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In probability theory and statistics, a **probability distribution** is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events. For instance, if the random variable X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 for *X* = heads, and 0.5 for *X* = tails (assuming the coin is fair). Examples of random phenomena can include the results of an experiment or survey.

**Probability theory** is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event.

**Statistics** is a branch of mathematics working with data collection, organization, analysis, interpretation and presentation. In applying statistics to, for example, a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments. See glossary of probability and statistics.

In probability theory, an **experiment** or **trial** is any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space. An experiment is said to be *random* if it has more than one possible outcome, and *deterministic* if it has only one. A random experiment that has exactly two possible outcomes is known as a Bernoulli trial.

- Introduction
- Terminology
- Basic terms
- Cumulative distribution function
- Discrete probability distribution
- Measure theoretic formulation
- Cumulative distribution function 2
- Delta-function representation
- Indicator-function representation
- Continuous probability distribution
- Some properties
- Kolmogorov definition
- Random number generation
- Applications
- Common probability distributions
- Related to real-valued quantities that grow linearly (e.g. errors, offsets)
- Related to positive real-valued quantities that grow exponentially (e.g. prices, incomes, populations)
- Related to real-valued quantities that are assumed to be uniformly distributed over a (possibly unknown) region
- Related to Bernoulli trials (yes/no events, with a given probability)
- Related to categorical outcomes (events with K possible outcomes, with a given probability for each outcome)
- Related to events in a Poisson process (events that occur independently with a given rate)
- Related to the absolute values of vectors with normally distributed components
- Related to normally distributed quantities operated with sum of squares (for hypothesis testing)
- Useful as conjugate prior distributions in Bayesian inference
- See also
- References
- Citations
- Sources
- External links

A probability distribution is specified in terms of an underlying sample space, which is the set of all possible outcomes of the random phenomenon being observed. The sample space may be the set of real numbers or a set of vectors, or it may be a list of non-numerical values; for example, the sample space of a coin flip would be {heads, tails} .

In probability theory, the **sample space** of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually denoted using set notation, and the possible ordered outcomes are listed as elements in the set. It is common to refer to a sample space by the labels *S*, Ω, or *U*.

In mathematics, a **set** is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2, 4, 6}. The concept of a set is one of the most fundamental in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics from set theory such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.

In probability theory, an **outcome** is a possible result of an experiment. Each possible outcome of a particular experiment is unique, and different outcomes are mutually exclusive. All of the possible outcomes of an experiment form the elements of a sample space.

Probability distributions are generally divided into two classes. A **discrete probability distribution** (applicable to the scenarios where the set of possible outcomes is discrete, such as a coin toss or a roll of dice) can be encoded by a discrete list of the probabilities of the outcomes, known as a probability mass function. On the other hand, a **continuous probability distribution** (applicable to the 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) is typically described by probability density functions (with the probability of any individual outcome actually being 0). The normal distribution is a commonly encountered continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.

In probability and statistics, a **probability mass function** (**PMF**) is a function that gives the probability that a discrete random variable is exactly equal to some value. The probability mass function is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose domain is discrete.

In probability theory, a **probability density function** (**PDF**), or **density** of a continuous random variable, is a function whose value at any given sample in the sample space can be interpreted as providing a *relative likelihood* that the value of the random variable would equal that sample. In other words, while the *absolute likelihood* for a continuous random variable to take on any particular value is 0, the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would equal one sample compared to the other sample.

In probability theory, the **normal****distribution** is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be **normally distributed** and is called a **normal deviate**.

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 of dimension 2 or more is called 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 list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.

In statistics, a **univariate distribution** is a probability distribution of only one random variable. This is in contrast to a multivariate distribution, the probability distribution of a random vector.

A **vector space** is a collection of objects called **vectors**, which may be added together and multiplied ("scaled") by numbers, called *scalars*. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called *axioms*, listed below.

