almost

/ˈɔːɫ.məʊst/

From Middle English almost, from Old English eallmǣst (“nearly all, almost, for the most part”), equivalent to al- (“all”) + most.

inh

almost only counts in horseshoes and hand grenades

almost everywhere

almost nowhere

almost certain

almost sure

(mathematics) Up to, except for a negligible set (where negligible is not universally but contextually defined).

(measure theory, probability theory) Up to a null set; except for a set of measure 0.


outcome

Three is a possible outcome of tossing a six-sided die.

(probability theory) The result of a random trial. An element of a sample space.


homogeneous

From Medieval Latin homogeneus, from Ancient Greek ὁμογενής (homogenḗs, “of the same race, family or kind”), from ὁμός (homós, “same”) + γένος (génos, “kind”). Compare homo- (“same”) and -ous (adjectival suffix).

der

(mathematics) In any of several technical senses uniform; scalable; having its behavior or form determined by, or the same as, its behavior on or form at a smaller component (of its domain of definition, of itself, etc.).

(probability theory, Fourier analysis, of a distribution S on Euclidean n-space (or on ℝⁿmathbf 0)) Informally: Determined by its restriction to the unit sphere. Formally: Such that, for all real t>0 and test functions ϕ( mathbf x), the equality S[t⁻ⁿϕ( mathbf x/t)]=t^(mS)[ϕ( mathbf x)] holds for some fixed real or complex m.


event

From Middle French event, from Latin ēventus (“an event, occurrence”), from ēveniō (“to happen, to fall out, to come out”), from ē (“out of, from”), short form of ex + veniō (“come”); related to venture, advent, convent, invent, convene, evene, etc.

If X is a random variable representing the toss of a six-sided die, then its sample space could be denoted as {1,2,3,4,5,6}. Examples of events could be: X=1, X=2, X>5,X̸=4, and X isin 1,3,5.

(probability theory) A set of some of the possible outcomes; a subset of the sample space.

ঘটনা

probability theory: a set of some of the possible outcomes


normal

en-adj

normal (comparative normaler or more normal, superlative normalest or most normal)

normaler (comparative)
more normal (comparative)
normalest (superlative)
most normal (superlative)

From Latin normālis (“made according to a carpenter's square; later: according to a rule”), from nōrma (“carpenter's square”), of uncertain origin; doublet of normale. The earliest meaning of the word in English was "perpendicular; forming a right angle" like something normālis (“made according to a carpenter's square”), but by Late Latin normālis had also come to mean "according to a rule", from which modern English senses of the word derive: in the 1800s, as people began to quantitatively study things like height, weight and blood pressure, the usual or most common values came to be called "normal", and by extension values regarded as healthy or desirable came to be called "normal" regardless of their usuality.

der

According to norms or rules or to a regular pattern.

(mathematics) Adhering to or being what is considered natural or regular in a particular field or context:

(probability theory, statistics, of a distribution) Which has a very specific bell curve shape; that is or has the qualities of a normal distribution.

(probability theory, statistics, of a random variable, etc.) Which has a normal distribution; which is associated with a random variable that has a normal distribution.


Markov chain

The probability density of the Bayseian posterior was estimated by Metropolis-coupled Markov chain Monte Carlo, with multiple incrementally heated chains.

2004 July 27, F. Keith Barker et al., “Phylogeny and diversification of the largest avian radiation”, in PNAS, page 11040, column 2:

(probability theory) A discrete-time stochastic process containing a Markov property.


filtration

en-noun

filtration (countable and uncountable, plural filtrations)

(mathematics, set theory, probability theory) A totally ordered collection of subsets.


bell curve

→ Turkish: çan eğrisi (calque)

Holonym: graph

All this is enhanced by well-chosen illustrations: the bell curve of Galton and similar frequency-curves,[…]

(probability theory) The bell-shaped curve that is typical of the normal distribution.

