4676:
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50:
721:
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3387:. This law is remarkable because it is not assumed in the foundations of probability theory, but instead emerges from these foundations as a theorem. Since it links theoretically derived probabilities to their actual frequency of occurrence in the real world, the law of large numbers is considered as a pillar in the history of statistical theory and has had widespread influence.
601:, the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events (events that contain no common results, e.g., the events {1,6}, {3}, and {2,4} are all mutually exclusive), the probability that any of these events occurs is given by the sum of the probabilities of the events.
1273:
604:
The probability that any one of the events {1,6}, {3}, or {2,4} will occur is 5/6. This is the same as saying that the probability of event {1,2,3,4,6} is 5/6. This event encompasses the possibility of any number except five being rolled. The mutually exclusive event {5} has a probability of 1/6, and
585:
of the sample space (or equivalently, the event space) is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results. One collection of possible results corresponds to getting an odd number. Thus, the subset {1,3,5} is
2030:
An example of such distributions could be a mix of discrete and continuous distributions—for example, a random variable that is 0 with probability 1/2, and takes a random value from a normal distribution with probability 1/2. It can still be studied to some extent by considering it to have a PDF of
2026:
The utility of the measure-theoretic treatment of probability is that it unifies the discrete and the continuous cases, and makes the difference a question of which measure is used. Furthermore, it covers distributions that are neither discrete nor continuous nor mixtures of the two.
420:
that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the
560:
Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The measure theory-based treatment of probability covers the discrete, continuous, a mix of the two, and more.
6329:
3348:
As the names indicate, weak convergence is weaker than strong convergence. In fact, strong convergence implies convergence in probability, and convergence in probability implies weak convergence. The reverse statements are not always true.
3084:
3740:
436:, probability theory is essential to many human activities that involve quantitative analysis of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in
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4172:
2570:
is used to define a density as the Radon-Nikodym derivative of the probability distribution of interest with respect to this dominating measure. Discrete densities are usually defined as this derivative with respect to a
3645:
3475:
3343:
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2583:. If a theorem can be proved in this general setting, it holds for both discrete and continuous distributions as well as others; separate proofs are not required for discrete and continuous distributions.
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every "event" a value between zero and one, with the requirement that the event made up of all possible results (in our example, the event {1,2,3,4,5,6}) be assigned a value of one. To qualify as a
1524:
1580:
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764:: Initially the probability of an event to occur was defined as the number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space: see
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Along with providing better understanding and unification of discrete and continuous probabilities, measure-theoretic treatment also allows us to work on probabilities outside
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Certain random variables occur very often in probability theory because they well describe many natural or physical processes. Their distributions, therefore, have gained
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590:. In this case, {1,3,5} is the event that the die falls on some odd number. If the results that actually occur fall in a given event, that event is said to have occurred.
2498:
2671:. They are listed below in the order of strength, i.e., any subsequent notion of convergence in the list implies convergence according to all of the preceding notions.
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by the CDF. This measure coincides with the pmf for discrete variables and PDF for continuous variables, making the measure-theoretic approach free of fallacies.
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is observed repeatedly during independent experiments, the ratio of the observed frequency of that event to the total number of repetitions converges towards
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620:. This function is usually denoted by a capital letter. In the case of a die, the assignment of a number to certain elementary events can be done using the
6312:
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has no positive probability for any single point, neither does it have a density. The modern approach to probability theory solves these problems using
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for modern probability theory; but, alternatives exist, such as the adoption of finite rather than countable additivity by
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of many independent and identically distributed random variables with finite variance tends towards a normal distribution
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For some classes of random variables, the classic central limit theorem works rather fast, as illustrated in the
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3375:. Furthermore, the more often the coin is tossed, the more likely it should be that the ratio of the number of
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For example, if the event is "occurrence of an even number when a dice is rolled", the probability is given by
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will approach unity. Modern probability theory provides a formal version of this intuitive idea, known as the
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373:, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of
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in nature, and this theorem, according to David
Williams, "is one of the great results of mathematics."
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815:, since 3 faces out of the 6 have even numbers and each face has the same probability of appearing.
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3079:{\displaystyle \lim _{n\rightarrow \infty }P\left(\left|X_{n}-X\right|\geq \varepsilon \right)=0}
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So, the probability of the entire sample space is 1, and the probability of the null event is 0.
