This is a list of probability topics, by Wikipedia page. It overlaps with the (alphabetical) list of statistical topics. There are also the outline of probability and

catalog of articles in probability theory This page lists articles related to probability theory. In particular, it lists many articles corresponding to specific probability distributions. Such articles are marked here by a code of the form (X:Y), which refers to number of random variab ...

. For distributions, see List of probability distributions. For journals, see list of probability journals. For contributors to the field, see list of mathematical probabilists and

list of statisticians This list of statisticians lists people who have made notable contributions to the theories or application of statistics, or to the related fields of probability or machine learning. Also included are actuaries and demographers. __NOTOC__ A ...

General aspects

Probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

Randomness In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual rand ...

Pseudorandomness A pseudorandom sequence of numbers is one that appears to be statistically random, despite having been produced by a completely deterministic and repeatable process. Background The generation of random numbers has many uses, such as for random ...

, Quasirandomness *

Randomization Randomization is the process of making something random. Randomization is not haphazard; instead, a random process is a sequence of random variables describing a process whose outcomes do not follow a deterministic pattern, but follow an evolution ...

hardware random number generator In computing, a hardware random number generator (HRNG) or true random number generator (TRNG) is a device that generates random numbers from a physical process, rather than by means of an algorithm. Such devices are often based on microscopi ...

Random number generation Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols that cannot be reasonably predicted better than by random chance is generated. This means that the particular outc ...

Random sequence The concept of a random sequence is essential in probability theory and statistics. The concept generally relies on the notion of a sequence of random variables and many statistical discussions begin with the words "let ''X''1,...,''Xn'' be indepen ...

Uncertainty Uncertainty refers to Epistemology, epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. Uncertainty arises in partially ...

Statistical dispersion In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartil ...

Observational error Observational error (or measurement error) is the difference between a measured value of a quantity and its true value.Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. In statistics, an error is not necessarily a "mistake ...

* Equiprobable **

Equipossible Equipossibility is a philosophical concept in possibility theory that is a precursor to the notion of equiprobability in probability theory. It is used to distinguish what ''can'' occur in a probability experiment. For example, it is the differenc ...

Average In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, ...

Probability interpretations The word probability has been used in a variety of ways since it was first applied to the mathematical study of games of chance. Does probability measure the real, physical, tendency of something to occur, or is it a measure of how strongly one be ...

* Markovian *

Statistical regularity Statistical regularity is a notion in statistics and probability theory that random events exhibit regularity when repeated enough times or that enough sufficiently similar random events exhibit regularity. It is an umbrella term that covers the law ...

Central tendency In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.Weisberg H.F (1992) ''Central Tendency and Variability'', Sage University Paper Series on Quantitative Applications ...

Bean machine The Galton board, also known as the Galton box or quincunx or bean machine, is a device invented by Sir Francis Galton to demonstrate the central limit theorem, in particular that with sufficient sample size the binomial distribution approximat ...

* Relative frequency *

Frequency probability Frequentist probability or frequentism is an interpretation of probability; it defines an event's probability as the limit of its relative frequency in many trials (the long-run probability). Probabilities can be found (in principle) by a re ...

Maximum likelihood In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed sta ...

Bayesian probability Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification o ...

Principle of indifference The principle of indifference (also called principle of insufficient reason) is a rule for assigning epistemic probabilities. The principle of indifference states that in the absence of any relevant evidence, agents should distribute their cre ...

* Credal set *

Cox's theorem Cox's theorem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws of probability theory from a certain set of postulates. This derivation justifies the so-called "logical" interpretation of probability, as the laws of pr ...

Principle of maximum entropy The principle of maximum entropy states that the probability distribution which best represents the current state of knowledge about a system is the one with largest entropy, in the context of precisely stated prior data (such as a proposition ...

Information entropy In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent to the variable's possible outcomes. Given a discrete random variable X, which takes values in the alphabet \ ...

* Urn problems * Extractor *

Aleatoric Aleatoricism or aleatorism, the noun associated with the adjectival aleatory and aleatoric, is a term popularised by the musical composer Pierre Boulez, but also Witold Lutosławski and Franco Evangelisti, for compositions resulting from "action ...

aleatoric music Aleatoric music (also aleatory music or chance music; from the Latin word ''alea'', meaning "dice") is music in which some element of the composition is left to chance, and/or some primary element of a composed work's realization is left to the ...

* Free probability * Exotic probability * Schrödinger method * Empirical measure * Glivenko–Cantelli theorem * Zero–one law **

Kolmogorov's zero–one law In probability theory, Kolmogorov's zero–one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, namely a ''tail event of independent σ-algebras'', will either almost surely happen or almost su ...

** Hewitt–Savage zero–one law *

Law of truly large numbers The law of truly large numbers (a statistical adage), attributed to Persi Diaconis and Frederick Mosteller, states that with a large enough number of independent samples, any highly implausible (i.e. unlikely in any single sample, but with cons ...