гаусова крива

bell shaped curve typical of the normal distribution

Gaußkurve

strictly the PDF of a normal distribution


PDF

PDF (countable and uncountable, plural PDFs)

Translingual: .pdf

probability density function
probability density function

(probability theory) Alternative letter-case form of pdf (“probability density function”)


hypergeometric distribution

hypergeometric distribution (plural hypergeometric distributions)

The hypergeometric distribution differs from the binomial distribution in that the random sample of n items is selected from a finite population of N items. With the hypergeometric distribution, there is no replacement. If N is large in respect to n (N>10n), the binomial distribution is a good approximation to the hypergeometric distribution.

2002, Bryan Dodson, Dennis Nolan, Reliability Engineering Handbook, QA Publishing, page 57:

2013, Alexandr A. Borovkov, unspecified translator, Probability Theory, [2009, Alexandr A. Borovkov, Teoriya Veroyatnostei, Knizhnyi dom Librokom, 5th Edition], Springer, page 1,
Sampling without replacement from a large population is considered, and convergence of the emerging hypergeometric distributions to the binomial one is formally proved.

Knowing that the asymptotic form approaches that of a hypergeometric distribution is not all that helpful since this is a complicated function.[…]To demonstrate analytically what the figure suggests graphically, it is better to work directly with the probability function (2.1.7) than with a moment generating function or a characteristic function, neither of which is easily calculable or useful in the case of a hypergeometric distribution.

2014, Mark P. Silverman, A Certain Uncertainty: Nature's Random Ways, Cambridge University Press, page 99:

probability distribution
probability distribution

(probability theory, statistics) A discrete probability distribution that describes the probability of k "successes" in a sequence of n draws without replacement from a finite population.


mutually exclusive

(logic, probability theory) Describing multiple events or propositions such that the occurrence of any one implies the nonoccurrence of all other events or propositions. Boolean function of NOR logic gate

узаемавыключа́льны

describing multiple events being such that the occurrence of any one implies the nonoccurrence of all others


stochastic matrix

stochastic matrix (plural stochastic matrices)

(linear algebra, probability theory, statistics) A matrix having the property that the entries in each column are non-negative, real and sum to 1.


likelihood

likelihood (countable and uncountable, plural likelihoods)

likelihoods (plural)
likelyhood (alternative)

From Middle English liklihode, equivalent to likely + -hood.

(statistics, probability theory) The probability that some fixed outcome was generated by a random distribution with a specific parameter.


stochastic process

stochastic process (plural stochastic processes)

A stochastic process #92;#123;#92;zeta#95;t#92;#125; is a family of random variables that depends on a parameter t, usually called time.[…]Two important properties stochastic processes may have are given in the following definition.

1994, Andrzej Lasota, Michael C. Mackey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, 2nd edition, Springer, page 254:

Hence even Bernoulli processes, often regarded as the most random stochastic processes, are simulated by deterministic systems in science when observed with specific observation functions. But then any attempt to separate such deterministic systems from the deterministic systems needed to simulate the stochastic processes in science must fail. The philosophical implication of this result is that the above guess is wrong, viz. the deterministic systems which simulate the stochastic processes in science include deterministic systems in science. This indicates that from a predictive viewpoint, stochastic processes and deterministic systems are very similar.

2009, Charlotte Werndl, “Deterministic Versus Indeterministic Descriptios: Not That Different After All”, in Alexander Hieke, Hannes Leitgeb, editors, Reduction - Abstraction - Analysis, page 71:

A unified treatment of all the aspects of Fourier theory relevant to the Fourier analysis of stochastic processes is not only unavoidable, but also intellectually satisfying, and in fact time-saving.

2014, Pierre Brémaud, Fourier Analysis and Stochastic Processes, Springer, page xi:

(mathematics, probability theory) A function that maps elements of an index set to elements of a collection of random variables; historically, a collection of random variables indexed by a set of numbers usually regarded as points in time.