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616:. A random variable is a function that assigns to each elementary event in the sample space a
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When doing calculations using the outcomes of an experiment, it is necessary that all those
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an element of the power set of the sample space of dice rolls. These collections are called
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4525:""The origins and legacy of Kolmogorov's Grundbegriffe", by Glenn Shafer and Vladimir Vovk"
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3735:{\displaystyle \displaystyle {\overline {X}}_{n}\,{\xrightarrow {\mathrm {a.\,s.} }}\,\mu }
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The first major treatise blending calculus with probability theory, originally in French:
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exists for all random variables (including discrete random variables) that take values in
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the two possible outcomes are "heads" and "tails". In this example, the random variable
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4188:. For example, the distributions with finite first, second, and third moment from the
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1293:: The classical definition breaks down when confronted with the continuous case. See
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533:
494:
published a book on the subject in 1657. In the 19th century, what is considered the
487:
449:
378:
294:
200:
99:
4637:
A modern introduction to probability and statistics : understanding why and how
4635:
4200:
variety, it works very slowly or may not work at all: in such cases one may use the
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119:
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4329: – When the occurrence of one event does not affect the likelihood of another
448:
was the probabilistic nature of physical phenomena at atomic scales, described in
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5926:
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5682:
5607:
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4167:{\displaystyle Z_{n}={\frac {\sum _{i=1}^{n}(X_{i}-\mu )}{\sigma {\sqrt {n}}}}\,}
2668:
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613:
401:
195:
146:
4610:"Leithner & Co Pty Ltd - Value Investing, Risk and Risk Management - Part I"
5843:
5442:
5369:
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4268: – List of definitions of terms and concepts in statistics and probability
4238:
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2130:
1605:
is said to have a continuous probability distribution if the corresponding CDF
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471:
210:
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for any CDF, and vice versa. The measure corresponding to a CDF is said to be
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Pairwise independence§Probability of the union of pairwise independent events
3972:
of the distribution followed by the original random variables. Formally, let
3940:
2310:
2166:
822:
747:
625:
605:
the event {1,2,3,4,5,6} has a probability of 1, that is, absolute certainty.
573:
that can produce a number of outcomes. The set of all outcomes is called the
510:
483:
83:
6180:
6059:
5838:
5749:
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5172:
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4495:"Probabilistic Expectation and Rationality in Classical Probability Theory"
3640:{\displaystyle \displaystyle {\overline {X}}_{n}\,{\xrightarrow {P}}\,\mu }
1308:
1250:
mapping a point in the sample space to the "probability" value is called a
826:
575:
545:
529:
390:
310:
220:
104:
4929:
3957:
The central limit theorem (CLT) explains the ubiquitous occurrence of the
2313:
on the set of real numbers, then there is a unique probability measure on
1272:
6291:
6207:
5374:
5038:
4961:
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Common intuition suggests that if a fair coin is tossed many times, then
2579:
distributions are usually defined as this derivative with respect to the
751:
617:
467:
413:
366:
362:
230:
71:
59:
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4432:
3470:{\displaystyle {\overline {X}}_{n}={\frac {1}{n}}{\sum _{k=1}^{n}X_{k}}}
3338:{\displaystyle \displaystyle X_{n}\,{\xrightarrow {\mathrm {a.s.} }}\,X}
6227:
5359:
5238:
5033:
4841:
4247: – Statistical measure of how far values spread from their average
4193:
433:
400:
Central subjects in probability theory include discrete and continuous
88:
34:
4291: – use of probability and logic to deal with uncertain situations
6275:
The
Unreasonable Effectiveness of Mathematics in the Natural Sciences
5590:
2917:{\displaystyle \displaystyle X_{n}\,{\xrightarrow {\mathcal {D}}}\,X}
1986:
736:
581:
524:
This culminated in modern probability theory, on foundations laid by
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3312:
3115:
2900:
2667:
In probability theory, there are several notions of convergence for
720:
5263:
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An
English translation by Nathan Morrison appeared under the title
4244:
4197:
4043:
3481:
417:
215:
4285: – Form of modelling that uses statistics to predict outcomes
544:
for probability theory in 1933. This became the mostly undisputed
5048:
4317: – Philosophical interpretation of the axioms of probability
3965:
445:
4634:
Dekking, Michel (2005). "Chapter 13: The law of large numbers".
4352: – Set of random variables of which any two are independent
6330:
European
Community on Computational Methods in Applied Sciences
5894:
5712:
4478:
Grinstead, Charles Miller; James Laurie Snell. "Introduction".
