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.

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} .

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.

A probability distribution whose sample space is the set of real numbers is called univariate, while a distribution whose sample space is a vector space 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.

To define probability distributions for the simplest cases, one needs 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

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.

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, ..., we have the sum of probabilities 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.

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. While for a discrete distribution an event with probability zero is impossible (e.g., rolling π on a standard die has probability zero and is impossible), 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 uncountably many other potential values exist 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 (3 cm, 4 cm) is nonzero. This apparent paradox is resolved by the fact that the probability that *X* attains some value within an infinite set, such as an interval, cannot be found by naively adding the probabilities for individual values. Formally, each value has an infinitesimally small probability, which statistically is equivalent to zero.

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.

As a 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.

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 common 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

**^**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 0521663490. OCLC 43953136.

- 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.

- Hazewinkel, Michiel, ed. (2001) [1994], "Probability distribution",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Field Guide to Continuous Probability Distributions, Gavin E. Crooks.

In probability theory and statistics, the **Bernoulli distribution**, named after Swiss mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable which takes the value 1 with probability and the value 0 with probability that is, the probability distribution of any single experiment that asks a yes–no question; the question results in a boolean-valued outcome, a single bit of information whose value is success/yes/true/one with probability *p* and failure/no/false/zero with probability *q*. It can be used to represent a (possibly biased) coin toss where 1 and 0 would represent "heads" and "tails" (or vice versa), respectively, and *p* would be the probability of the coin landing on heads or tails, respectively. In particular, unfair coins would have

The Bernoulli distribution is a special case of the binomial distribution where a single trial is conducted (so *n* would be 1 for such a binomial distribution). It is also a special case of the **two-point distribution**, for which the possible outcomes need not be 0 and 1.

In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own boolean-valued outcome: a random variable containing a single bit of information: success/yes/true/one (with probability p) or failure/no/false/zero (with probability q = 1 − p).

A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.

The binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacement from a population of size N. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one. However, for N much larger than n, the binomial distribution remains a good approximation, and is widely used.

Cumulative distribution functionIn 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 the case of a continuous distribution, it gives the area under the probability density function from minus infinity to . Cumulative distribution functions are also used to specify the distribution of multivariate random variables.

Degenerate distributionIn mathematics, a **degenerate distribution** is a probability distribution in a space (discrete or continuous) with support only on a space of lower dimension. If the degenerate distribution is univariate (involving only a single random variable) 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 the case of a real-valued random variable, the degenerate distribution is localized at a point *k*_{0} on the real line. The probability mass function equals 1 at this point and 0 elsewhere.

The degenerate univariate distribution can be viewed as the limiting case of a continuous distribution whose variance goes to 0 causing the probability density function to be a delta function at *k*_{0}, with infinite height there but area equal to 1.

The cumulative distribution function of the univariate degenerate distribution is:

Entropy (information theory)

Information entropy is the average rate at which information is produced by a stochastic source of data.

The measure of information entropy associated with each possible data value is the negative logarithm of the probability mass function for the value:

S

=

−

∑

i

P

i

log

P

i

S=-\sum _{i}P_{i}\log {P_{i}}

.When the data source produces a low-probability value (i.e., when a low-probability event occurs), the event carries more "information" ("surprisal") than when the source data produces a high-probability value. The amount of information conveyed by each event defined in this way becomes a random variable whose expected value is the information entropy. Generally, entropy refers to disorder or uncertainty, and the definition of entropy used in information theory is directly analogous to the definition used in statistical thermodynamics. The concept of information entropy was introduced by Claude Shannon in his 1948 paper "A Mathematical Theory of Communication".The basic model of a data communication system is composed of three elements, a source of data, a communication channel, and a receiver, and – as expressed by Shannon – the "fundamental problem of communication" is for the receiver to be able to identify what data was generated by the source, based on the signal it receives through the channel. The entropy provides an absolute limit on the shortest possible average length of a lossless compression encoding of the data produced by a source, and if the entropy of the source is less than the channel capacity of the communication channel, the data generated by the source can be reliably communicated to the receiver (at least in theory, possibly neglecting some practical considerations such as the complexity of the system needed to convey the data and the amount of time it may take for the data to be conveyed).

Information entropy is typically measured in bits (alternatively called "shannons") or sometimes in "natural units" (nats) or decimal digits (called "dits", "bans", or "hartleys"). The unit of the measurement depends on the base of the logarithm that is used to define the entropy.