隨機過程 /随机过程

a function mapping to set of random variables


sample space

(probability theory) The set of all possible outcomes of a game, experiment or other situation.


probability density function

probability density function (plural probability density functions)

(probability theory) Any function whose integral over a set gives the probability that a random variable has a value in that set.


probabilistic

en-adj

probabilistic (comparative more probabilistic, superlative most probabilistic)

more probabilistic (comparative)
most probabilistic (superlative)

LL-Q1860 (eng)-Naomi Persephone Amethyst (NaomiAmethyst)-probabilistic.wav

From probabil(ity) + -istic and probabil(ism) + -istic.

Richard Hanania—one of those weird right-wing randos who appear on the scene out of nowhere every few months, suddenly raised to Twitter prominence for some occult algorithmic/dark-money reason that we will never get a good answer about—wrote a long newsletter in which he endorses opposing The Current Thing as a “heuristic”: “I would argue that a probabilistic approach suggests that we should be anti-current thing.”

2022 May 9, Max Read, “What Is “The Current Thing”?”, in Slate:

Johnson explained that absolute gauging assumed that everything that could go wrong would go wrong at the same time, resulting in the train hitting the platform. In contrast, probabilistic gauging says that everything can go wrong, but not at the same time. For the Pitsea '357', probabilistic gauging estimated a 0.00001% chance of the train hitting the platform. That seems much closer to the daily life of Pitsea Platform 2.

2023 April 19, Philip Haigh, “Gauging ways to ensure rail freight gains more paths”, in RAIL, number 981, page 45:

(mathematics, probability theory) Of, pertaining to, or derived using probability.


birthday problem

(probability theory) The mathematics problem that concerns the probability there exist two people in a random group who share a birthday.


binomial distribution

en-noun

binomial distribution (plural binomial distributions)

The first two have long histories in statistical theory, the binomial distribution having been discovered by James Bernoulli about 1700 and the Poisson distribution, which is a limiting form of the binomial distribution, by S. D. Poisson in 1837.

1979, M. G. Bulmer, Principles of Statistics, Dover, page 81:

The number of hits by a player in a given number of at-bats can be represented by a binomial distribution where the probability of a hit is the player's "true" batting average.

2003, Jim Albert, Teaching Statistics Using Baseball, Mathematical Association of America, page 263:

In any application of the binomial distribution, we must clearly specify which outcome is the success and which is the failure. The binomial distribution model allows us to compute the probability of observing a specified number of successes when the process is repeated a specific number of times (e.g., in a set of patients) and the outcome for a given patient is either a success or a failure.

2008, Lisa Marie Sullivan, Essentials of Biostatistics, Jones and Bartlett Publishers, page 67:

(probability theory, statistics) The discrete probability distribution of the number of successes in a sequence of n independent trials, each of which yields success with probability p.


state space

(computer science, probability theory) A (mathematical) space of all possible states of a system or process.


expected value

expected value (countable and uncountable, plural expected values)

(probability theory) The weighted average of outcome values, using probability as the weighting function.


probability density

probability density (plural probability densities)

(probability theory) probability density function


Markov process

Markov process (plural Markov processes)

Named after Russian mathematician Andrey Markov (Андре́й Андре́евич Ма́рков) (1856—1922), who researched the processes.

It has been remarked in 1 that a Markov process with time reversal is again a Markov process.

2007, N. G. Van Kampen, Stochastic Processes in Physics and Chemistry, 3rd edition, page 81:

The second part is written for probabilists and it studies the case where fast motions are certain Markov processes such as random evolutions and, in particular, diffusions.

2009, Yuri Kifer, “Large Deviations and Adiabatic Transitions for Dynamical Systems and Markov Processes in Fully Coupled Averaging”, in Memoirs of the American Mathematical Society, number 944, page vii:

2013, M. G. Shur, Markov Process, entry in Michiel Hazewinkel (editor), Encyclopaedia of Mathematics, Volume 6, page 102,
In the theory of Markov processes most attention is given to homogeneous (in time) processes.

conditional probability distribution
conditional probability distribution

(probability theory) Any stochastic process for which the conditional probability distribution of future states depends only on the current state (and not on past states).


binary distribution

binary distribution (plural binary distributions)

Bernoulli distribution
Bernoulli distribution

(statistics, probability theory) Bernoulli distribution.