1078:
4856:
A lively introduction to probability theory for the beginner.
4640:. Library Genesis. London : Springer. pp. 180–194.
4274: – Function related to statistics and probability theory
2566:
When it is convenient to work with a dominating measure, the
2462:{\displaystyle P(E)=\int _{\omega \in E}\mu _{F}(d\omega )\,}
377:. Typically these axioms formalise probability in terms of a
6325:
International
Council for Industrial and Applied Mathematics
4830:
2nd ed. Springer Series in
Statistics. (2002). 650 pp.
2125:
Other distributions may not even be a mix, for example, the
4804:(Chelsea, New York) in 1950, with a second edition in 1956.
4433:"A Brief Look at the History of Probability and Statistics"
4020:
1287:
deals with events that occur in a continuous sample space.
755:
743:
3130:{\displaystyle \displaystyle X_{n}\,{\xrightarrow {P}}\,X}
973:{\displaystyle f(x)\in {\mbox{ for all }}x\in \Omega \,;}
3774:
It follows from the LLN that if an event of probability
3486:
independent and identically distributed random variables
3272:{\displaystyle P(\lim _{n\rightarrow \infty }X_{n}=X)=1}
1434:
The CDF necessarily satisfies the following properties.
393:. Any specified subset of the sample space is called an
4888:
4739:
4582:
Introduction to
Probability and Mathematical Statistics
4553:(8th ed.). Pearson Prentice Hall. pp. 26–27.
4477:
4340:
Pages displaying short descriptions of redirect targets
4331:
Pages displaying short descriptions of redirect targets
4234:
Pages displaying short descriptions of redirect targets
2021:
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is attached, which satisfies the following properties:
5573:
4192:; on the other hand, for some random variables of the
2586:
2575:
over the set of all possible outcomes. Densities for
1519:{\displaystyle \lim _{x\rightarrow -\infty }F(x)=0\,;}
951:
794:
779:
638:
5944:
Numerical methods for ordinary differential equations
4663:, "Probability with martingales", Cambridge 1991/2008
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where the integration is with respect to the measure
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1575:{\displaystyle \lim _{x\rightarrow \infty }F(x)=1\,.}
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have a number assigned to them. This is done using a
6320:
Société de Mathématiques
Appliquées et Industrielles
6313:
Japan
Society for Industrial and Applied Mathematics
5949:
Numerical methods for partial differential equations
5663:
4293:
Pages displaying wikidata descriptions as a fallback
4261:
Pages displaying wikidata descriptions as a fallback
4211:
632:
could assign to the outcome "heads" the number "0" (
6136:
4811:
1939:{\displaystyle P(X\in E)=\int _{x\in E}f(x)\,dx\,.}
4867:. Springer -Verlag, New York (2005). 510 pp.
4865:Probabilistic Symmetries and Invariance Principles
4298:Probabilistic proofs of non-probabilistic theorems
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2563:, probability is defined on a space of functions.
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2209:
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1333:) or a subset thereof, then a function called the
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4819:. New York, Toronto, London: John Wiley and Sons.
1953:exists only for continuous random variables, the
6370:
3229:
3191:is said to converge towards the random variable
3010:
2978:is said to converge towards the random variable
1856:{\displaystyle P(X\in E)=\int _{x\in E}dF(x)\,.}
1535:
1476:
505:Initially, probability theory mainly considered
1866:In case the PDF exists, this can be written as
1047:) lies between zero and one for every value of
808:{\displaystyle {\tfrac {3}{6}}={\tfrac {1}{2}}}
624:. This does not always work. For example, when
6308:Society for Industrial and Applied Mathematics
4766:
4579:
517:considerations compelled the incorporation of
5559:
4945:
4884:Probability: Theory and Examples, 5th edition
4772:Grundbegriffe der Wahrscheinlichkeitsrechnung
4612:. Leithner.com.au. 2000-09-15. Archived from
4338: – Physics of many interacting particles
1029:{\displaystyle \sum _{x\in \Omega }f(x)=1\,.}
669:) and to the outcome "tails" the number "1" (
486:in the seventeenth century (for example the "
335:
6126:Supersymmetric theory of stochastic dynamics
4253: – System for reasoning about vagueness
412:(which provide mathematical abstractions of
27:Branch of mathematics concerning probability
4959:
1725:{\displaystyle f(x)={\frac {dF(x)}{dx}}\,.}
1303:: If the sample space of a random variable
853:. It is then assumed that for each element
5566:
5552:
4952:
4938:
4840:
4241: – Average value of a random variable
1207:{\displaystyle P(E)=\sum _{x\in E}f(x)\,.}
385:taking values between 0 and 1, termed the
342:
328:
4720:Learn how and when to remove this message
4363:Catalog of articles in probability theory
4311: – Foundations of probability theory
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444:. A great discovery of twentieth-century
4802:Foundations of the Theory of Probability
4683:This article includes a list of general
4430:
3946:
2601:in probability theory. Some fundamental
1271:
719:
4877:
4633:
4418:The Stanford Encyclopedia of Philosophy
3352:
2082:{\displaystyle (\delta +\varphi (x))/2}
1755:{\displaystyle E\subseteq \mathbb {R} }
1280:, a continuous probability distribution
369:. Although there are several different
14:
6371:
4580:Bain, Lee J.; Engelhardt, Max (1992).