** Littlewood's law **

Infinite monkey theorem The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, such as the complete works of William Shakespeare. In fact, the monkey would ...

Littlewood–Offord problem In mathematical field of combinatorial geometry, the Littlewood–Offord problem is the problem of determining the number of subsums of a set of vectors that fall in a given convex set. More formally, if ''V'' is a vector space of dimension '' ...

Inclusion–exclusion principle In combinatorics, a branch of mathematics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as : , A \c ...

* Impossible event *

Information geometry Information geometry is an interdisciplinary field that applies the techniques of differential geometry to study probability theory and statistics. It studies statistical manifolds, which are Riemannian manifolds whose points correspond to p ...

Talagrand's concentration inequality In the probability theory field of mathematics , Talagrand's concentration inequality is an isoperimetric-type inequality for product probability spaces. It was first proved by the French mathematician Michel Talagrand. The inequality is one of th ...

Foundations of probability theory

Probability theory 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 o ...

Probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...

Sample space In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually de ...

Standard probability space In probability theory, a standard probability space, also called Lebesgue–Rokhlin probability space or just Lebesgue space (the latter term is ambiguous) is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin i ...

Random element 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 who commented that the “development of probability theory and expansio ...

*** Random compact set ** Dynkin system *

Probability axioms The Kolmogorov axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933. These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probabili ...

Normalizing constant The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. The normalizing constant is used to reduce any probability function to a probability density function with total probability of one. ...

Event (probability theory) In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. A single outcome may be an element of many different events, and different events in an experiment are us ...

** Complementary event *

Elementary event In probability theory, an elementary event, also called an atomic event or sample point, is an event which contains only a single outcome in the sample space. Using set theory terminology, an elementary event is a singleton. Elementary events a ...

Mutually exclusive In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails ...

Boole's inequality In probability theory, Boole's inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the indivi ...

Probability density function In 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) c ...

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

* Law of total cumulance *

Law of total expectation The proposition in probability theory known as the law of total expectation, the law of iterated expectations (LIE), Adam's law, the tower rule, and the smoothing theorem, among other names, states that if X is a random variable whose expected v ...

Law of total probability In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. It expresses the total probability of an outcome which can be realized via several distinct ev ...

* Law of total variance *

Almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0 ...

* Bayesianism *

Prior probability In 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 ...

Posterior probability The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior p ...

* Borel's paradox * Bertrand's paradox *

Coherence (philosophical gambling strategy) In a thought experiment proposed by the Italian probabilist Bruno de Finetti in order to justify Bayesian probability, an array of wagers is coherent precisely if it does not expose the wagerer to certain loss regardless of the outcomes of events ...

Dutch book In gambling, a Dutch book or lock is a set of odds and bets, established by the bookmaker, that ensures that the bookmaker will profit—at the expense of the gamblers—regardless of the outcome of the event (a horse race, for example) on which ...

Algebra of random variables The algebra of random variables in statistics, provides rules for the symbolic manipulation of random variables, while avoiding delving too deeply into the mathematically sophisticated ideas of probability theory. Its symbolism allows the treatm ...

Belief propagation A belief is an attitude that something is the case, or that some proposition is true. In epistemology, philosophers use the term "belief" to refer to attitudes about the world which can be either true or false. To believe something is to take i ...

* Transferable belief model *

Dempster–Shafer theory The theory of belief functions, also referred to as evidence theory or Dempster–Shafer theory (DST), is a general framework for reasoning with uncertainty, with understood connections to other frameworks such as probability, possibility and i ...

Possibility theory Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. It uses measures of possibility and necessity between 0 and 1, ranging from impossible to possible and unneces ...

Random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
s

Discrete random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...

Probability mass function In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...

Constant random variable In mathematics, a degenerate distribution is, according to some, a probability distribution in a space with support only on a manifold of lower dimension, and according to others a distribution with support only at a single point. By the latter ...

Expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...

Jensen's inequality In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier ...

Variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...

** Standard deviation **

Geometric standard deviation In probability theory and statistics, the geometric standard deviation (GSD) describes how spread out are a set of numbers whose preferred average is the geometric mean. For such data, it may be preferred to the more usual standard deviation. N ...

Multivariate random variable In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its valu ...

Joint probability distribution Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considere ...

Marginal distribution In 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 ...

** Kirkwood approximation *

Independent identically-distributed random variables In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usua ...

** Independent and identically-distributed random variables * Statistical independence **

Conditional independence In probability theory, conditional independence describes situations wherein an observation is irrelevant or redundant when evaluating the certainty of a hypothesis. Conditional independence is usually formulated in terms of conditional probabil ...

