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= Probability distribution =

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}.

== Terminology [edit | edit source] ==

Functions for Discrete Variables[edit | edit source]

 * Probability function: describes the probability distribution of a discrete random variables
 * Probability mass function (PMF): function that gives the probability that a discrete random variable is equal to some value
 * Frequency distribution: 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.
 * Discrete probability distribution function: general term to indicate the way the total probability of 1 is distributed over all various possible outcomes (i.e. over entire population) for discrete random variable
 * Cumulative distribution function: function evaluating the probability that  will take a value less than or equal to  for a discrete random variable
 * Categorical distribution: for discrete random variables with a finite set of values.

Functions for Continuous Variables[edit | edit source]

 * Probability density function (PDF): 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
 * Continuous probability distribution function: most often reserved for continuous random variables
 * Cumulative distribution function: function evaluating the probability that  will take a value less than or equal to  for continuous variable

Basic terms[edit | edit source]

 * 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.

Kolmogorov definition[edit | edit source]
Main articles: Probability space and Probability measure

In the measure-theoretic formalization of probability theory, a random variable is defined as a measurable function X from a probability space  to a measurable space. Given that the probability of the measurable function (or event) X satisfies Kolmogorov's probability axioms, the probability distribution of X is hence defined as the pushforward measure X*P  of X, which is a probability measure on  satisfying X*P = PX −1.

Common probability distributions and their applications[edit | edit source]
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.

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.)

All of the univariate distributions below are singly peaked; that is, it is assumed that the values cluster around a single point. In practice, actually observed quantities may cluster around multiple values. Such quantities can be modeled using a mixture distribution.

Linear growth (e.g. errors, offsets)[edit | edit source]

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

Exponential growth (e.g. prices, incomes, populations)[edit | edit source]

 * 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

Uniformly distributed quantities[edit | edit source]

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

Bernoulli trials (yes/no events, with a given probability)[edit | edit source]

 * 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 Pólya urn model (in some sense, the "opposite" of sampling without replacement)

Categorical outcomes (events with K possible outcomes, with a given probability for each outcome)[edit | edit source]

 * 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 process (events that occur independently with a given rate)[edit | edit source]

 * 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

Absolute values of vectors with normally distributed components[edit | edit source]

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

Normally distributed quantities operated with sum of squares (for hypothesis testing)[edit | edit source]

 * 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)

As a conjugate prior distributions in Bayesian inference[edit | edit source]
Main article: Conjugate prior


 * 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

Some specialized applications of probability distributions

 * 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.
 * Prediction of natural phenomena occurrences based on previous frequency distributions such as tropical cyclones, hail, time in between events, etc.