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| {{About|probability distribution|generalized functions in mathematical analysis|Distribution (mathematics)|other uses|Distribution (disambiguation)}}
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| {{No footnotes|date=July 2011}}
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| {{refimprove|date=July 2011}}
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| In [[probability and statistics]], a '''probability distribution''' assigns a [[probability]] to each [[measure (mathematics)|measurable subset]] of the possible outcomes of a random [[Experiment (probability theory)|experiment]], [[Survey methodology|survey]], or procedure of [[statistical inference]]. Examples are found in experiments whose [[sample space]] is non-numerical, where the distribution would be a [[categorical distribution]]; experiments whose sample space is encoded by discrete [[random variables]], where the distribution can be specified by a [[probability mass function]]; and experiments with sample spaces encoded by continuous random variables, where the distribution can be specified by a [[probability density function]]. More complex experiments, such as those involving [[stochastic processes]] defined in [[continuous time]], may demand the use of more general [[probability measure]]s.
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| In [[applied probability]], a probability distribution can be specified in a number of different ways, often chosen for mathematical convenience:
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| *by supplying a valid [[#Terminology|probability mass function]] or [[#Terminology|probability density function]]
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| *by supplying a valid [[cumulative distribution function]] or [[survival function]]
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| *by supplying a valid [[hazard function]]
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| *by supplying a valid [[Characteristic function (probability theory)|characteristic function]]
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| *by supplying a rule for constructing a new random variable from other random variables whose [[joint probability distribution]] is known.
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| A probability distribution can either be [[Univariate distribution|univariate]] or [[Multivariate distribution|multivariate]]. A univariate distribution gives the probabilities of a single [[random variable]] taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a [[random vector]]—a set 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]]. The [[multivariate normal distribution]] is a commonly encountered multivariate distribution.
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| ==Introduction==
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| [[File:Dice Distribution (bar).svg|thumb|250px|right|The [[probability mass function]] (pmf) ''p''(''S'') specifies the probability distribution for the sum ''S'' of counts from two [[dice]]. For example, the figure shows that ''p''(11) = 1/18. The pmf allows the computation of probabilities of events such as ''P''(''S'' > 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.]]
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| To define probability distributions for the simplest cases, one needs to distinguish between '''discrete''' and '''continuous''' [[random variable]]s. In the discrete case, one can easily assign a probability to each possible value: for example, when throwing a {{dice}}, each of the six values ''1'' to ''6'' has the probability 1/6. In contrast, when a random variable takes values from a continuum, probabilities can be nonzero only if they refer to intervals: in quality control one might demand that the probability of a "500 g" package containing between 490 g and 510 g should be no less than 98%.
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| [[File:Standard deviation diagram.svg|right|thumb|250px|The [[probability density function]] (pdf) of the [[normal distribution]], also called Gaussian or "bell curve", the most important continuous random distribution. As notated on the figure, the probabilities of intervals of values corresponds to the area under the curve.]]
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| If the random variable is real-valued (or more generally, if a [[total order]] is defined for its possible values), the cumulative distribution function (CDF) gives the probability that the random variable is no larger than a given value; in the real-valued case, the CDF is the [[integral]] of the [[probability density function]] (pdf) provided that this function exists.
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| ==Terminology==
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| As probability theory is used in quite diverse applications, terminology is not uniform and sometimes confusing. The following terms are used for non-cumulative probability distribution functions:
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| * '''Probability mass''', [[Probability mass function]], '''p.m.f.''': for discrete random variables.
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| * [[Categorical distribution]]: for discrete random variables with a finite set of values.
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| * '''Probability density''', [[Probability density function]], '''p.d.f''': most often reserved for continuous random variables.
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| The following terms are somewhat ambiguous as they can refer to non-cumulative or cumulative distributions, depending on authors' preferences:
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| * '''Probability distribution function''': continuous or discrete, non-cumulative or cumulative.
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| * '''Probability function''': even more ambiguous, can mean any of the above or other things.
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| Finally,
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| * '''Probability distribution''': sometimes the same as ''probability distribution function'', but usually refers to the more complete assignment of probabilities to all measurable subsets of outcomes, not just to specific outcomes or ranges of outcomes.