FS

first success distribution
first success distribution#English

(probability theory) Initialism of first success distribution.


memoryless

(probability theory) Of a probability distribution, such that any derived probability from a set of random samples is distinct and has no information (i.e. "memory") of earlier samples.


memorylessness

(probability theory) The property of being memoryless.


random sequence

random sequence (plural random sequences)

(statistics, probability theory) A sequence of identically distributed random variables.


iid

(statistics, probability theory) Initialism of independent and identically distributed.

НЕР

initialism of independent and identically distributed.


multichain

multichain Markov chain

(probability theory) Of a Markov chain, consisting of two or more closed sets of recurrent states plus (optionally) a set of states that do not belong to any of the closed sets.


multichain

A multichain with transient states is called a reducible multichain.

2016, Theodore J. Sheskin, Markov Chains and Decision Processes for Engineers and Managers, page 24:

(probability theory) A multichain Markov chain.


Poisson process

Poisson process (plural Poisson processes)

From the related concept of Poisson distribution (specifying a bounded subset of the domain of a Poisson process defines a random variable that has a Poisson distribution). See also stochastic process.

These properties are readily apparent when one considers that the Poisson process is derived from the binomial processes, which, as seen in Section 6.1, can be viewed in terms of coin tosses.

1995, Randolph Nelson, Probability, Stochastic Processes, and Queueing Theory: The Mathematics of Computer Performance Modeling, Springer, page 252:

2012, Peter Guttorp, Thordis L. Thorarinsdottir, Chapter 4: Bayesian Inference for Non-Markovian Point Processes, Emilio Porcu, José–María Montero, Martin Schlather (editors), Advances and Challenges in Space-time Modelling of Natural Events, Springer, Lecture Notes in Statistics 207, page 88,
The doubly stochastic Poisson process, introduced by Cox in [9] and so named by Bartlett in [6] is obtained by letting the rate λ(t) of the Poisson process vary according to a stochastic process, say Λ(t). […] There are instances of doubly stochastic Poisson processes that are identical to cluster processes (for example, the shot noise process driven by a stationary Poisson process is identical to a Neyman-Scott Poisson cluster process, see p.171-172 in [12]).

University buses arrive at the Students Center to take students to their classes according to a Poisson process with an average rate of 5 buses per hour.

2014, Oliver Ibe, Fundamentals of Applied Probability and Random Processes, Elsevier (Academic Press), page 415:

(mathematics, statistics, probability theory) A stochastic process in which events occur continually and independently of one another.

Poisson point field

stochastic process of continuous, independent events


detailed balance

detailed balance (plural detailed balances)

where P is the Markov transition matrix and πᵢ and πⱼ are the equilibrium probabilities of being in states i and j, respectively.

(mathematics, probability theory) a Markov process is said to show detailed balance if the transition rates between each pair of states i and j in the state space obey


characteristic function

characteristic function (plural characteristic functions)

(mathematics, probability theory) A complex function completely defining the probability distribution of a real-valued random variable

характеристична функция

a complex function completely defining a probability distribution


error function

normally distributed
normal distribution

(mathematics, statistics, probability theory) A sigmoid function giving the probability that a measurement, under the influence of normally distributed errors with standard deviation, is within a certain distance from the mean value.


quasireversibility

(mathematics, probability theory) A property of some queues, differing from reversibility in that a stronger condition is imposed on arrival rates and a weaker condition is applied on probability fluxes.


Bayes' theorem

Named after English mathematician Thomas Bayes (1701–1761), who developed an early formulation. The modern expression of the theorem is due to Pierre-Simon Laplace, who extended Bayes's work but was apparently unaware of it.

Bayes theorem (alternative)
Bayes's theorem (alternative)

The starting point for many techniques in probabilistic classification is Bayes' theorem, which provides a way of relating evidence to a hypothesis.

2010, Jonathan Harrington, Phonetic Analysis of Speech Corpora, page 327:

Bayes's theorem is a relationship between the conditional probabilities of two events.