4492:
4414:"Quantum Logic and Probability Theory"
4266:Glossary of probability and statistics
4079:Then the sequence of random variables
3889:{\displaystyle {\textrm {E}}(Y_{i})=p}
3279:. Strong convergence is also known as
1985:These concepts can be generalized for
821:: The modern definition starts with a
728:, a discrete probability distribution.
5547:
4933:
4019:be independent random variables with
3394:(LLN) states that the sample average
3371:, and the other half it will turn up
2721:{\displaystyle X_{1},X_{2},\dots ,\,}
470:has its roots in attempts to analyze
5514:
4669:
4546:
4012:{\displaystyle X_{1},X_{2},\dots \,}
3184:{\displaystyle X_{1},X_{2},\dots \,}
2971:{\displaystyle X_{1},X_{2},\dots \,}
2793:{\displaystyle F_{1},F_{2},\dots \,}
2022:Measure-theoretic probability theory
2018:and other continuous sample spaces.
1262:Continuous probability distributions
528:. Kolmogorov combined the notion of
509:events, and its methods were mainly
5526:
4909:
4757:Théorie Analytique des Probabilités
4072:{\displaystyle \sigma ^{2}>0.\,}
3538:, provided that the expectation of
2587:Classical probability distributions
1268:Continuous probability distribution
880:, an intrinsic "probability" value
766:Classical definition of probability
699:{\displaystyle X({\text{tails}})=1}
496:classical definition of probability
416:or uncertain processes or measured
24:
5575:Industrial and applied mathematics
4828:Foundations of Modern Probability,
4689:it lacks sufficient corresponding
4373:List of publications in statistics
3755:
3717:
3710:
3660:
3320:
3314:
3239:
3020:
2902:
2867:. Weak convergence is also called
2379:
2322:
2293:
2259:
2226:
2177:
2147:
1545:
1489:
1108:
1039:That is, the probability function
999:
963:
866:
840:
829:, which relates to the set of all
710:Discrete probability distributions
466:The modern mathematical theory of
389:, to a set of outcomes called the
25:
6390:
5805:Stochastic differential equations
4202:Generalized Central Limit Theorem
3842:taking values 1 with probability
3831:{\displaystyle Y_{1},Y_{2},...\,}
3367:half of the time it will turn up
3144:The sequence of random variables
2931:The sequence of random variables
2753:if their respective CDF converges
1408:{\displaystyle F(x)=P(X\leq x)\,}
716:Discrete probability distribution
478:in the sixteenth century, and by
432:As a mathematical foundation for
6121:Supersymmetric quantum mechanics
5525:
5513:
5502:
5501:
5489:
4745:Analytical Theory of Probability
4674:
4214:
3578:It is in the different forms of
3288:Most common shorthand notation:
3091:Most common shorthand notation:
2876:Most common shorthand notation:
2548:{\displaystyle \mathbb {R} ^{n}}
2388:{\displaystyle {\mathcal {F}}\,}
2331:{\displaystyle {\mathcal {F}}\,}
2302:{\displaystyle {\mathcal {F}}\,}
2235:{\displaystyle {\mathcal {F}}\,}
2186:{\displaystyle {\mathcal {F}}\,}
2011:{\displaystyle \mathbb {R} ^{n}}
1336:cumulative distribution function
735:deals with events that occur in
662:{\textstyle X({\text{heads}})=0}
48:
6003:Stochastic variational calculus
5795:Ordinary differential equations
5410:Computational complexity theory
4654:
4627:
4177:converges in distribution to a
3580:convergence of random variables
3514:converges towards their common
2678:A sequence of random variables
2663:Convergence of random variables
2657:Convergence of random variables
2275:{\displaystyle P(\Omega )=1.\,}
1976:{\displaystyle \mathbb {R} \,.}
1423:) returns the probability that
833:in classical sense, denoted by
5800:Partial differential equations