Pairwise independence In probability theory, a pairwise independent collection of random variables is a set of random variables any two of which are independent. Any collection of mutually independent random variables is pairwise independent, but some pairwise independe ...

Covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the le ...

Covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements o ...

De Finetti's theorem In probability theory, de Finetti's theorem states that exchangeable observations are conditionally independent relative to some latent variable. An epistemic probability distribution could then be assigned to this variable. It is named in hono ...

Correlation In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statisti ...

Uncorrelated In probability theory and statistics, two real-valued random variables, X, Y, are said to be uncorrelated if their covariance, \operatorname ,Y= \operatorname Y- \operatorname \operatorname /math>, is zero. If two variables are uncorrelated, there ...

Correlation function A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables re ...

Canonical correlation In statistics, canonical-correlation analysis (CCA), also called canonical variates analysis, is a way of inferring information from cross-covariance matrices. If we have two vectors ''X'' = (''X''1, ..., ''X'n'') and ''Y' ...

Convergence of random variables 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 ...

** Weak convergence of measures ***

Helly–Bray theorem In probability theory, the Helly–Bray theorem relates the weak convergence of cumulative distribution functions to the convergence of expectations of certain measurable functions. It is named after Eduard Helly and Hubert Evelyn Bray. Let '' ...

***

Slutsky's theorem In probability theory, Slutsky’s theorem extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables. The theorem was named after Eugen Slutsky. Slutsky's theorem is also attributed ...

** Skorokhod's representation theorem ** Lévy's continuity theorem **

Uniform integrability In mathematics, uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales. Measure-theoretic definition Uniform integrability is an extension to the ...

Markov's inequality In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function of a random variable is greater than or equal to some positive constant. It is named after the Russian mathematician Andrey Marko ...

Chebyshev's inequality In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from th ...

Chernoff bound In probability theory, the Chernoff bound gives exponentially decreasing bounds on tail distributions of sums of independent random variables. Despite being named after Herman Chernoff, the author of the paper it first appeared in, the result is d ...

Chernoff's inequality In probability theory, the Chernoff bound gives exponentially decreasing bounds on tail distributions of sums of independent random variables. Despite being named after Herman Chernoff, the author of the paper it first appeared in, the result is ...

Bernstein inequalities (probability theory) In probability theory, Bernstein inequalities give bounds on the probability that the sum of random variables deviates from its mean. In the simplest case, let ''X''1, ..., ''X'n'' be independent Bernoulli trial, Bernoulli random variab ...

** Hoeffding's inequality * Kolmogorov's inequality * Etemadi's inequality *

Chung–Erdős inequality In probability theory 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 express ...

Khintchine inequality In mathematics, the Khintchine inequality, named after Aleksandr Khinchin and spelled in multiple ways in the Latin alphabet, is a theorem from probability, and is also frequently used in mathematical analysis, analysis. Heuristically, it says that ...

* Paley–Zygmund inequality * Laws of large numbers ** Asymptotic equipartition property ** Typical set **

Law of large numbers In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials sho ...

Kolmogorov's two-series theorem In probability theory, Kolmogorov's two-series theorem is a result about the convergence of random series. It follows from Kolmogorov's inequality and is used in one proof of the strong law of large numbers. Statement of the theorem Let \left( ...

Random field In physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as \mathbb^n). That is, it is a function f(x) that takes on a random value at each point x \in \mathbb^n(or some other ...

Conditional random field Conditional random fields (CRFs) are a class of statistical modeling methods often applied in pattern recognition and machine learning and used for structured prediction. Whereas a classifier predicts a label for a single sample without consi ...

Borel–Cantelli lemma In probability theory, the Borel–Cantelli lemma is a theorem about sequences of events. In general, it is a result in measure theory. It is named after Émile Borel and Francesco Paolo Cantelli, who gave statement to the lemma in the first ...

* Wick product

Conditional probability In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occu ...

* Conditioning (probability) *

Conditional expectation In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given ...

Conditional probability distribution In probability theory and statistics, given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value; in some cases the c ...

Regular conditional probability In probability theory, regular conditional probability is a concept that formalizes the notion of conditioning on the outcome of a random variable. The resulting conditional probability distribution is a parametrized family of probability measures c ...

Disintegration theorem In mathematics, the disintegration theorem is a result in measure theory and probability theory. It rigorously defines the idea of a non-trivial "restriction" of a measure to a measure zero subset of the measure space in question. It is related ...

Bayes' theorem In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For exa ...

de Finetti's theorem In probability theory, de Finetti's theorem states that exchangeable observations are conditionally independent relative to some latent variable. An epistemic probability distribution could then be assigned to this variable. It is named in hono ...

Exchangeable random variables In statistics, an exchangeable sequence of random variables (also sometimes interchangeable) is a sequence ''X''1, ''X''2, ''X''3, ... (which may be finitely or infinitely long) whose joint probability distribution does not change wh ...