2011, Allen Downey, Think Stats, O'Reilly, page 56:

We have now seen how Bayes' theorem enables us to correctly update a prior probability for some unknown event when we see evidence about the event.

2013, Norman Fenton, Martin Neil, Risk Assessment and Decision Analysis with Bayesian Networks, Taylor & Francis (CRC Press), page 131:

conditional probability
conditional probability

(probability theory) A theorem expressed as an equation that describes the conditional probability of an event or state given prior knowledge of another event.

Bayes' theorem
Pierre-Simon Laplace
Thomas Bayes


almost surely

(probability theory) With probability equal to one.


density function

density function (plural density functions)

probability density function
probability density function#English

(probability theory) Clipping of probability density function.


comonotonic

(probability theory) Of a random vector, having components that can be represented as increasing functions of a single random variable.


gamma distribution

gamma distribution (plural gamma distributions)

The gamma distribution is positively skewed and along with the Weibull, lognormal, and inverse Gaussian (IG) distributions is also available as a model in reliability and life-span studies.

1988, A. Clifford Cohen, Betty Jones Whitten, Parameter Estimation in Reliability and Life Span Models, Marcel Dekker, page 93:

1999 [1977, Wiley], Jean Dickinson Gibbons, Ingram Olkin, Milton Sobel, Selecting and Ordering Populations: A New Statistical Methodology, SIAM, Republished unabridged and corrected, page 328,
The ranking and selection problems in this chapter apply to populations that follow the gamma distribution. The gamma distribution model is used widely in many fields of applications, particularly in engineering and the social and physical sciences.

The negative exponential distribution is a special case of the gamma distribution in which k#61;1. Another important special case of a gamma distribution with parameters k and #92;theta#61;2 is the χ² distribution G(k,2) with n#61;2k degrees of freedom.[…]Gamma distributions have an additivity property: the sum of two random variables following gamma distributions G(k#95;1,#92;theta) and G(k#95;2,#92;theta) is a random variable following a gamma distribution G(k#95;1#43;k#95;2,#92;theta) with parameters k#95;1#43;k#95;2 and #92;theta[…].

2005, M. T. Todinov, Reliability and Risk Models: Setting Reliability Requirements, Wiley, page 36:

exponential distribution
exponential distribution

chi-square distribution
chi-square distribution

(probability theory, statistics) Any of a family of two-parameter continuous probability distributions, of which the common exponential distribution and chi-square distribution are special cases.


Markov property

Markov property (plural Markov properties)

Introduced by Andrey Markov (Russian: Андре́й Ма́рков; 1856–1922), Russian mathematician.

(probability theory) The memoryless property of Markov models, according to which the future states depend only on the current state and not those that preceded it.

марковски свойство

the property of future states depending only on the current state


secretary problem

secretary problem (plural secretary problems)

(probability theory) The problem of selecting the best out of n applicants for a position, under the condition that all applicants are interviewed in random order, the quality of the applicants is not known before interviewing, and once rejected an applicant cannot be recalled.


Rademacher distribution

(probability theory, statistics) A discrete probability distribution where a random variate has a 50% chance of being +1 and a 50% chance of being -1.


conditional probability distribution

conditional probability distribution (plural conditional probability distributions)

conditional probability distributions (plural)

1999, Anders L. Madsen, Bruce D'Ambrosio, Lazy Propagation and Independence of Causal Influence, Anthony Hunter, Simon Parsons, Symbolic and Quantitative Approaches to Reasoning and Uncertainty, Springer, page 293,
A Bayesian network consists of a directed, acyclic graph and a set of conditional probability distributions. For each variable in the graph there is a conditional probability distribution of the variable given its parents.

2010, Philipp Koehn, Statistical Machine Translation, Cambridge University Press, page 69,
Note that for two independent random variables X and Y, the conditional probability distribution p(x|y) is the same as simply p(y).

