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Inprobability theory andstatistics, aprobability distribution is afunction that gives the probabilities of occurrence of possibleevents for anexperiment.[1][2] It is a mathematical description of arandom phenomenon in terms of itssample space and theprobabilities ofevents (subsets of the sample space).[3]
For instance, ifX is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution ofX would take the value 0.5 (1 in 2 or 1/2) forX = heads, and 0.5 forX = tails (assuming thatthe 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.
A probability distribution is a mathematical description of the probabilities of events, subsets of thesample space. The sample space, often represented in notation by is theset of all possibleoutcomes of a random phenomenon being observed. The sample space may be any set: a set ofreal numbers, a set of descriptive labels, a set ofvectors, a set of arbitrary non-numerical values, etc. For example, the sample space of a coin flip could beΩ ={"heads", "tails"}.
To define probability distributions for the specific case ofrandom variables (so the sample space can be seen as a numeric set), it is common to distinguish betweendiscrete andcontinuousrandom variables. In the discrete case, it is sufficient to specify aprobability mass function assigning a probability to each possible outcome (e.g. when throwing a fairdie, each of the six digits“1” to“6”, corresponding to the number of dots on the die, has probability The probability of anevent 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" isIn 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 weighsexactly 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 thecumulative distribution function, which describes the probability that the random variable is no larger than a given value (i.e.,P(X ≤x) for somex. The cumulative distribution function is the area under theprobability density function from-∞ tox, as shown in figure 1.[4]
Most continuous probability distributions encountered in practice are not only continuous but alsoabsolutely continuous. Such distributions can be described by theirprobability density function. Informally, the probability density of a random variable describes theinfinitesimal probability that takes any value — that is as becomes is arbitrarily small. The probability that lies in a given interval can be computed rigorously byintegrating the probability density function over that interval.[5]
Let be aprobability space, be ameasurable space, and be a-valued random variable. Then theprobability distribution of is thepushforward measure of the probability measure onto induced by. Explicitly, this pushforward measure on is given by for
Any probability distribution is aprobability measure on (in general different from, unless happens to be the identity map).[citation needed]
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 whoseinput space is aσ-algebra, and gives areal numberprobability as its output, particularly, a number in.
The probability function can take as argument subsets of the sample space itself, as in the coin toss example, where the function was defined so thatP(heads) = 0.5 andP(tails) = 0.5. However, because of the widespread use ofrandom variables, which transform the sample space into a set of numbers (e.g.,,), it is more common to study probability distributions whose argument are subsets of these particular kinds of sets (number sets),[6] and all probability distributions discussed in this article are of this type. It is common to denote as the probability that a certain value of the variable belongs to a certain event.[7][8]
The above probability function only characterizes a probability distribution if it satisfies all theKolmogorov axioms, that is:
The concept of probability function is made more rigorous by defining it as the element of aprobability space, where is the set of possible outcomes, is the set of all subsets whose probability can be measured, and is the probability function, orprobability measure, that assigns a probability to each of these measurable subsets.[9]
Probability distributions usually belong to one of two classes. Adiscrete probability distribution is applicable to the scenarios where the set of possible outcomes isdiscrete (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 asprobability 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 aprobability density function, and the probability distribution is by definition the integral of the probability density function.[7][5][8] Thenormal distribution is a commonly encountered absolutely continuous probability distribution. More complex experiments, such as those involvingstochastic processes defined incontinuous time, may demand the use of more generalprobability measures.
A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is calledunivariate, while a distribution whose sample space is avector space of dimension 2 or more is calledmultivariate. A univariate distribution gives the probabilities of a singlerandom variable taking on various different values; a multivariate distribution (ajoint probability distribution) gives the probabilities of arandom vector – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include thebinomial distribution, thehypergeometric distribution, and thenormal distribution. A commonly encountered multivariate distribution is themultivariate normal distribution.
Besides the probability function, the cumulative distribution function, the probability mass function and the probability density function, themoment generating function and thecharacteristic function also serve to identify a probability distribution, as they uniquely determine an underlying cumulative distribution function.[10]
Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below.[1]
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 with regard to a probability distribution is defined as
The cumulative distribution function of any real-valued random variable has the properties:
Conversely, any function that satisfies the first four of the properties above is the cumulative distribution function of some probability distribution on the real numbers.[13]
Any probability distribution can be decomposed as themixture of adiscrete, anabsolutely continuous and asingular continuous distribution,[14] and thus any cumulative distribution function admits a decomposition as theconvex sum of the three according cumulative distribution functions.
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Adiscrete probability distribution is the probability distribution of a random variable that can take on only a countable number of values[15] (almost surely)[16] which means that the probability of any event can be expressed as a (finite orcountably infinite) sum:where is a countable set with. Thus the discrete random variables (i.e. random variables whose probability distribution is discrete) are exactly those with aprobability mass function. 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 for, the sum of probabilities would be.
Well-known discrete probability distributions used in statistical modeling include thePoisson distribution, theBernoulli distribution, thebinomial distribution, thegeometric distribution, thenegative binomial distribution andcategorical distribution.[3] When asample (a set of observations) is drawn from a larger population, the sample points have anempirical distribution that is discrete, and which provides information about the population distribution. Additionally, thediscrete uniform distribution is commonly used in computer programs that make equal-probability random selections between a number of choices.