In probability and statistics, a **random variable**, **random quantity**, **aleatory variable**, or **stochastic variable** is described informally as a variable whose values depend on outcomes of a random phenomenon. The formal mathematical treatment of random variables is a topic in probability theory. In that context, a random variable is understood as a measurable function defined on a sample space whose outcomes are typically real numbers.

To define probability distributions for the simplest cases, it is necessary to distinguish between **discrete** and **continuous** random variables. In the discrete case, it is sufficient to specify a probability mass function assigning a probability to each possible outcome: for example, when throwing a fair dice, each of the six values 1 to 6 has the probability 1/6. The probability of an event is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event "the dice rolls an even value" is

**Dice** are small throwable objects that can rest in multiple positions, used for generating random numbers. Dice are commonly used in tabletop games—including dice games, board games, and role-playing games—and for gambling.

In probability theory, an **event** is a set of outcomes of an experiment to which a probability is assigned. A single outcome may be an element of many different events, and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. An event defines a complementary event, namely the complementary set, and together these define a Bernoulli trial: did the event occur or not?

In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, the probability that a given object weighs *exactly* 500 g is zero, because the probability of measuring exactly 500 g tends to zero as the accuracy of our measuring instruments increases. Nevertheless, 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%, and this demand is less sensitive to the accuracy of measurement instruments.

Continuous probability distributions can be described in several ways. The probability density function describes the infinitesimal probability of any given value, and the probability that the outcome lies in a given interval can be computed by integrating the probability density function over that interval. On the other hand, the cumulative distribution function describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval. The cumulative distribution function is the antiderivative of the probability density function provided that the latter function exists.

In mathematics, **infinitesimals** are things so small that there is no way to measure them. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were quantitatively small. The word *infinitesimal* comes from a 17th-century Modern Latin coinage *infinitesimus*, which originally referred to the "infinite-th" item in a sequence. Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" means "extremely small". To give it a meaning, it usually must be compared to another infinitesimal object in the same context. Infinitely many infinitesimals are summed to produce an integral.

In probability theory and statistics, the **cumulative distribution function** (**CDF**) of a real-valued random variable , or just **distribution function** of , evaluated at , is the probability that will take a value less than or equal to .

In calculus, an **antiderivative**, **primitive function**, **primitive integral** or **indefinite integral** of a function *f* is a differentiable function *F* whose derivative is equal to the original function *f*. This can be stated symbolically as . The process of solving for antiderivatives is called **antidifferentiation** and its opposite operation is called differentiation, which is the process of finding a derivative.

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:

**Frequency distribution**: A frequency distribution is a table that displays the frequency of various outcomes**in a sample**.**Relative frequency distribution**: A frequency distribution where each value has been divided (normalized) by a number of outcomes in a sample i.e. sample size.**Probability distribution**: Sometimes used as an alias for*Relative frequency distribution*but most books use it as a limit to which*Relative frequency distribution*tends when sample size tends to population size. It's a general term to indicate the way the total probability of 1 is distributed over**all**various possible outcomes (i.e. over entire population). It may for instance refer to a table that displays the probabilities of various outcomes in a finite population or to the probability density of an uncountably infinite population.**Cumulative distribution function**: is a general functional form to describe a probability distribution.**Probability distribution function**: somewhat ambiguous term sometimes referring to a functional form of probability distribution table. Could be called a "normalized frequency distribution function", where area under the graph equals to 1.**Probability mass**, Probability mass function,**p.m.f.**,**Discrete probability distribution function**: for discrete random variables.- Categorical distribution: for discrete random variables with a finite set of values.
**Probability density**, Probability density function,**p.d.f.**,**Continuous probability distribution function**: most often reserved for continuous random variables.

The following terms are somewhat ambiguous as they can refer to non-cumulative or cumulative distributions, depending on authors' preferences:

**Probability distribution function**: continuous or discrete, non-cumulative or cumulative.**Probability function**: even more ambiguous, can mean any of the above or other things.