2015, Iman Behmanesh, Babak Moaveni, Geert Lombaert, Costas Papadimitriou, Chapter 6: Hierarchical Bayesian Model Updating for Probabilistic Damage Identification, H. Sezer Atamturktur, Babak Moaveni, Costas Papadimitriou, Tyler Schoenherr (editors), Model Validation and Uncertainty Quantification, Volume 3: Proceedings of the 33rd IMAC, Springer, page 58,
Alternatively, Laplace asymptotic approximation can be used to approximate the conditional probability distributions of θ₁ in Eq. (6.12) as a Gaussian distribution to simplify the sampling process.

(probability theory) The probability distribution of a random variable given the known outcome of another random variable.

условно разпределение

probability distribution given a random variable


normal random variable

normal random variable (plural normal random variables)

Since this density function cannot be integrated directly, in order to find probabilities involving normal random variables, we must resort to tables giving us various percentile points of the distribution.

1986, Mark C. K. Yang, David H. Robinson, Understanding and Learning Statistics by Computer, World Scientific, page 28:

Although normal random variables are continuous, they are often used to model discrete quantities.[…]The probability density function of a normal random variable depends on two parameters, denoted μ and σ, which are, respectively, the mean and standard deviation of the random variable.

2016, Geoffrey C. Berresford, Andrew M. Rockett, Applied Calculus, 7th edition, Cengage Learning, page 731:

A special type of continuous random variable, known as a normal random variable, is studied.[…]We present the additive property of normal random variables. The percentiles of normal random variables are studied.

2017, Sheldon M. Ross, Introductory Statistics, 4th edition, Elsevier (Academic Press), page 263:

(probability theory, statistics) A random variable whose probability distribution is a normal distribution.


standard normal random variable

standard normal random variable (plural standard normal random variables)

standard normal random variables (plural)

Suppose X#92;simN(#92;mu,#92;sigma). Then
Z#61;#92;frac#123;x-#92;mu#125;#123;#92;sigma#125;
is a standard normal random variable. In other words, any normal random variable can be standardized by subtracting the mean from the variable and dividing the difference by the standard deviation.

2000, George E. Monahan, Management Decision Making: Spreadsheet modeling, analysis, and applications, volume 1, Cambridge University Press, page 405:

This technique is based on a polynomial transformation that proceeds by taking the sum of a linear combination of a standard normal random variable, its square, and its cube.

2010, Todd C. Headrick, Statistical Simulation: Power Method Polynomials and Other Transformations, Taylor & Francis (CRC Press), page 1:

2016, Maria Dolores Ugarte, Ana F. Militino, Alan T. Arnholt, Probability and Statistics with R, 2nd Edition, Taylor & Francis (CRC Press), page 297,
A normal random variable with μ = 0 and σ = 1, often denoted Z, is called a standard normal random variable. […] Further, any normal random variable can be converted to a standard normal random variable using (4.25).

standard normal distribution
standard normal distribution

(probability theory, statistics) A random variable whose probability distribution is a standard normal distribution.


hypergeometric random variable

hypergeometric random variable (plural hypergeometric random variables)

hypergeometric random variables (plural)

Computer generation of classical hypergeometric random variables has been discussed in detail by Kachitvichyanukul and Schmeiser (1985).

1992, Norman Lloyd Johnson, Samuel Kotz, Adrienne W. Kemp, Univariate Discrete Distributions, page 67:

Chapter 6 introduces testing of hypotheses immediately after the study of binomial and hypergeometric random variables.

2005, Martin Buntinas, Gerald Marlowe Funk, Statistics for the Sciences, page xv:

If #92;textstyleX is a hypergeometric random variable, then the variance is #92;textstyleVar(X)#61;#92;left(#92;frac#123;M#125;#123;N#125;#92;right)n#92;left(1-#92;frac#123;M#125;#123;N#125;#92;right)#92;left(1-#92;frac#123;n-1#125;#123;N-1#125;#92;right).

2007, Purna Chandra Biswal, Probability and Statistics, India: Prentice-Hall, published 2008, page 87:

Thus, whereas the expected value of the hypergeometric random variable with parameters n, N, p is the same as that of the binomial random variable with parameters n, p, its variance is smaller than that of the binomial by the factor (N − n)/(N − 1).