Rule of succession In probability theory, the rule of succession is a formula introduced in the 18th century by Pierre-Simon Laplace in the course of treating the sunrise problem. The formula is still used, particularly to estimate underlying probabilities when ...

Conditional event algebra A standard, Boolean algebra of events is a set of events related to one another by the familiar operations ''and'', ''or'', and ''not''. A conditional event algebra (CEA) contains not just ordinary events but also conditional events, which have the ...

** Goodman–Nguyen–van Fraassen algebra

Theory of probability distributions

Probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...

* Probability distribution function *

* Quantile * Moment (mathematics) **

Moment about the mean In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random ...

** Standardized moment ***

Skewness In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined. For a unimo ...

***

Kurtosis In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kur ...

***

Locality Locality may refer to: * Locality (association), an association of community regeneration organizations in England * Locality (linguistics) * Locality (settlement) * Suburbs and localities (Australia), in which a locality is a geographic subdivi ...

Cumulant In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will hav ...

Factorial moment In probability theory, the factorial moment is a mathematical quantity defined as the expectation or average of the falling factorial of a random variable. Factorial moments are useful for studying non-negative integer-valued random variables,D. J. ...

*** Law of the unconscious statistician **

Second moment method In mathematics, the second moment method is a technique used in probability theory and analysis to show that a random variable has positive probability of being positive. More generally, the "moment method" consists of bounding the probability th ...

***

Coefficient of variation In probability theory and statistics, the coefficient of variation (CV), also known as relative standard deviation (RSD), is a standardized measure of dispersion of a probability distribution or frequency distribution. It is often expressed ...

***

Variance-to-mean ratio In probability theory and statistics, the index of dispersion, dispersion index, coefficient of dispersion, relative variance, or variance-to-mean ratio (VMR), like the coefficient of variation, is a normalized measure of the dispersion of a pro ...

Covariance function In probability theory and statistics, the covariance function describes how much two random variables change together (their ''covariance'') with varying spatial or temporal separation. For a random field or stochastic process ''Z''(''x'') on a d ...

** An inequality on location and scale parameters **

Taylor expansions for the moments of functions of random variables In probability theory, it is possible to approximate the moments of a function ''f'' of a random variable ''X'' using Taylor expansions, provided that ''f'' is sufficiently differentiable and that the moments of ''X'' are finite. First moment ...

** Moment problem ***

Hamburger moment problem In mathematics, the Hamburger moment problem, named after Hans Ludwig Hamburger, is formulated as follows: given a sequence (''m''0, ''m''1, ''m''2, ...), does there exist a positive Borel measure ''μ'' (for instance, the measure determined by ...

**** Carleman's condition ***

Hausdorff moment problem In mathematics, the Hausdorff moment problem, named after Felix Hausdorff, asks for necessary and sufficient conditions that a given sequence be the sequence of moments :m_n = \int_0^1 x^n\,d\mu(x) of some Borel measure supported on the clo ...

*** Trigonometric moment problem ***

Stieltjes moment problem In mathematics, the Stieltjes moment problem, named after Thomas Joannes Stieltjes, seeks necessary and sufficient conditions for a sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and ...

Prior probability distribution In 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 int ...

Total variation distance In probability theory, the total variation distance is a distance measure for probability distributions. It is an example of a statistical distance metric, and is sometimes called the statistical distance, statistical difference or variational dist ...

Hellinger distance In probability and statistics, the Hellinger distance (closely related to, although different from, the Bhattacharyya distance) is used to quantify the similarity between two probability distributions. It is a type of ''f''-divergence. The Hel ...

Wasserstein metric In mathematics, the Wasserstein distance or Kantorovich–Rubinstein metric is a distance function defined between probability distributions on a given metric space M. It is named after Leonid Vaseršteĭn. Intuitively, if each distribution is ...

* Lévy–Prokhorov metric ** Lévy metric * Continuity correction *

Heavy-tailed distribution In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded: that is, they have heavier tails than the exponential distribution. In many applications it is the right tail of the distrib ...

Truncated distribution In statistics, a truncated distribution is a conditional distribution that results from restricting the domain of some other probability distribution. Truncated distributions arise in practical statistics in cases where the ability to record, or ...

Infinite divisibility Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...

* Stability (probability) * Indecomposable distribution *

Power law In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one qua ...

Anderson's theorem In mathematics, Anderson's theorem is a result in real analysis and geometry which says that the integral of an integrable, symmetric, unimodal, non-negative function ''f'' over an ''n''-dimensional convex body ''K'' does not decrease if ''K'' is ...

Probability bounds analysis Probability bounds analysis (PBA) is a collection of methods of uncertainty propagation for making qualitative and quantitative calculations in the face of uncertainties of various kinds. It is used to project partial information about random varia ...