The logarithm of the probability distribution is useful as a measure of entropy because it is additive for independent sources. For instance, the entropy of a fair coin toss is 1 bit, and the entropy of m tosses is m bits. In a straightforward representation, log2(n) bits are needed to represent a variable that can take one of n values if n is a power of 2. If these values are equally probable, the entropy (in bits) is equal to this number. If one of the values is more probable to occur than the others, an observation that this value occurs is less informative than if some less common outcome had occurred. Conversely, rarer events provide more information when observed. Since observation of less probable events occurs more rarely, the net effect is that the entropy (thought of as average information) received from non-uniformly distributed data is always less than or equal to log2(n). Entropy is zero when one outcome is certain to occur. The entropy quantifies these considerations when a probability distribution of the source data is known. The meaning of the events observed (the meaning of messages) does not matter in the definition of entropy. Entropy only takes into account the probability of observing a specific event, so the information it encapsulates is information about the underlying probability distribution, not the meaning of the events themselves.

Joint probability distributionGiven 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**.

The joint probability distribution can be expressed either in terms of a joint cumulative distribution function or in terms of a joint probability density function (in the case of continuous variables) or joint probability mass function (in the case of discrete variables). These in turn can be used to find two other types of distributions: the marginal distribution giving the probabilities for any one of the variables with no reference to any specific ranges of values for the other variables, and the conditional probability distribution giving the probabilities for any subset of the variables conditional on particular values of the remaining variables.

Marginal distributionIn probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables in the subset without reference to the values of the other variables. This contrasts with a conditional distribution, which gives the probabilities contingent upon the values of the other variables.

Marginal variables are those variables in the subset of variables being retained. These concepts are "marginal" because they can be found by summing values in a table along rows or columns, and writing the sum in the margins of the table. The distribution of the marginal variables (the marginal distribution) is obtained by marginalizing – that is, focusing on the sums in the margin – over the distribution of the variables being discarded, and the discarded variables are said to have been marginalized out.

The context here is that the theoretical studies being undertaken, or the data analysis being done, involves a wider set of random variables but that attention is being limited to a reduced number of those variables. In many applications, an analysis may start with a given collection of random variables, then first extend the set by defining new ones (such as the sum of the original random variables) and finally reduce the number by placing interest in the marginal distribution of a subset (such as the sum). Several different analyses may be done, each treating a different subset of variables as the marginal variables.

Maximum entropy probability distributionIn statistics and information theory, a maximum entropy probability distribution has entropy that is at least as great as that of all other members of a specified class of probability distributions. According to the principle of maximum entropy, if nothing is known about a distribution except that it belongs to a certain class (usually defined in terms of specified properties or measures), then the distribution with the largest entropy should be chosen as the least-informative default. The motivation is twofold: first, maximizing entropy minimizes the amount of prior information built into the distribution; second, many physical systems tend to move towards maximal entropy configurations over time.

MeanThere are several kinds of **mean** in various branches of mathematics (especially statistics).

For a data set, the arithmetic mean, also called the mathematical expectation or average, is the central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers *x*_{1}, *x*_{2}, ..., *x _{n}* is typically denoted by , pronounced "

In probability and statistics, the population **mean**, or expected value, are a measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution. In the case of a discrete probability distribution of a random variable *X*, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value *x* of *X* and its probability *p*(*x*), and then adding all these products together, giving . An analogous formula applies to the case of a continuous probability distribution. Not every probability distribution has a defined mean; see the Cauchy distribution for an example. Moreover, for some distributions the mean is infinite.

For a finite population, the **population mean** of a property is equal to the arithmetic mean of the given property while considering every member of the population. For example, the population mean height is equal to the sum of the heights of every individual divided by the total number of individuals. The sample mean may differ from the population mean, especially for small samples. The law of large numbers dictates that the larger the size of the sample, the more likely it is that the sample mean will be close to the population mean.

Outside probability and statistics, a wide range of other notions of "mean" are often used in geometry and analysis; examples are given below.

Poisson distributionIn probability theory and statistics, the Poisson distribution (French pronunciation: [pwasɔ̃]; in English often rendered ), named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

For instance, an individual keeping track of the amount of mail they receive each day may notice that they receive an average number of 4 letters per day. If receiving any particular piece of mail does not affect the arrival times of future pieces of mail, i.e., if pieces of mail from a wide range of sources arrive independently of one another, then a reasonable assumption is that the number of pieces of mail received in a day obeys a Poisson distribution. Other examples that may follow a Poisson include the number of phone calls received by a call center per hour and the number of decay events per second from a radioactive source.