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| ===Basic terms===
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| * [[Mode (statistics)|Mode]]: for a discrete random variable, the value with highest probability (the location at which the probability mass function has its peak); for a continuous random variable, the location at which the probability density function has its peak.
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| * [[Support (mathematics)|Support]]: the smallest closed set whose [[Complement (set theory)|complement]] has probability zero.
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| * [[Heavy-tailed distribution|Head]]: the range of values where the pmf or pdf is relatively high.
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| * [[Heavy-tailed distribution|Tail]]: the complement of the head within the support; the large set of values where the pmf or pdf is relatively low.
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| * [[Expected value]] or '''mean''': the [[weighted average]] of the possible values, using their probabilities as their weights; or the continuous analog thereof.
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| * [[Median]]: the value such that the set of values less than the median has a probability of one-half.
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| * [[Variance]]: the second moment of the pmf or pdf about the mean; an important measure of the [[Statistical dispersion|dispersion]] of the distribution.
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| * [[Standard deviation]]: the square root of the variance, and hence another measure of dispersion.
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| * '''Symmetry''': a property of some distributions in which the portion of the distribution to the left of a specific value is a mirror image of the portion to its right.
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| * [[Skewness]]: a measure of the extent to which a pmf or pdf "leans" to one side of its mean.
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| ==Cumulative distribution function==
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| Because a probability distribution Pr on the real line is determined by the probability of a [[Scalar (mathematics)|scalar]] random variable ''X'' being in a half-open interval <nowiki>(</nowiki>-∞, ''x''<nowiki>]</nowiki>, the probability distribution is completely characterized by its [[cumulative distribution function]]:
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| : <math> F(x) = \Pr \left[ X \le x \right] \qquad \text{ for all } x \in \mathbb{R}.</math>
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| ==Discrete probability distribution==
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| {{See also|Probability mass function|Categorical distribution}}
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| [[File:Discrete probability distrib.svg|right|thumb|The probability mass function of a discrete probability distribution. The probabilities of the [[Singleton (mathematics)|singleton]]s {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.]]
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| [[File:Discrete probability distribution.svg|right|thumb|The [[cumulative distribution function|cdf]] of a discrete probability distribution, ...]]
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| [[File:Normal probability distribution.svg|right|thumb|... of a continuous probability distribution, ...]]
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| [[File:Mixed probability distribution.svg|right|thumb|... of a distribution which has both a continuous part and a discrete part.]]
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| A '''discrete probability distribution''' shall be understood as a ''probability distribution'' characterized by a [[probability mass function]]. Thus, the distribution of a [[random variable]] ''X'' is discrete, and ''X'' is then called a '''discrete random variable''', if
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| :<math>\sum_u \Pr(X=u) = 1</math>
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| as ''u'' runs through the set of all possible values of ''X''. It follows that such a random variable can assume only a [[finite set|finite]] or [[countable|countably infinite]] number of values. For the number of potential values to be countably infinite even though their probabilities sum to 1 requires that the probabilities decline to zero fast enough: for example, if <math>\Pr(X=n) = \tfrac{1}{2^n}</math> for ''n'' = 1, 2, ..., we have the sum of probabilities 1/2 + 1/4 + 1/8 + ... = 1.
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| Among the most well-known discrete probability distributions that are used for statistical modeling are the [[Poisson distribution]], the [[Bernoulli distribution]], the [[binomial distribution]], the [[geometric distribution]], and the [[negative binomial distribution]]. In addition, the [[Uniform distribution (discrete)|discrete uniform distribution]] is commonly used in computer programs that make equal-probability random selections between a number of choices.
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| ===Cumulative density===
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| Equivalently to the above, a discrete random variable can be defined as a random variable whose [[cumulative distribution function]] (cdf) increases only by [[jump discontinuity|jump discontinuities]]—that is, its cdf increases only where it "jumps" to a higher value, and is constant between those jumps. The points where jumps occur are precisely the values which the random variable may take.
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| ===Delta-function representation===
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| Consequently, a discrete probability distribution is often represented as a generalized [[probability density function]] involving [[Dirac delta function]]s, which substantially unifies the treatment of continuous and discrete distributions. This is especially useful when dealing with probability distributions involving both a continuous and a discrete part.