* Probability box

Properties of probability distributions

Central limit theorem In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables thems ...

** Illustration of the central limit theorem ** Concrete illustration of the central limit theorem ** Berry–Esséen theorem ** Berry–Esséen theorem **

De Moivre–Laplace theorem In probability theory, the de Moivre–Laplace theorem, which is a special case of the central limit theorem, states that the normal distribution may be used as an approximation to the binomial distribution under certain conditions. In particu ...

** Lyapunov's central limit theorem ** Martingale central limit theorem ** Infinite divisibility (probability) ** Method of moments (probability theory) ** Stability (probability) **

Stein's lemma Stein's lemma, named in honor of Charles Stein, is a theorem of probability theory that is of interest primarily because of its applications to statistical inference — in particular, to James–Stein estimation and empirical Bayes methods & ...

Characteristic function (probability theory) In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is ...

Lévy continuity theorem Levy, Lévy or Levies may refer to: People * Levy (surname), people with the surname Levy or Lévy * Levy Adcock (born 1988), American football player * Levy Barent Cohen (1747–1808), Dutch-born British financier and community worker * Levy Fi ...

* Darmois–Skitovich theorem *

Edgeworth series The Gram–Charlier A series (named in honor of Jørgen Pedersen Gram and Carl Charlier), and the Edgeworth series (named in honor of Francis Ysidro Edgeworth) are series that approximate a probability distribution in terms of its cumulants. The s ...

* Kac–Bernstein theorem *

Location parameter In geography, location or place are used to denote a region (point, line, or area) on Earth's surface or elsewhere. The term ''location'' generally implies a higher degree of certainty than ''place'', the latter often indicating an entity with an ...

Maxwell's theorem In probability theory, Maxwell's theorem, named in honor of James Clerk Maxwell, states that if the probability distribution of a vector-valued random variable ''X'' = ( ''X''1, ..., ''X'n'' )''T'' is the same as the distribution of ''GX'' for ...

Moment-generating function In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compar ...

** Factorial moment generating function *

Negative probability The probability of the outcome of an experiment is never negative, although a quasiprobability distribution allows a negative probability, or quasiprobability for some events. These distributions may apply to unobservable events or conditional prob ...

Probability-generating function In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. Probability generating functions are oft ...

* Vysochanskiï–Petunin inequality *

Mutual information In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the " amount of information" (in units such as ...

Kullback–Leibler divergence In mathematical statistics, the Kullback–Leibler divergence (also called relative entropy and I-divergence), denoted D_\text(P \parallel Q), is a type of statistical distance: a measure of how one probability distribution ''P'' is different fro ...

Normally distributed and uncorrelated does not imply independent In probability theory, although simple examples illustrate that linear uncorrelatedness of two random variables does not in general imply their independence, it is sometimes mistakenly thought that it does imply that when the two random variables a ...

* Le Cam's theorem * Large deviations theory ** Contraction principle (large deviations theory) **

Varadhan's lemma In mathematics, Varadhan's lemma is a result from large deviations theory named after S. R. Srinivasa Varadhan. The result gives information on the asymptotic distribution of a statistic ''φ''(''Z'ε'') of a family of random variables ''Z'� ...

Tilted large deviation principle In mathematics — specifically, in large deviations theory — the tilted large deviation principle is a result that allows one to generate a new large deviation principle from an old one by "tilting", i.e. integration against an expon ...

Rate function In mathematics — specifically, in large deviations theory — a rate function is a function used to quantify the probabilities of rare events. It is required to have several properties which assist in the formulation of the large devia ...

** Laplace principle (large deviations theory) **

Exponentially equivalent measures In mathematics, exponential equivalence of measures is how two sequences or families of probability measures are "the same" from the point of view of large deviations theory. Definition Let (M,d) be a metric space and consider two one-parameter ...

** Cramér's theorem (second part)

Applied probability

* ''Empirical findings'' **

Benford's law Benford's law, also known as the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation that in many real-life sets of numerical data, the leading digit is likely to be small.Arno Berger and Theodore P ...

** Pareto principle ** Zipf's law *

Boy or Girl paradox The Boy or Girl paradox surrounds a set of questions in probability theory, which are also known as The Two Child Problem, Mr. Smith's Children and the Mrs. Smith Problem. The initial formulation of the question dates back to at least 1959, when&n ...

Stochastic processes

Adapted process In the study of stochastic processes, an adapted process (also referred to as a non-anticipating or non-anticipative process) is one that cannot "see into the future". An informal interpretation is that ''X'' is adapted if and only if, for every rea ...

Basic affine jump diffusion In mathematics probability theory, a basic affine jump diffusion (basic AJD) is a stochastic process Z of the form : dZ_t=\kappa (\theta -Z_t)\,dt+\sigma \sqrt\,dB_t+dJ_t,\qquad t\geq 0, Z_\geq 0, where B is a standard Brownian motion, and ...