- 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, a location at which the probability density function has a local peak.
- Support: the smallest closed set whose complement has probability zero.
- Head: the range of values where the pmf or pdf is relatively high.
- Tail: the complement of the head within the support; the large set of values where the pmf or pdf is relatively low.
- Expected value or
**mean**: the weighted average of the possible values, using their probabilities as their weights; or the continuous analog thereof. - Median: the value such that the set of values less than the median, and the set greater than the median, each have probabilities no greater than one-half.
- Variance: the second moment of the pmf or pdf about the mean; an important measure of the dispersion of the distribution.
- Standard deviation: the square root of the variance, and hence another measure of dispersion.
- Symmetry: a property of some distributions in which the portion of the distribution to the left of a specific value 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.

Because a probability distribution *P* on the real line is determined by the probability of a scalar random variable *X* being in a half-open interval (−∞, *x*], the probability distribution is completely characterized by its cumulative distribution function:

A **discrete probability distribution** is a probability distribution characterized by a probability mass function. Thus, the distribution of a random variable *X* is discrete, and *X* is called a **discrete random variable**, if

as *u* runs through the set of all possible values of *X*. A discrete random variable can assume only a finite or countably infinite number of values.^{ [1] } For the number of potential values to be countably infinite, even though their probabilities sum to 1, the probabilities have to decline to zero fast enough. For example, if for *n* = 1, 2, ..., the sum of probabilities would be 1/2 + 1/4 + 1/8 + ... = 1.

Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, and the negative binomial distribution. Additionally, the discrete uniform distribution is commonly used in computer programs that make equal-probability random selections between a number of choices.

When a sample (a set of observations) is drawn from a larger population, the sample points have an empirical distribution that is discrete and that provides information about the population distribution.

A measurable function between a probability space and a measurable space is called a **discrete random variable** provided that its image is a countable set. In this case measurability of means that the pre-images of singleton sets are measurable, i.e., for all . The latter requirement induces a probability mass function via . Since the pre-images of disjoint sets are disjoint,

This recovers the definition given above.

Equivalently to the above, a discrete random variable can be defined as a random variable whose cumulative distribution function (cdf) increases only by 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.

Consequently, a discrete probability distribution is often represented as a generalized probability density function involving Dirac delta functions, 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.

For a discrete random variable *X*, let *u*_{0}, *u*_{1}, ... be the values it can take with non-zero probability. Denote

These are disjoint sets, and for such sets

It follows that the probability that *X* takes any value except for *u*_{0}, *u*_{1}, ... is zero, and thus one can write *X* as

except on a set of probability zero, where is the indicator function of *A*. This may serve as an alternative definition of discrete random variables.

A **continuous probability distribution** is a probability distribution that has a cumulative distribution function that is continuous. Most often they are generated by having a probability density function. Mathematicians call distributions with probability density functions **absolutely continuous**, since their cumulative distribution function is 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, uniform, chi-squared, and others.

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 [*a*, *b*] is given by the integral

In particular, the probability for *X* to take any single value *a* (that is *a* ≤ *X* ≤ *a*) is zero, because an integral with coinciding upper and lower limits is always equal to zero.

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 distributions, 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.

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.

By one convention, a probability distribution is called *continuous* if its cumulative distribution function is continuous and, therefore, the probability measure of singletons for all .

Another convention reserves the term *continuous probability distribution* for absolutely continuous distributions. These distributions can be characterized by a probability density function: a non-negative Lebesgue integrable function defined on the real numbers such that

Discrete distributions and some continuous distributions (like the Cantor distribution) do not admit such a density.

- The probability distribution of the sum of two independent random variables is the
**convolution**of each of their distributions. - Probability distributions are not a vector space—they are not closed under linear combinations, 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).

In the measure-theoretic formalization of probability theory, a random variable is defined as a measurable function *X* from a probability space to measurable space . A **probability distribution of X** is the pushforward measure

A frequent problem in statistical simulations (the Monte Carlo method) is the generation of 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 half-open interval [0,1). These random variates *X* are then transformed via some algorithm to create a new random variate having the required probability distribution.