5673:Arbitrary-precision arithmetic
4912:Probability: A Graduate Course
4602:
4573:
4540:
4517:
4486:
4471:
4424:
4406:
4395:
4145:
4126:
3929:{\displaystyle {\bar {Y}}_{n}}
3914:
3877:
3864:
3752:
3657:
3561:
3546:
3260:
3236:
3225:
3017:
2455:
2446:
2414:
2408:
2262:
2256:
2105:
2099:
2068:
2065:
2059:
2050:
2044:
2038:
1922:
1916:
1891:
1879:
1846:
1840:
1812:
1800:
1704:
1698:
1683:
1677:
1559:
1553:
1542:
1503:
1497:
1483:
1427:will be less than or equal to
1401:
1389:
1380:
1374:
1236:
1230:
1197:
1191:
1166:
1160:
1013:
1007:
947:
935:
929:
923:
896:
890:
873:{\displaystyle x\in \Omega \,}
687:
679:
650:
642:
115:Collectively exhaustive events
13:
1:
5688:Interactive geometry software
4550:A First Course in Probability
4383:
3764:{\displaystyle n\to \infty .}
1285:Continuous probability theory
564:
526:Andrey Nikolaevich Kolmogorov
4511:10.1016/0315-0860(80)90025-7
4493:Daston, Lorraine J. (1980).
4388:
4323: – Mathematical concept
4232: – Branch of statistics
3964:The theorem states that the
3691:
3666:{\displaystyle n\to \infty }
3608:
3410:
1657:probability density function
1460:monotonically non-decreasing
1326:{\displaystyle \mathbb {R} }
555:
7:
5740:Computational number theory
5703:Numerical-analysis software
4588:: Brooks/Cole. p. 53.
4480:Introduction to Probability
4431:LIGHTNER, JAMES E. (1991).
4315:Probability interpretations
4207:
2869:convergence in distribution
733:Discrete probability theory
521:variables into the theory.
371:probability interpretations
10:
6395:
5460:Films about mathematicians
4893:Cambridge University Press
4732:
4378:List of statistical topics
4368:List of probability topics
3950:
3840:Bernoulli random variables
3356:
2928:Convergence in probability
2660:
2590:
2493:{\displaystyle \mu _{F}\,}
1265:
713:
459:
6338:
6300:
6284:
6246:
6198:
6146:Algebra of physical space
6011:
5969:
5763:
5725:
5613:Automated theorem proving
5581:
5483:
5433:
5390:
5300:
5262:
5229:
5181:
5153:
5100:
5047:
5029:Philosophy of mathematics
5004:
4969:
4846:Understanding Probability
4780:10.1007/978-3-642-49888-6
3846:and 0 with probability 1-
2593:Probability distributions
2154:{\displaystyle \Omega \,}
1252:probability mass function
1115:{\displaystyle \Omega \,}
406:probability distributions
5939:Numerical linear algebra
5465:Recreational mathematics
4848:. Cambridge Univ. Press.
4356:
4327:Statistical independence
4303:Probability distribution
2631:continuous distributions
2559:. For example, to study
599:probability distribution
285:Law of total probability
280:Conditional independence
169:Exponential distribution
154:Probability distribution
5678:Finite element analysis
5628:Constraint satisfaction
5350:Mathematical statistics
5340:Mathematical psychology
5310:Engineering mathematics
5244:Algebraic number theory
4817:Probability and Measure
4741:Pierre Simon de Laplace
4704:more precise citations.
4437:The Mathematics Teacher
4278:Notation in probability
4230:Mathematical Statistics
3568:{\displaystyle |X_{k}|}
3281:almost sure convergence
2732:to the random variable
2627:geometric distributions
2111:{\displaystyle \delta }
846:{\displaystyle \Omega }
823:finite or countable set
579:of the experiment. The
264:Conditional probability
6233:Mathematical economics
5907:Multivariable calculus
5790:Differential equations
5633:Constraint programming
5623:Computational geometry
5496:Mathematics portal
5345:Mathematical sociology
5325:Mathematical economics
5320:Mathematical chemistry
5249:Analytic number theory
5130:Differential equations
4547:Ross, Sheldon (2010).