Bernoulli process In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) 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. Th ...

** Bernoulli scheme *

Branching process In probability theory, a branching process is a type of mathematical object known as a stochastic process, which consists of collections of random variables. The random variables of a stochastic process are indexed by the natural numbers. The ori ...

Point process In statistics and probability theory, a point process or point field is a collection of mathematical points randomly located on a mathematical space such as the real line or Euclidean space. Kallenberg, O. (1986). ''Random Measures'', 4th edition ...

Chapman–Kolmogorov equation In mathematics, specifically in the theory of Markovian stochastic processes in probability theory, the Chapman–Kolmogorov equation(CKE) is an identity relating the joint probability distributions of different sets of coordinates on a stochastic p ...

Chinese restaurant process In probability theory, the Chinese restaurant process is a discrete-time stochastic process, analogous to seating customers at tables in a restaurant. Imagine a restaurant with an infinite number of circular tables, each with infinite capacity. Cu ...

Coupling (probability) In probability theory, coupling is a proof technique that allows one to compare two unrelated random variables (distributions) and by creating a random vector whose marginal distributions correspond to and respectively. The choice of is ge ...

Ergodic theory Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...

** Maximal ergodic theorem ** Ergodic (adjective) * Galton–Watson process * Gauss–Markov process *

Gaussian process In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. ...

** Gaussian random field **

Gaussian isoperimetric inequality In mathematics, the Gaussian isoperimetric inequality, proved by Boris Tsirelson and Vladimir Sudakov, and later independently by Christer Borell, states that among all sets of given Gaussian measure in the ''n''-dimensional Euclidean space, half-s ...

** Large deviations of Gaussian random functions *

Girsanov's theorem In probability theory, the Girsanov theorem tells how stochastic processes change under changes in measure. The theorem is especially important in the theory of financial mathematics as it tells how to convert from the physical measure which desc ...

* Hawkes process * Increasing process * Itô's lemma *

Jump diffusion Jump diffusion is a stochastic process that involves jumps and diffusion. It has important applications in magnetic reconnection, coronal mass ejections, condensed matter physics, option pricing, and pattern theory and computational vision. In ...

Law of the iterated logarithm In probability theory, the law of the iterated logarithm describes the magnitude of the fluctuations of a random walk. The original statement of the law of the iterated logarithm is due to A. Ya. Khinchin (1924). Another statement was given by A ...

* Lévy flight *

Lévy process In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which dis ...

* Loop-erased random walk *

Markov chain A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happen ...

** Examples of Markov chains **

Detailed balance The principle of detailed balance can be used in kinetic systems which are decomposed into elementary processes (collisions, or steps, or elementary reactions). It states that at equilibrium, each elementary process is in equilibrium with its reve ...

Markov property In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process. It is named after the Russian mathematician Andrey Markov. The term strong Markov property is similar to the Markov prop ...

Hidden Markov model A hidden Markov model (HMM) is a statistical Markov model in which the system being modeled is assumed to be a Markov process — call it X — with unobservable ("''hidden''") states. As part of the definition, HMM requires that there be an ob ...

** Maximum-entropy Markov model ** Markov chain mixing time *

Markov partition A Markov partition in mathematics is a tool used in dynamical systems theory, allowing the methods of symbolic dynamics to be applied to the study of hyperbolic dynamics. By using a Markov partition, the system can be made to resemble a discret ...

Markov process A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happen ...

Continuous-time Markov process A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of a ...

Piecewise-deterministic Markov process In probability theory, a piecewise-deterministic Markov process (PDMP) is a process whose behaviour is governed by random jumps at points in time, but whose evolution is deterministically governed by an ordinary differential equation between those ...

Martingale Martingale may refer to: * Martingale (probability theory), a stochastic process in which the conditional expectation of the next value, given the current and preceding values, is the current value * Martingale (tack) for horses * Martingale (coll ...

** Doob martingale **

Optional stopping theorem In probability theory, the optional stopping theorem (or sometimes Doob's optional sampling theorem, for American probabilist Joseph Doob) says that, under certain conditions, the expected value In probability theory, the expected value (a ...

** Martingale representation theorem ** Azuma's inequality ** Wald's equation *

Poisson process In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...

** Poisson random measure * Population process * Process with independent increments * Progressively measurable process *

Queueing theory Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the ...

** Erlang unit *

Random walk In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb ...

* Random walk Monte Carlo *

Renewal theory Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. Instead of exponentially distributed holding times, a renewal process may have any independent and identically distributed (IID) ho ...

Skorokhod's embedding theorem In mathematics and probability theory, Skorokhod's embedding theorem is either or both of two theorems that allow one to regard any suitable collection of random variables as a Wiener process ( Brownian motion) evaluated at a collection of stopping ...

Stationary process In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. ...