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 numbers are often inadequate for describing a quantity, while probability distributions are often more appropriate.

Several more specific example of an application:

- The cache language models and other statistical language models used in natural language processing to assign probabilities to the occurrence of particular words and word sequences do so by means of probability distributions.
- Probabilistic load flow in power-flow study explains the uncertainties of input variables as probability distribution and provide the power flow calculation also in term of probability distribution.
^{ [3] }

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.)

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.

- Normal distribution (Gaussian distribution), for a single such quantity; the most commonly used continuous distribution

- Log-normal distribution, for a single such quantity whose log is normally distributed
- Pareto distribution, for a single such quantity whose log is exponentially distributed; the prototypical power law distribution

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

- Basic distributions:
- Bernoulli distribution, for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no)
- Binomial distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed total number of independent occurrences
- Negative binomial distribution, for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs
- Geometric distribution, for binomial-type observations but where the quantity of interest is the number of failures before the first success; a special case of the negative binomial distribution

- Related to sampling schemes over a finite population:
- Hypergeometric distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, using sampling without replacement
- Beta-binomial distribution, 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)

- 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
- Multivariate hypergeometric distribution, similar to the multinomial distribution, but using sampling without replacement; a generalization of the hypergeometric distribution

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

- 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.

- 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)
- 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 of normally distributed samples with unknown variance (see Student's t-test)
- 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)

- Beta distribution, for a single probability (real number between 0 and 1); conjugate to the Bernoulli distribution and binomial distribution
- Gamma distribution, for a non-negative scaling parameter; conjugate to the rate parameter of a Poisson distribution or exponential distribution, the precision (inverse variance) of a normal distribution, 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

- List of probability distributions
- Copula (statistics)
- Empirical probability
- Histogram
- Joint probability distribution
- Likelihood function
- List of statistical topics
- Kirkwood approximation
- Moment-generating function
- Quasiprobability distribution
- Riemann–Stieltjes integral application to probability theory

In probability theory, the **expected value** of a random variable, intuitively, is the long-run average value of repetitions of the **same experiment** it represents. For example, the expected value in rolling a six-sided die is 3.5, because the average of all the numbers that come up is 3.5 as the number of rolls approaches infinity. In other words, the law of large numbers states that the arithmetic mean of the values almost surely converges to the expected value as the number of repetitions approaches infinity. The expected value is also known as the **expectation**, **mathematical expectation**, **EV**, **average**, **mean value**, **mean**, or **first moment**.

In probability theory, two events are **independent**, **statistically independent**, or **stochastically independent** if the occurrence of one does not affect the probability of occurrence of the other. Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other.

In probability theory, a **probability space** or a **probability triple** is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind. One proposes that each time a situation of that kind arises, the set of possible outcomes is the same and the probabilities are also the same.

In probability theory, there exist several different notions of **convergence of random variables**. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as **stochastic convergence** and they formalize the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that is essentially unchanging when items far enough into the sequence are studied. The different possible notions of convergence relate to how such a behavior can be characterized: two readily understood behaviors are that the sequence eventually takes a constant value, and that values in the sequence continue to change but can be described by an unchanging probability distribution.

In mathematics, a **degenerate distribution** is a probability distribution in a space with support only on a space of lower dimension. If the degenerate distribution is univariate it is a **deterministic distribution** and takes only a single value. Examples include a two-headed coin and rolling a die whose sides all show the same number. This distribution satisfies the definition of "random variable" even though it does not appear random in the everyday sense of the word; hence it is considered degenerate.

In probability and statistics, a **Bernoulli process** is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The component **Bernoulli variables***X*_{i} are identically distributed and independent. Prosaically, a Bernoulli process is a repeated coin flipping, possibly with an unfair coin. Every variable *X*_{i} in the sequence is associated with a Bernoulli trial or experiment. They all have the same Bernoulli distribution. Much of what can be said about the Bernoulli process can also be generalized to more than two outcomes ; this generalization is known as the Bernoulli scheme.

