Editing Probability distribution (section)

== Common probability distributions and their applications ==
{{Main list|List of probability distributions}}

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 theory of gases|kinetic properties of gases]] to the [[quantum mechanical]] description of [[fundamental particles]]. For these and many other reasons, simple [[number]]s are often inadequate for describing a quantity, while probability distributions are often more appropriate.

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, absolutely continuous, multivariate, etc.)

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

=== Linear growth (e.g. errors, offsets) ===

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

=== Exponential growth (e.g. prices, incomes, populations) ===

* [[Log-normal distribution]], for a single such quantity whose log is [[Normal distribution|normally]] distributed
* [[Pareto distribution]], for a single such quantity whose log is [[Exponential distribution|exponentially]] distributed; the prototypical [[power law]] distribution

=== Uniformly distributed quantities ===

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

=== Bernoulli trials (yes/no events, with a given probability) ===

* 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 (statistics)|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 {{mvar|K}} possible outcomes) ===

* [[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) ===

* [[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 ===

* [[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 ===

* [[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 distribution|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 distribution|chi squared]] variables; useful e.g. for inferences that involve comparing variances or involving [[R-squared]] (the squared [[Pearson product-moment correlation coefficient|correlation coefficient]])

=== As conjugate prior distributions in Bayesian inference ===
{{Main|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 (statistics)|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]]<ref>{{Cite book|title=Pattern recognition and machine learning|last=Bishop, Christopher M.|date=2006|publisher=Springer|isbn=0-387-31073-8|location=New York| oclc=71008143}}</ref>

=== Some specialized applications of probability distributions ===

* The [[cache language model]]s and other [[Statistical Language Model|statistical language models]] used in [[natural language processing]] to assign probabilities to the occurrence of particular words and word sequences do so by means of probability distributions.
* In quantum mechanics, the probability density of finding the particle at a given point is proportional to the square of the magnitude of the particle's [[wavefunction]] at that point (see [[Born rule]]). Therefore, the probability distribution function of the position of a particle is described by <math display="inline">P_{a\le x\le b} (t) = \int_a^b d x\,|\Psi(x,t)|^2 </math>, probability that the particle's position {{math|''x''}} will be in the interval {{math|''a'' ≤ ''x'' ≤ ''b''}} in dimension one, and a similar [[triple integral]] in dimension three. This is a key principle of quantum mechanics.<ref>{{Cite book| title=Physical chemistry for the chemical sciences|last=Chang, Raymond.|others=Thoman, John W., Jr., 1960-| year=2014| isbn=978-1-68015-835-9 |location=[Mill Valley, California]|pages=403–406|oclc=927509011}}</ref>
* Probabilistic load flow in [[power-flow study]] explains the uncertainties of input variables as probability distribution and provides the power flow calculation also in term of probability distribution.<ref>{{Cite book|title=2008 Third International Conference on Electric Utility Deregulation and Restructuring and Power Technologies|last1=Chen|first1=P.| last2=Chen|first2=Z.| last3=Bak-Jensen|first3=B.|date=April 2008|isbn=978-7-900714-13-8|pages=1586–1591|chapter=Probabilistic load flow: A review| doi=10.1109/drpt.2008.4523658|s2cid=18669309}}</ref>
* Prediction of natural phenomena occurrences based on previous [[frequency distribution]]s such as [[tropical cyclone]]s, hail, time in between events, etc.<ref>{{Cite book|title=Statistical methods in hydrology and hydroclimatology|last=Maity | first = Rajib| isbn=978-981-10-8779-0|location=Singapore|oclc=1038418263|date = 2018-04-30}}</ref>