4168:
4125:
4073:
4036:
4013:
3930:
3890:
3832:
3765:
3736:
3667:
3641:
3569:
3532:
3508:
3471:
3455:
3339:
3273:
3206:
3185:
3131:
3080:
2993:
2972:
2918:
2857:
2836:
2815:
2794:
2747:
2722:
2603:discrete distributions
2555:, as in the theory of
2549:
2518:
2494:
2463:
2389:
2364:
2332:
2303:
2276:
2236:
2211:
2187:
2155:
2112:
2083:
2012:
1977:
1940:
1857:
1781:
1756:
1726:
1640:
1619:
1599:
1576:
1520:
1452:
1409:
1358:
1327:
1281:
1244:
1243:{\displaystyle f(x)\,}
1208:
1141:
1116:
1095:
1030:
974:
904:
903:{\displaystyle f(x)\,}
874:
847:
809:
729:
700:
663:
462:History of probability
456:History of probability
206:Continuous or discrete
159:Bernoulli distribution
6186:Supersymmetry algebra
6171:Representation theory
6166:Renormalization group
5812:Differential geometry
5693:Optimization software
5665:Mathematical software
5475:Mathematics education
5405:Theory of computation
5125:Hypercomplex analysis
4169:
4105:
4074:
4037:
4014:
3953:Central limit theorem
3947:Central limit theorem
3931:
3891:
3833:
3766:
3737:
3668:
3642:
3590:law of large numbers
3570:
3533:
3509:
3507:{\displaystyle X_{k}}
3472:
3435:
3340:
3274:
3207:
3186:
3132:
3081:
2994:
2973:
2919:
2858:
2837:
2816:
2800:converges to the CDF
2795:
2748:
2723:
2577:absolutely continuous
2568:Radon-Nikodym theorem
2550:
2519:
2495:
2464:
2390:
2365:
2333:
2304:
2277:
2237:
2212:
2188:
2156:
2113:
2084:
2013:
1978:
1941:
1858:
1782:
1757:
1727:
1648:absolutely continuous
1641:
1620:
1600:
1577:
1521:
1453:
1410:
1359:
1328:
1275:
1245:
1209:
1142:
1117:
1096:
1031:
975:
905:
875:
848:
810:
723:
701:
664:
442:sequential estimation
438:statistical mechanics
427:central limit theorem
164:Binomial distribution
6238:Mathematical finance
6223:Social choice theory
6138:Algebraic structures
6087:in quantum mechanics
6023:Analytical mechanics
5989:Stochastic processes
5961:Variational calculus
5773:Approximation theory
5698:Statistical software
5455:Informal mathematics
5335:Mathematical physics
5330:Mathematical finance
5315:Mathematical biology
5254:Diophantine geometry
4499:Historia Mathematica
4449:10.5951/MT.84.8.0623
4283:Predictive modelling
4257:Fuzzy measure theory
4186:Berry–Esseen theorem
4086:
4049:
4035:{\displaystyle \mu }
4026:
3976:
3904:
3854:
3789:
3746:
3681:
3651:
3598:
3542:
3531:{\displaystyle \mu }
3522:
3491:
3401:
3392:law of large numbers
3385:law of large numbers
3359:Law of large numbers
3353:Law of large numbers
3292:
3219:
3195:
3148:
3095:
3006:
2982:
2935:
2880:
2846:
2825:
2804:
2757:
2736:
2682:
2557:stochastic processes
2530:
2517:{\displaystyle F\,.}
2504:
2476:
2402:
2374:
2353:
2317:
2288:
2250:
2221:
2200:
2172:
2144:
2120:Dirac delta function
2093:
2035:
1993:
1961:
1873:
1794:
1770:
1738:
1671:
1629:
1609:
1589:
1585:The random variable
1531:
1472:
1441:
1368:
1347:
1315:
1291:Classical definition
1224:
1154:
1130:
1105:
1101:of the sample space
1084:
1067:in the sample space
1051:in the sample space
985:
917:
884:
857:
837:
775:
762:Classical definition
726:Poisson distribution
673:
636:
423:law of large numbers
410:stochastic processes
359:probability calculus
290:Law of large numbers
259:Marginal probability
184:Poisson distribution
33:Part of a series on
6213:Operations research
6082:Perturbation theory
5880:Multilinear algebra
5851:Functional analysis
5708:Numerical libraries
5640:Computational logic
5470:Mathematics and art
5380:Operations research
5135:Functional analysis
4914:. Springer-Verlag.
