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 distribution In statistics, the frequency or absolute frequency of an Event (probability theory), event i is the number n_i of times the observation has occurred/been recorded in an experiment or study. These frequencies are often depicted graphically or tabu ...
    s such as
    tropical cyclone A tropical cyclone is a rapidly rotating storm system with a low-pressure area, a closed low-level atmospheric circulation, strong winds, and a spiral arrangement of thunderstorms that produce heavy rain and squalls. Depending on its locat ...
    s, hail, time in between events, etc.


    Fitting


    See also

    *
    Conditional probability distribution In probability theory and statistics, the conditional probability distribution is a probability distribution that describes the probability of an outcome given the occurrence of a particular event. Given two jointly distributed random variables X ...
    * Empirical probability distribution *
    Histogram A histogram is a visual representation of the frequency distribution, distribution of quantitative data. To construct a histogram, the first step is to Data binning, "bin" (or "bucket") the range of values— divide the entire range of values in ...
    *
    Joint probability distribution A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
    *
    Probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
    * Quasiprobability distribution * 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.
    Distinguishing probability measure, function and distribution
    Math Stack Exchange {{DEFAULTSORT:Probability Distribution Mathematical and quantitative methods (economics) it:Variabile casuale#Distribuzione di probabilitÃ