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| ===Indicator-function representation===
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| For a discrete random variable ''X'', let ''u''<sub>0</sub>, ''u''<sub>1</sub>, ... be the values it can take with non-zero probability. Denote
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| :<math>\Omega_i=\{\omega: X(\omega)=u_i\},\, i=0, 1, 2, \dots</math>
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| These are [[disjoint set]]s, and by formula (1)
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| :<math>\Pr\left(\bigcup_i \Omega_i\right)=\sum_i \Pr(\Omega_i)=\sum_i\Pr(X=u_i)=1.</math>
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| It follows that the probability that ''X'' takes any value except for ''u''<sub>0</sub>, ''u''<sub>1</sub>, ... is zero, and thus one can write ''X'' as
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| :<math>X=\sum_i u_i 1_{\Omega_i}</math>
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| except on a set of probability zero, where <math>1_A</math> is the [[indicator function]] of ''A''. This may serve as an alternative definition of discrete random variables.
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| ==Continuous probability distribution==
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| {{See also|Probability density function}}
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| A '''continuous probability distribution''' is a ''probability distribution'' that has a [[probability density function]]. Mathematicians also call such a distribution '''absolutely continuous''', since its [[cumulative distribution function]] is [[absolute continuity|absolutely continuous]] with respect to the [[Lebesgue measure]] ''λ''. If the distribution of ''X'' is continuous, then ''X'' is called a '''continuous random variable'''. There are many examples of continuous probability distributions: [[normal distribution|normal]], [[Uniform distribution (continuous)|uniform]], [[Chi-squared distribution|chi-squared]], and [[List of probability distributions#Continuous distributions|others]].
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| Intuitively, a continuous random variable is the one which can take a continuous range of values — as opposed to a [[discrete distribution]], where the set of possible values for the random variable is at most [[countable set|countable]]. While for a discrete distribution an [[event (probability theory)|event]] with [[probability]] zero is impossible (e.g. rolling 3½ on a standard die is impossible, and has probability zero), this is not so in the case of a continuous random variable. For example, if one measures the width of an oak leaf, the result of 3½ cm is possible, however it has probability zero because there are uncountably many other potential values even between 3 cm and 4 cm. Each of these individual outcomes has probability zero, yet the probability that the outcome will fall into the [[interval (mathematics)|interval]] {{nowrap|(3 cm, 4 cm)}} is nonzero. This apparent [[paradox]] is resolved by the fact that the probability that ''X'' attains some value within an [[Infinity|infinite]] set, such as an interval, [[integral|cannot be found by naively adding]] the probabilities for individual values. Formally, each value has an [[infinitesimal]]ly small probability, which [[almost surely|statistically is equivalent]] to zero.
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| Formally, if ''X'' is a continuous random variable, then it has a [[probability density function]] ''ƒ''(''x''), and therefore its probability of falling into a given interval, say {{nowrap|[''a'', ''b'']}} is given by the integral
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| : <math> | |
| \Pr[a\le X\le b] = \int_a^b f(x) \, dx
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| </math>
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| In particular, the probability for ''X'' to take any single value ''a'' (that is {{nowrap|''a'' ≤ ''X'' ≤ ''a''}}) is zero, because an [[integral]] with coinciding upper and lower limits is always equal to zero.
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| The definition states that a continuous probability distribution must possess a density, or equivalently, its cumulative distribution function be absolutely continuous. This requirement is stronger than simple continuity of the cumulative distribution function, and there is a special class of distributions, [[singular distribution]]s, which are neither continuous nor discrete nor a mixture of those. An example is given by the [[Cantor distribution]]. Such singular distributions however are never encountered in practice.
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| Note on terminology: some authors use the term "continuous distribution" to denote the distribution with continuous cumulative distribution function. Thus, their definition includes both the (absolutely) continuous and singular distributions.
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| By one convention, a probability distribution <math>\,\mu</math> is called ''continuous'' if its cumulative distribution function <math>F(x)=\mu(-\infty,x]</math> is [[continuous function|continuous]] and, therefore, the probability measure of singletons <math>\mu\{x\}\,=\,0</math> for all <math>\,x</math>.