4910:Gut, Allan (2005).
4813:Patrick Billingsley
4586:Belmont, California
4402:Inferring From Data
4336:Statistical physics
4289:Probabilistic logic
4272:Likelihood function
3959:normal distribution
3724:
3629:
3582:that separates the
3327:
3205:{\displaystyle X\,}
3119:
3086:for every ε > 0.
2992:{\displaystyle X\,}
2906:
2856:{\displaystyle F\,}
2835:{\displaystyle X\,}
2814:{\displaystyle F\,}
2746:{\displaystyle X\,}
2363:{\displaystyle E\,}
2244:probability measure
2210:{\displaystyle P\,}
2127:Cantor distribution
1780:{\displaystyle E\,}
1639:{\displaystyle F\,}
1451:{\displaystyle F\,}
1364:exists, defined by
1357:{\displaystyle F\,}
1278:normal distribution
1140:{\displaystyle E\,}
1094:{\displaystyle E\,}
953: for all
746:, experiments with
742:Examples: Throwing
387:probability measure
249:Complementary event
191:Probability measure
179:Pareto distribution
174:Normal distribution
6379:Probability theory
6350:Mathematics portal
6247:Other applications
5971:Probability theory
5954:Validated numerics
5934:Numerical analysis
5827:Geometric analysis
5817:Differential forms
5650:Information theory
5415:Numerical analysis
5024:Mathematical logic
5019:Information theory
4309:Probability axioms
4222:Mathematics portal
4190:exponential family
4164:
4069:
4032:
4009:
3926:
3886:
3828:
3761:
3732:
3731:
3663:
3637:
3636:
3565:
3528:
3504:
3467:
3335:
3334:
3269:
3243:
3202:
3181:
3141:Strong convergence
3127:
3126:
3076:
3024:
2989:
2968:
2914:
2913:
2853:
2832:
2811:
2790:
2743:
2718:
2651:beta distributions
2635:continuous uniform
2599:special importance
2545:
2514:
2490:
2459:
2385:
2360:
2328:
2299:
2272:
2232:
2207:
2183:
2151:
2108:
2079:
2008:
1973:
1936:
1853:
1777:
1752:
1722:
1654:is said to have a
1636:
1625:is continuous. If
1615:
1595:
1572:
1549:
1516:
1493:
1448:
1405:
1354:
1323:
1295:Bertrand's paradox
1282:
1240:
1204:
1187:
1137:
1112:
1091:
1077:is defined as any
1071:is equal to 1. An
1063:) over all values
1026:
1003:
970:
955:
900:
870:
843:
805:
803:
788:
730:
696:
659:
540:and presented his
492:Christiaan Huygens
381:, which assigns a
355:Probability theory
300:Boole's inequality
236:Stochastic process
125:Mutual exclusivity
42:Probability theory
18:Probability Theory
6366:
6365:
6200:Decision sciences
6194:
6193:
6176:Spacetime algebra
5868:Harmonic analysis
5834:Dynamical systems
5778:Clifford analysis
5755:Discrete geometry
5721:
5720:
5541:
5540:
5140:Harmonic analysis
4863:Olav Kallenberg;
4789:978-3-642-49888-6
4730:
4729:
4722:
4647:978-1-85233-896-1
4595:978-0-534-38020-5
4560:978-0-13-603313-4
4420:. 10 August 2021.
4321:Probability space
4181:random variable.
4161:
4158:
3917:
3861:
3725:
3694:
3630:
3611:
3518:(expected value)
3432:
3413:
3379:to the number of
3328:
3228:
3120:
3009:
2907:
2619:negative binomial
2370:in the σ-algebra
2135:probability space
1716:
1618:{\displaystyle F}
1598:{\displaystyle X}
1534:
1475:
1301:Modern definition
1172:
1055:, and the sum of
988:
954:
831:possible outcomes
819:Modern definition
802:
787:
685:
648:
622:identity function
610:elementary events
593:Probability is a
534:Richard von Mises
498:was completed by
488:problem of points
450:quantum mechanics
414:non-deterministic
379:probability space
361:is the branch of
352:
351:
254:Joint probability
201:Bernoulli process
100:Probability space
16:(Redirected from
6386:
6151:Feynman integral
6134:
6133:
6094:Potential theory
5983:random variables
5873:Fourier analysis
5856:Operator algebra
5783:Clifford algebra
5735:Computer algebra
5661:
5660:
5568:
5561:
5554:
5545:
5544:
5529:
5528:
5517:
5516:
5505:
5504:
5494:
5493:
5425:Computer algebra
5400:Computer science
5120:Complex analysis
4954:
4947:
4940:
4931:
4930:
4925:
4906:
4849:
4820:
4793:
4748:
4725:
4718:
4714:
4711:
4705:
4700:this article by
4691:inline citations
4678:
4677:
4670:
4664:
4658:
4652:
4651:
4631:
4625:
4624:
4622:
4621:
4606:
4600:
4599:
4584:(2nd ed.).