A real-valued discrete random variable can equivalently be defined as a random variable whose cumulative distribution function increases only byjump 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 formThe 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.
A discrete probability distribution is often represented withDirac measures, also called one-point distributions (see below), the probability distributions ofdeterministic random variables. For any outcome, let be the Dirac measure concentrated at. Given a discrete probability distribution, there is a countable set with and a probability mass function. If is any event, thenor in short,
Similarly, discrete distributions can be represented with theDirac delta function as ageneralizedprobability density function, wherewhich meansfor any event[17]
For a discrete random variable, let be the values it can take with non-zero probability. DenoteThese aredisjoint sets, and for such setsIt follows that the probability that takes any value except for is zero, and thus one can write asexcept on a set of probability zero, where is the indicator function of. This may serve as an alternative definition of discrete random variables.
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 has a one-point distribution if it has a possible outcome such that[18] All other possible outcomes then have probability 0. Its cumulative distribution function jumps immediately from 0 before to 1 at. 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.
Anabsolutely 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.[19] More precisely, a real random variable has anabsolutely continuous probability distribution if there is a function such that for each interval the probability of belonging to is given by the integral of over:[20][21]This is the definition of aprobability density function, so that absolutely continuous probability distributions are exactly those with a probability density function.In particular, the probability for to take any single value (that is,) is zero, because anintegral with coinciding upper and lower limits is always equal to zero.If the interval is replaced by any measurable set, the according equality still holds:
Anabsolutely continuous random variable is a random variable whose probability distribution is absolutely continuous.
There are many examples of absolutely continuous probability distributions:normal,uniform,chi-squared, andothers.
Absolutely continuous probability distributions as defined above are precisely those with anabsolutely continuous cumulative distribution function.In this case, the cumulative distribution function has the formwhere is a density of the random variable with regard to the distribution.
Note on terminology: Absolutely continuous distributions ought to be distinguished fromcontinuous 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 existsingular distributions, which are neither absolutely continuous nor discrete nor a mixture of those, and do not have a density. An example is given by theCantor distribution. Some authors however use the term "continuous distribution" to denote all distributions whose cumulative distribution function isabsolutely continuous, i.e. refer to absolutely continuous distributions as continuous distributions.[7]
For a more general definition of density functions and the equivalent absolutely continuous measures seeabsolutely continuous measure.
In themeasure-theoretic formalization ofprobability theory, arandom variable is defined as ameasurable function from aprobability space to ameasurable space. Given that probabilities of events of the form satisfyKolmogorov's probability axioms, theprobability distribution of is theimage measure of , which is aprobability measure on satisfying.[22][23][24]
Absolutely continuous and discrete distributions with support on or are extremely useful to model a myriad of phenomena,[7][4] since most practical distributions are supported on relatively simple subsets, such ashypercubes orballs. However, this is not always the case, and there exist phenomena with supports that are actually complicated curves within some space 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.[25]
One example is shown in the figure to the right, which displays the evolution of asystem of differential equations (commonly known as theRabinovich–Fabrikant equations) that can be used to model the behaviour ofLangmuir waves inplasma.[26] 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.[27][25]
This kind of complicated support appears quite frequently indynamical systems. It is not simple to establish that the system has a probability measure, and the main problem is the following. Let be instants in time and a subset of the support; if the probability measure exists for the system, one would expect the frequency of observing states inside set would be equal in interval and, which might not happen; for example, it could oscillate similar to a sine,, whose limit when 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.[28] The branch of dynamical systems that studies the existence of a probability measure isergodic 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.
Most algorithms are based on apseudorandom number generator that produces numbers that are uniformly distributed in thehalf-open interval[0, 1). Theserandom variates 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.[29]
For example, supposeU has a uniform distribution between 0 and 1. To construct a random Bernoulli variable for some0 < p < 1, defineWe thus haveTherefore, the random variableX has a Bernoulli distribution with parameterp.[29]
This method can be adapted to generate real-valued random variables with any distribution: for be any cumulative distribution functionF, letFinv be the generalized left inverse of also known in this context as thequantile function orinverse distribution function:Then,Finv(p) ≤x if and only ifp ≤F(x). As a result, ifU is uniformly distributed on[0, 1], then the cumulative distribution function ofX =Finv(U) isF.
For example, suppose we want to generate a random variable having an exponential distribution with parameter — that is, with cumulative distribution functionso, and ifU has a uniform distribution on[0, 1) then has an exponential distribution with parameter[29]
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 theMonte Carlo method) are used.
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 thekinetic properties of gases to thequantum mechanical description offundamental particles. For these and many other reasons, simplenumbers 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, seelist 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 amixture distribution.
Probability distribution fitting or simply distribution fitting is the fitting of a probability distribution to a series of data concerning the repeated measurement of a variable phenomenon.The aim of distribution fitting is topredict theprobability or toforecast thefrequency of occurrence of the magnitude of the phenomenon in a certain interval.
There are many probability distributions (seelist of probability distributions) of which some can be fitted more closely to the observed frequency of the data than others, depending on the characteristics of the phenomenon and of the distribution. The distribution giving a close fit is supposed to lead to good predictions.
In distribution fitting, therefore, one needs to select a distribution that suits the data well.
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