2017, Sheldon M. Ross, Introductory Statistics, Elsevier (Academic Press), page 250:

hypergeometric distribution
hypergeometric distribution

(probability theory, statistics) A random variable whose probability distribution is a hypergeometric distribution.

хипергеометрична случайна величина

random variable with hypergeometric distribution


Kelly criterion

Described by J. L. Kelly, Jr, a researcher at Bell Labs, in 1956.

(probability theory) A formula used to determine the optimal size of a series of bets in order to maximize the logarithm of wealth.


branching process

branching process (plural branching processes)

(mathematics, probability theory) A stochastic process which typically models reproduction of objects, with the assumption that the reproductions of individual particles are mutually independent.


birthday paradox

LL-Q1860 (eng)-Flame, not lame-birthday paradox.wav

(probability theory) The counterintuitive observation that a random group of people needs to have only 23 members before there is a 50% chance of two of them having the same birthday.


Popoviciu's inequality

Popoviciu's inequality
Popoviciu's inequality on variances
Tiberiu Popoviciu

Named after Romanian mathematician Tiberiu Popoviciu (1906–1975).

named-after

(probability theory) An upper bound on the variance of any bounded probability distribution.


π-system

π-systems (plural)
pi-system (alternative)

To see this, first check that #92;sigma(X#95;0,X#95;1,#92;dots)#61;#92;sigma(#92;mathcalF#95;0), where #92;textstyle#92;mathcalF#95;0#58;#61;#92;bigcup#92;infty#95;#123;k#61;0#125;#92;sigma(X#95;0,#92;dots,X#95;k) is a field and, in particular, a #92;boldsymbol#92;pi-system.

2007, Rabi Bhattacharya, Edward C. Waymire, A Basic Course in Probability Theory, Springer, page 49:

We start with a basis of simple roots #92;Delta of #92;Phi. Then we apply all possible elementary transformations and add the resulting #92;boldsymbol#92;pi-systems to the list. Of course, if #92;Gamma is a #92;boldsymbol#92;pi-system, and #92;Gamma' is a #92;boldsymbol#92;pi-system obtained from it by an elementary transformation and the diagrams of #92;Gamma and #92;Gamma' are the same, the root subsystems they span are the same, and therefore we do not add #92;Gamma'.

2017, Willem Adriaan de Graaf, Computation with Linear Algebraic Groups, Taylor & Francis (CRC Press), page 221:

Clearly the definitions for a #92;boldsymbol#92;pi-system and a #92;lambda-system are both satisfied by a #92;sigma-algebra.[…]
Proposition 4.1.8 Let #92;Omega be a set and B be a collection of subsets of #92;Omega. The collection B is a #92;sigma-algebra if and only if B is a #92;lambda-system and a #92;boldsymbol#92;pi-system.

2021, Jeremy J. Becnel, Tools for Infinite Dimensional Analysis, Taylor & Francis (CRC Press):

(set theory, measure theory, probability theory) A non-empty collection of subsets of a given set Ώ that is closed under non-empty finite intersections.

π-система

collection of subsets closed under non-empty finite intersections


multi-armed bandit

en-noun

multi-armed bandit (plural multi-armed bandits)

From one-armed bandit, by analogy with a gambler at a row of slot machines who has to decide how best to play them.

(probability theory, machine learning) An algorithm that allocates a fixed limited set of resources between competing alternative choices so as to maximize the expected gain, when each choice's properties are only partially known at the time of allocation, and may become better understood as time passes or allocations are made.


impossible event

impossible event (plural impossible events)

(probability theory) An event that contains no possible outcomes and thus cannot possibly occur; an empty set.


certain event

Measure 0 is assigned to an impossible event, and measure 1 is assigned to a certain (sure) event. A certain event is denoted as #92;Omega; an impossible event, as #92;empty.