In probability theory and statistics, given two jointly distributed random variables and , the **conditional probability distribution** of *Y* given *X* is the probability distribution of when is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value of as a parameter. When both and are categorical variables, a conditional probability table is typically used to represent the conditional probability. The conditional distribution contrasts with the marginal distribution of a random variable, which is its distribution without reference to the value of the other variable.

In probability and statistics, a **mixture distribution** is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection according to given probabilities of selection, and then the value of the selected random variable is realized. The underlying random variables may be random real numbers, or they may be random vectors, in which case the mixture distribution is a multivariate distribution.

Given random variables , that are defined on a probability space, the **joint probability distribution** for is a probability distribution that gives the probability that each of falls in any particular range or discrete set of values specified for that variable. In the case of only two random variables, this is called a **bivariate distribution**, but the concept generalizes to any number of random variables, giving a **multivariate distribution**.

Probability theory and statistics have some commonly used conventions, in addition to standard mathematical notation and mathematical symbols.

*Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.*

In probability theory, especially in mathematical statistics, a **location–scale family** is a family of probability distributions parametrized by a location parameter and a non-negative scale parameter. For any random variable whose probability distribution function belongs to such a family, the distribution function of also belongs to the family. Moreover, if and are two random variables whose distribution functions are members of the family, and assuming 1) existence of the first two moments and 2) has zero mean and unit variance, then can be written as , where and are the mean and standard deviation of .

In probability theory, **random element** is a generalization of the concept of random variable to more complicated spaces than the simple real line. The concept was introduced by Maurice Fréchet (1948) who commented that the “development of probability theory and expansion of area of its applications have led to necessity to pass from schemes where (random) outcomes of experiments can be described by number or a finite set of numbers, to schemes where outcomes of experiments represent, for example, vectors, functions, processes, fields, series, transformations, and also sets or collections of sets.”

In probability theory, particularly information theory, the **conditional mutual information** is, in its most basic form, the expected value of the mutual information of two random variables given the value of a third.

In probability theory, **conditional probability** is a measure of the probability of an event given that another event has occurred. If the event of interest is *A* and the event *B* is known or assumed to have occurred, "the conditional probability of *A* given *B*", or "the probability of *A* under the condition *B*", is usually written as P(*A* | *B*), or sometimes *P*_{B}(*A*) or P(*A* / *B*). For example, the probability that any given person has a cough on any given day may be only 5%. But if we know or assume that the person has a cold, then they are much more likely to be coughing. The conditional probability of coughing by the unwell might be 75%, then: P(Cough) = 5%; P(Cough | Sick) = 75%

- ↑ 1941-, Çınlar, E. (Erhan) (2011).
*Probability and stochastics*. New York: Springer. p. 51. ISBN 9780387878591. OCLC 710149819. - ↑ W., Stroock, Daniel (1999).
*Probability theory : an analytic view*(Rev. ed.). Cambridge [England]: Cambridge University Press. p. 11. ISBN 978-0521663496. OCLC 43953136. - ↑ Chen, P.; Chen, Z.; Bak-Jensen, B. (April 2008). "Probabilistic load flow: A review".
*2008 Third International Conference on Electric Utility Deregulation and Restructuring and Power Technologies*. pp. 1586–1591. doi:10.1109/drpt.2008.4523658. ISBN 978-7-900714-13-8.

- B. S. Everitt:
*The Cambridge Dictionary of Statistics*, Cambridge University Press, Cambridge (3rd edition, 2006). ISBN 0-521-69027-7 - Bishop:
*Pattern Recognition and Machine Learning*, Springer, ISBN 0-387-31073-8. - den Dekker, A. J.; Sijbers, J. (2014). "Data distributions in magnetic resonance images: A review".
*Physica Medica*.**30**(7): 725–741. doi:10.1016/j.ejmp.2014.05.002. PMID 25059432.

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