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| Another convention reserves the term ''continuous probability distribution'' for [[absolute continuity|absolutely continuous]] distributions. These distributions can be characterized by a [[probability density function]]: a non-negative [[Lebesgue integration|Lebesgue integrable]] function <math>\,f</math> defined on the real numbers such that
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| :<math>
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| F(x) = \mu(-\infty,x] = \int_{-\infty}^x f(t)\,dt.
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| </math>
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| Discrete distributions and some continuous distributions (like the [[Cantor distribution]]) do not admit such a density.
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| ==Some properties==
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| * The probability distribution of the sum of two independent random variables is the '''[[convolution]]''' of each of their distributions.
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| * Probability distributions are not a [[vector space]]—they are not closed under [[linear combination]]s, as these do not preserve non-negativity or total integral 1—but they are closed under [[convex combination]], thus forming a [[convex subset]] of the space of functions (or measures).
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| ==Kolmogorov definition==
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| {{Main|Probability space|Probability measure}}
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| In the [[measure theory|measure-theoretic]] formalization of [[probability theory]], a [[random variable]] is defined as a [[measurable function]] ''X'' from a [[probability space]] <math>\scriptstyle (\Omega, \mathcal{F}, \operatorname{P})</math> to measurable space <math>\scriptstyle (\mathcal{X},\mathcal{A})</math>. A '''probability distribution''' is the [[pushforward measure]], P, satisfying ''X''<sub>*</sub>P = P''X''<sup> −1</sup> on <math>\scriptstyle (\mathcal{X},\mathcal{A})</math>.{{clarify|reason=unexplained notation, terminology|date=July 2012}}
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| ==Random number generation==
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| {{Main|Pseudo-random number sampling}}
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| A frequent problem in statistical simulations (the [[Monte Carlo method]]) is the generation of [[Pseudorandomness|pseudo-random numbers]] that are distributed in a given way. Most algorithms are based on a [[pseudorandom number generator]] that produces numbers ''X'' that are uniformly distributed in the interval [0,1). These [[random variate]]s ''X'' are then transformed via some algorithm to create a new random variate having the required probability distribution.
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| ==Applications==
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| 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 [[measurement error|intrinsic error]]; in [[physics]] many processes are described probabilistically, from the [[kinetic theory|kinetic properties of gases]] to the [[quantum mechanical]] description of [[fundamental particles]]. For these and many other reasons, simple [[number]]s are often inadequate for describing a quantity, while probability distributions are often more appropriate.
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| As a more specific example of an application, the [[cache language model]]s and other [[Statistical Language Model|statistical language models]] used in [[natural language processing]] to assign probabilities to the occurrence of particular words and word sequences do so by means of probability distributions.
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| ==Common probability distributions==
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| {{Main|List of probability distributions}}
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| The following is a list of some of the most common probability distributions, grouped by the type of process that they are related to. For a more complete list, see [[list of probability distributions]], which groups by the nature of the outcome being considered (discrete, continuous, multivariate, etc.)
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| Note also that 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]].