1995, Michael Pecht, Product Reliability, Maintainability, and Supportability Handbook, CRC Press, page 17:

Hence the event A#95;1#92;capA#95;2#92;cap#92;dots#92;capA#95;k will be a certain event too, i.e. P(A#95;1#92;capA#95;2#92;cap#92;dots#92;capA#95;k)#61;1.

2002, Andrew G. Bronevich, Alexander N. Karkishchenko, “The structure of fuzzy measure families induced by upper and lower probabilities”, in Carlo Bertoluzza, María Á. Gil, Dan A. Ralescu, editors, Statistical Modeling, Analysis and Management of Fuzzy Data, Springer, page 165:

In probability theory, events are sets with a non-negative length that is at most one, since we assume that the sample space #92;Omega is the certain event and should be assigned the largest length, i.e., 1 (probability 100%).

2018, Athanasios Christou Micheas, Theory of Stochastic Objects, Taylor & Francis (CRC Press / Chapman & Hall), unnumbered page:

(probability theory) An event that contains all of the possible outcomes and thus is known a priori to be certain to occur;

An event that contains all of the possible outcomes and thus is known a priori to be certain to occur;
(more formally) the entire sample space (which by definition has measure 1 and probability 1); said sample space with the exclusion of, at most, a set of measure 0 (thus retaining measure 1).


sample point

(probability theory) A point in the sample space representing some concrete non-decomposable result of an individual experiment.


Little's law

The theorem was proposed by John Little.

(probability theory) A theorem which states that the long-term average number L of customers in a stationary system is equal to the long-term average effective arrival rate λ multiplied by the average time W that a customer spends in the system.


Gumbel distribution

Gumbel distribution (plural Gumbel distributions)

Named after Emil Julius Gumbel (1891–1966), who described the distribution.

(probability theory, statistics) A model of the distribution of the maximum (or the minimum) of a number of samples of various distributions.


Chinese restaurant process

Chinese restaurant process (plural Chinese restaurant processes)

The restaurant analogy first appeared in a 1985 write-up by David Aldous, where it was attributed to Jim Pitman (who additionally credits Lester Dubins).

(probability theory) A discrete-time stochastic process, analogous to seating customers at tables in a restaurant, such that, at time n, the n customers have been partitioned among m ≤ n tables (or blocks of a partition). It has applications in population genetics, linguistic analysis, and image recognition.


Lindley equation

Introduced by English statistician Dennis Lindley.

(probability theory) A discrete-time stochastic process Aₙ where n takes integer values and A_(n + 1) = max(0, Aₙ + Bₙ). It is useful in queueing theory.


sojourn time

sojourn time (countable and uncountable, plural sojourn times)

(probability theory, statistics) The time spent in a state of a stochastic process.


Buffon's needle problem

(mathematics, probability theory) An experiment that determines the probability that a needle, when dropped, will lie across a line between two parallel strips of wood, each of the same width; often used to estimate π.


Markov renewal process

Markov renewal process (countable and uncountable, plural Markov renewal processes)

(mathematics, probability theory, statistics) A class of random processes that generalize the class of Markov jump processes.


discrete-time Markov chain

discrete-time Markov chain (plural discrete-time Markov chains)

(mathematics, probability theory) A sequence of random variables, known as a stochastic process, in which the value of the next variable depends only on the value of the current variable, and not any variables in the past.


mean hitting time

mean hitting time (countable and uncountable, plural mean hitting times)

(mathematics, probability theory) The average number of steps that a Markov chain takes to go from a specified starting state to a specified target state or set of states.


semi-Markov process

semi-Markov process (plural semi-Markov processes)

(mathematics, probability theory) A generalization of the Markov chain in which the distribution of the amount of time that the process spends in a given state is permitted to be nonexponentially (generally) distributed.


unichain

unichain Markov chain

(probability theory) Of a Markov chain, consisting of a single recurrent class plus a possibly empty set of transient states.


iid-ness

(statistics, probability theory) The state of being iid (independent and identically distributed).


tree diagram

(probability theory) A visual representation used to represent a probability space.