Posterior probabilityIn Bayesian statistics, the posterior probability of a random event or an uncertain proposition is the conditional probability that is assigned after the relevant evidence or background is taken into account. Similarly, the posterior probability distribution is the probability distribution of an unknown quantity, treated as a random variable, conditional on the evidence obtained from an experiment or survey. "Posterior", in this context, means after taking into account the relevant evidence related to the particular case being examined. For instance, there is a ("non-posterior") probability of a person finding buried treasure if they dig in a random spot, and a posterior probability of finding buried treasure if they dig in a spot where their metal detector rings.

Prior probabilityIn Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity before some evidence is taken into account. For example, the prior could be the probability distribution representing the relative proportions of voters who will vote for a particular politician in a future election. The unknown quantity may be a parameter of the model or a latent variable rather than an observable variable.

Bayes' theorem calculates the renormalized pointwise product of the prior and the likelihood function, to produce the posterior probability distribution, which is the conditional distribution of the uncertain quantity given the data.

Similarly, the prior probability of a random event or an uncertain proposition is the unconditional probability that is assigned before any relevant evidence is taken into account.

Priors can be created using a number of methods. A prior can be determined from past information, such as previous experiments. A prior can be elicited from the purely subjective assessment of an experienced expert. An uninformative prior can be created to reflect a balance among outcomes when no information is available. Priors can also be chosen according to some principle, such as symmetry or maximizing entropy given constraints; examples are the Jeffreys prior or Bernardo's reference prior. When a family of conjugate priors exists, choosing a prior from that family simplifies calculation of the posterior distribution.

Parameters of prior distributions are a kind of hyperparameter. For example, if one uses a beta distribution to model the distribution of the parameter p of a Bernoulli distribution, then:

p is a parameter of the underlying system (Bernoulli distribution), and

α and β are parameters of the prior distribution (beta distribution); hence hyperparameters.Hyperparameters themselves may have hyperprior distributions expressing beliefs about their values. A Bayesian model with more than one level of prior like this is called a hierarchical Bayes model.

Probability density functionIn probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) 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 (since there are an infinite set of possible values to begin with), 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 a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value. This probability is given by the integral of this variable's PDF over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one.

The terms "probability distribution function" and "probability function" have also sometimes been used to denote the probability density function. However, this use is not standard among probabilists and statisticians. In other sources, "probability distribution function" may be used when the probability distribution is defined as a function over general sets of values, or it may refer to the cumulative distribution function, or it may be a probability mass function (PMF) rather than the density. "Density function" itself is also used for the probability mass function, leading to further confusion. In general though, the PMF is used in the context of discrete random variables (random variables that take values on a discrete set), while PDF is used in the context of continuous random variables.

Probability distribution functionA probability distribution function is some function that may be used to define a particular probability distribution. Depending upon which text is consulted, the term may refer to:

a cumulative distribution function,

a probability mass function

a probability density function.The similar term probability function may mean any of the above and, in addition,

a probability measure function, as in a probability space, where the domain of the function is the set of events.

Sampling distributionIn statistics, a sampling distribution or finite-sample distribution is the probability distribution of a given random-sample-based statistic. If an arbitrarily large number of samples, each involving multiple observations (data points), were separately used in order to compute one value of a statistic (such as, for example, the sample mean or sample variance) for each sample, then the sampling distribution is the probability distribution of the values that the statistic takes on. In many contexts, only one sample is observed, but the sampling distribution can be found theoretically.

Sampling distributions are important in statistics because they provide a major simplification en route to statistical inference. More specifically, they allow analytical considerations to be based on the probability distribution of a statistic, rather than on the joint probability distribution of all the individual sample values.

Shape of a probability distributionIn statistics, the concept of the shape of a probability distribution arises in questions of finding an appropriate distribution to use to model the statistical properties of a population, given a sample from that population. The shape of a distribution may be considered either descriptively, using terms such as "J-shaped", or numerically, using quantitative measures such as skewness and kurtosis.

Considerations of the shape of a distribution arise in statistical data analysis, where simple quantitative descriptive statistics and plotting techniques such as histograms can lead on to the selection of a particular family of distributions for modelling purposes.

Singular distributionIn probability, a singular distribution is a probability distribution concentrated on a set of Lebesgue measure zero, where the probability of each point in that set is zero.

Statistical modelA statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of some sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form, the data-generating process.

A statistical model is usually specified as a mathematical relationship between one or more random variables and other non-random variables. As such, a statistical model is "a formal representation of a theory" (Herman Adèr quoting Kenneth Bollen).All statistical hypothesis tests and all statistical estimators are derived via statistical models. More generally, statistical models are part of the foundation of statistical inference.

Uniform distribution (continuous)In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by the two parameters, a and b, which are its minimum and maximum values. The distribution is often abbreviated U(a,b). It is the maximum entropy probability distribution for a random variable X under no constraint other than that it is contained in the distribution's support.

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