Stochastic calculus Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created ...

Itô calculus Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations. The centra ...

** Malliavin calculus ** Stratonovich integral *

Time series analysis In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...

Autoregressive model In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...

Moving average model In time series analysis, the moving-average model (MA model), also known as moving-average process, is a common approach for modeling univariate time series. The moving-average model specifies that the output variable is cross-correlated with ...

Autoregressive moving average model In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...

Autoregressive integrated moving average model In statistics and econometrics, and in particular in time series analysis, an autoregressive integrated moving average (ARIMA) model is a generalization of an autoregressive moving average (ARMA) model. Both of these models are fitted to time ser ...

** Anomaly time series *

Voter model In the mathematical theory of probability, the voter model is an interacting particle system introduced by Richard A. Holley and Thomas M. Liggett in 1975. One can imagine that there is a "voter" at each point on a connected graph, where the ...

Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It i ...

Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...

Geometric Brownian motion A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. ...

** Donsker's theorem **

Empirical process In probability theory, an empirical process is a stochastic process that describes the proportion of objects in a system in a given state. For a process in a discrete state space a population continuous time Markov chain or Markov population model ...

Wiener equation A simple mathematical representation of Brownian motion, the Wiener equation, named after Norbert Wiener, assumes the current velocity of a fluid particle fluctuates randomly: :\mathbf = \frac = g(t), where v is velocity, x is position, ''d/dt'' ...

** Wiener sausage

Geometric probability

* Buffon's needle *

Integral geometry In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformat ...

* Hadwiger's theorem *

Wendel's theorem In geometric probability theory, Wendel's theorem, named after James G. Wendel, gives the probability that ''N'' points distributed uniformly at random on an (n-1)-dimensional hypersphere all lie on the same "half" of the hypersphere. In other w ...

Gambling Gambling (also known as betting or gaming) is the wagering of something of Value (economics), value ("the stakes") on a Event (probability theory), random event with the intent of winning something else of value, where instances of strategy (ga ...

Luck Luck is the phenomenon and belief that defines the experience of improbable events, especially improbably positive or negative ones. The naturalistic interpretation is that positive and negative events may happen at any time, both due to rand ...

Game of chance A game of chance is in contrast with a game of skill. It is a game whose outcome is strongly influenced by some randomizing device. Common devices used include dice, spinning tops, playing cards, roulette wheels, or numbered balls drawn from a ...

Odds Odds provide a measure of the likelihood of a particular outcome. They are calculated as the ratio of the number of events that produce that outcome to the number that do not. Odds are commonly used in gambling and statistics. Odds also have ...

* Gambler's fallacy * Inverse gambler's fallacy *

Parrondo's paradox Parrondo's paradox, a paradox in game theory, has been described as: ''A combination of losing strategies becomes a winning strategy''. It is named after its creator, Juan Parrondo, who discovered the paradox in 1996. A more explanatory descript ...

* Pascal's wager *

Gambler's ruin The gambler's ruin is a concept in statistics. It is most commonly expressed as follows: A gambler playing a game with negative expected value will eventually go broke, regardless of their betting system. The concept was initially stated: A pers ...

Poker probability In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands. History Probability and gambling have been ideas since long before the invention of poker. The d ...

** Poker probability (Omaha) **

Poker probability (Texas hold 'em) In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands. History Probability and gambling have been ideas since long before the invention of poker. The de ...

Pot odds In poker, pot odds are the ratio of the current size of the pot to the cost of a contemplated call. Pot odds are compared to the odds of winning a hand with a future card in order to estimate the call's expected value. The purpose of this is to s ...

Roulette Roulette is a casino game named after the French word meaning ''little wheel'' which was likely developed from the Italian game Biribi''.'' In the game, a player may choose to place a bet on a single number, various groupings of numbers, the ...

Martingale (betting system) A martingale is a class of betting strategies that originated from and were popular in 18th-century France. The simplest of these strategies was designed for a game in which the gambler wins the stake if a coin comes up heads and loses if it com ...

** The man who broke the bank at Monte Carlo *

Lottery A lottery is a form of gambling that involves the drawing of numbers at random for a prize. Some governments outlaw lotteries, while others endorse it to the extent of organizing a national or state lottery. It is common to find some degree o ...

Lottery machine A lottery is a form of gambling that involves the drawing of numbers at random for a prize. Some governments outlaw lotteries, while others endorse it to the extent of organizing a national or state lottery. It is common to find some degree of ...

Pachinko is a mechanical game originating in Japan that is used as an arcade game, and much more frequently for gambling. Pachinko fills a niche in Japanese gambling comparable to that of the slot machine in the West as a form of low-stakes, low- ...

* Coupon collector's problem

Coincidence

Birthday paradox In probability theory, the birthday problem asks for the probability that, in a set of randomly chosen people, at least two will share a birthday. The birthday paradox is that, counterintuitively, the probability of a shared birthday exceeds 5 ...