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| ===Related to real-valued quantities that grow linearly (e.g. errors, offsets)===
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| *[[Normal distribution]] ([[Gaussian distribution]]), for a single such quantity; the most common continuous distribution
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| ===Related to positive real-valued quantities that grow exponentially (e.g. prices, incomes, populations)===
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| *[[Log-normal distribution]], for a single such quantity whose log is [[normal distribution|normally]] distributed
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| *[[Pareto distribution]], for a single such quantity whose log is [[exponential distribution|exponentially]] distributed; the prototypical [[power law]] distribution
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| ===Related to real-valued quantities that are assumed to be uniformly distributed over a (possibly unknown) region===
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| *[[Discrete uniform distribution]], for a finite set of values (e.g. the outcome of a fair die)
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| *[[Continuous uniform distribution]], for continuously distributed values
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| ===Related to Bernoulli trials (yes/no events, with a given probability)===
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| *Basic distributions:
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| **[[Bernoulli distribution]], for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no)
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| **[[Binomial distribution]], for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed total number of [[independent (statistics)|independent]] occurrences
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| **[[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
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| **[[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]]
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| *Related to sampling schemes over a finite population:
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| **[[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]]
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| **[[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 [[Polya urn scheme]] (in some sense, the "opposite" of [[sampling without replacement]])
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| ===Related to categorical outcomes (events with ''K'' possible outcomes, with a given probability for each outcome)===
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| *[[Categorical distribution]], for a single categorical outcome (e.g. yes/no/maybe in a survey); a generalization of the [[Bernoulli distribution]]
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| *[[Multinomial distribution]], for the number of each type of categorical outcome, given a fixed number of total outcomes; a generalization of the [[binomial distribution]]
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| *[[Multivariate hypergeometric distribution]], similar to the [[multinomial distribution]], but using [[sampling without replacement]]; a generalization of the [[hypergeometric distribution]]
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| ===Related to events in a Poisson process (events that occur independently with a given rate)===
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| *[[Poisson distribution]], for the number of occurrences of a Poisson-type event in a given period of time
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| *[[Exponential distribution]], for the time before the next Poisson-type event occurs
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| *[[Gamma distribution]], for the time before the next k Poisson-type events occur
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| ===Related to the absolute values of vectors with normally distributed components ===
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| *[[Rayleigh distribution]], for the distribution of vector magnitudes with Gaussian distributed orthogonal components. Rayleight distributions are found in RF signals with Gaussian real and imaginary components.
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| *[[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.
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| ===Related to normally distributed quantities operated with sum of squares (for hypothesis testing)===
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| *[[Chi-squared distribution]], the distribution of a sum of squared [[standard normal]] variables; useful e.g. for inference regarding the [[sample variance]] of normally distributed samples (see [[chi-squared test]])
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| *[[Student's t distribution]], the distribution of the ratio of a [[standard normal]] variable and the square root of a scaled [[chi squared distribution|chi squared]] variable; useful for inference regarding the [[mean]] of normally distributed samples with unknown variance (see [[Student's t-test]])
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| *[[F-distribution]], the distribution of the ratio of two scaled [[chi squared distribution|chi squared]] variables; useful e.g. for inferences that involve comparing variances or involving [[R-squared]] (the squared [[Pearson product-moment correlation coefficient|correlation coefficient]])
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| ===Useful as conjugate prior distributions in Bayesian inference===
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| {{main|Conjugate prior}}
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| *[[Beta distribution]], for a single probability (real number between 0 and 1); conjugate to the [[Bernoulli distribution]] and [[binomial distribution]]
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| *[[Gamma distribution]], for a non-negative scaling parameter; conjugate to the rate parameter of a [[Poisson distribution]] or [[exponential distribution]], the [[precision (statistics)|precision]] (inverse [[variance]]) of a [[normal distribution]], etc.
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| *[[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]]
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| *[[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]]
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| ==See also==
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| {{Portal|Statistics}}
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| * [[Copula (statistics)]]
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| * [[Histogram]]
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| * [[Likelihood function]]
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| * [[List of statistical topics]]
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| * [[Kirkwood approximation]]
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| * [[Moment-generating function]]
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| * [[Quasiprobability distribution]]
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| * [[Riemann–Stieltjes integral#Application to probability theory|Riemann–Stieltjes integral application to probability theory]]
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| ==References==
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| {{Reflist}}
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| * B. S. Everitt: ''The Cambridge Dictionary of Statistics'', [[Cambridge University Press]], Cambridge (3rd edition, 2006). ISBN 0-521-69027-7
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| * Bishop: ''Pattern Recognition and Machine Learning'', [[Springer Publishing|Springer]], ISBN 0-387-31073-8
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| ==External links==
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| {{commons|Probability distribution|Probability distribution}}
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| *{{springer|title=Probability distribution|id=p/p074900}}
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| {{ProbDistributions}}
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| {{Theory of probability distributions}}
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| {{Statistics|descriptive}}
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| {{DEFAULTSORT:Probability Distribution}}
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| [[Category:Probability distributions|*]]
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| {{Link GA|fr}}
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| [[it:Variabile casuale#Distribuzione di probabilità]]
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