Birthday problem In probability theory, the birthday problem asks for the probability that, in a set of randomly chosen people, at least two will share a birthday. The birthday paradox is that, counterintuitively, the probability of a shared birthday exceeds 5 ...

Index of coincidence In cryptography, coincidence counting is the technique (invented by William F. Friedman) of putting two texts side-by-side and counting the number of times that identical letters appear in the same position in both texts. This count, either as a ...

* Bible code *

Spurious relationship In statistics, a spurious relationship or spurious correlation is a mathematical relationship in which two or more events or variables are associated but '' not'' causally related, due to either coincidence or the presence of a certain third, ...

Monty Hall problem The Monty Hall problem is a brain teaser, in the form of a probability puzzle, loosely based on the American television game show ''Let's Make a Deal'' and named after its original host, Monty Hall. The problem was originally posed (and solved) ...

Algorithmics

Probable prime In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers. Different types of probable primes have different specific con ...

Probabilistic algorithm A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performan ...

= Randomised algorithm *

Monte Carlo method Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deter ...

Las Vegas algorithm In computing, a Las Vegas algorithm is a randomized algorithm that always gives correct results; that is, it always produces the correct result or it informs about the failure. However, the runtime of a Las Vegas algorithm differs depending on the ...

Probabilistic Turing machine In theoretical computer science, a probabilistic Turing machine is a non-deterministic Turing machine that chooses between the available transitions at each point according to some probability distribution. As a consequence, a probabilistic Turin ...

Stochastic programming In the field of mathematical optimization, stochastic programming is a framework for modeling optimization problems that involve uncertainty. A stochastic program is an optimization problem in which some or all problem parameters are uncertain ...

* Probabilistically checkable proof * Box–Muller transform * Metropolis algorithm *

Gibbs sampling In statistics, Gibbs sampling or a Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm for obtaining a sequence of observations which are approximated from a specified multivariate probability distribution, when direct sampling is diff ...

Inverse transform sampling method Inverse transform sampling (also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, Smirnov transform, or the golden ruleAalto University, N. Hyvönen, Computational methods in inverse probl ...

* Walk-on-spheres method

Financial mathematics Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require ...

Risk In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environme ...

Value at risk Value at risk (VaR) is a measure of the risk of loss for investments. It estimates how much a set of investments might lose (with a given probability), given normal market conditions, in a set time period such as a day. VaR is typically used by ...

Market risk Market risk is the risk of losses in positions arising from movements in market variables like prices and volatility. There is no unique classification as each classification may refer to different aspects of market risk. Nevertheless, the mos ...

Risk-neutral measure In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or '' equivalent martingale measure'') is a probability measure such that each share price is exactly equal to the discounted expectation of the share price un ...

* Volatility *

Technical analysis In finance, technical analysis is an analysis methodology for analysing and forecasting the direction of prices through the study of past market data, primarily price and volume. Behavioral economics and quantitative analysis use many of the sa ...

Kelly criterion In probability theory, the Kelly criterion (or Kelly strategy or Kelly bet), is a formula that determines the optimal theoretical size for a bet. It is valid when the expected returns are known. The Kelly bet size is found by maximizing the expec ...

Genetics Genetics is the study of genes, genetic variation, and heredity in organisms.Hartl D, Jones E (2005) It is an important branch in biology because heredity is vital to organisms' evolution. Gregor Mendel, a Moravian Augustinian friar worki ...

Punnett square The Punnett square is a square diagram that is used to predict the genotypes of a particular cross or breeding experiment. It is named after Reginald C. Punnett, who devised the approach in 1905. The diagram is used by biologists to determine ...

Hardy–Weinberg principle In population genetics, the Hardy–Weinberg principle, also known as the Hardy–Weinberg equilibrium, model, theorem, or law, states that allele and genotype frequencies in a population will remain constant from generation to generation in t ...

Ewens's sampling formula In population genetics, Ewens's sampling formula, describes the probabilities associated with counts of how many different alleles are observed a given number of times in the sample. Definition Ewens's sampling formula, introduced by Warren Ewe ...

Population genetics Population genetics is a subfield of genetics that deals with genetic differences within and between populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as adaptation, speciation, and pop ...

Historical

History of probability Probability has a dual aspect: on the one hand the likelihood of hypotheses given the evidence for them, and on the other hand the behavior of stochastic processes such as the throwing of dice or coins. The study of the former is historically olde ...

* ''

The Doctrine of Chances ''The Doctrine of Chances'' was the first textbook on probability theory, written by 18th-century French mathematician Abraham de Moivre and first published in 1718.. De Moivre wrote in English because he resided in England at the time, havi ...

'' {{DEFAULTSORT:Probability * Mathematics-related lists Statistics-related lists Lists of topics