4577:
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4544:
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4529:
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4484:
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4422:
4421:
4410:
4404:
4399:
4345:Subjective logic
4341:
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4000:
3988:
3987:
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3919:
3918:
3910:
3895:
3893:
3892:
3887:
3876:
3875:
3863:
3862:
3859:
3838:are independent
3837:
3835:
3834:
3829:
3814:
3813:
3801:
3800:
3785:For example, if
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3704:
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2750:
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2725:
2724:
2719:
2707:
2706:
2694:
2693:
2675:Weak convergence
2669:random variables
2607:discrete uniform
2581:Lebesgue measure
2573:counting measure
2554:
2552:
2551:
2546:
2544:
2543:
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2181:
2180:
2160:
2158:
2157:
2152:
2117:
2115:
2114:
2109:
2088:
2086:
2085:
2080:
2075:
2017:
2015:
2014:
2009:
2007:
2006:
2001:
1987:multidimensional
1982:
1980:
1979:
1974:
1968:
1945:
1943:
1942:
1937:
1912:
1911:
1862:
1860:
1859:
1854:
1833:
1832:
1786:
1784:
1783:
1778:
1761:
1759:
1758:
1753:
1751:
1731:
1729:
1728:
1723:
1717:
1715:
1707:
1690:
1645:
1643:
1642:
1637:
1624:
1622:
1621:
1616:
1604:
1602:
1601:
1596:
1581:
1579:
1578:
1573:
1548:
1525:
1523:
1522:
1517:
1492:
1464:right-continuous
1457:
1455:
1454:
1449:
1414:
1412:
1411:
1406:
1363:
1361:
1360:
1355:
1332:
1330:
1329:
1324:
1322:
1249:
1247:
1246:
1241:
1213:
1211:
1210:
1205:
1186:
1146:
1144:
1143:
1138:
1121:
1119:
1118:
1113:
1100:
1098:
1097:
1092:
1035:
1033:
1032:
1027:
1002:
979:
977:
976:
971:
956:
952:
909:
907:
906:
901:
879:
877:
876:
871:
852:
850:
849:
844:
814:
812:
811:
806:
804:
795:
789:
780:
705:
703:
702:
697:
686:
683:
668:
666:
665:
660:
649:
646:
595:way of assigning
550:Bruno de Finetti
532:, introduced by
480:Pierre de Fermat
476:Gerolamo Cardano
402:random variables
344:
337:
330:
120:Elementary event
52:
30:
29:
21:
6394:
6393:
6389:
6388:
6387:
6385:
6384:
6383:
6369:
6368:
6367:
6362:
6334:
6296:
6280:
6242:
6190:
6156:Poisson algebra
6132:
6014:
6007:
5965:
5861:Operator theory
5759:
5717:
5683:Tensor software
5659:
5608:Automata theory
5577:
5572:
5542:
5537:
5488:
5479:
5429:
5386:
5365:Systems science
5296:
5292:Homotopy theory
5258:
5225:
5177:
5149:
5096:
5043:
5014:Category theory
5000:
4965:
4958:
4928:
4922:
4903:
4824:Olav Kallenberg
4790:
4735:
4726:
4715:
4709:
4706:
4696:Please help to
4695:
4679:
4675:
4668:
4667:
4659:
4655:
4648:
4632:
4628:
4619:
4617:
4608:
4607:
4603:
4596:
4578:
4574:
4565:
4563:
4561:
4545:
4541:
4532:
4530:
4527:
4523:
4522:
4518:
4491:
4487:
4482:. pp. vii.
4476:
4472:
4429:
4425:
4412:
4411:
4407:
4400:
4396:
4391:
4386:
4359:
4339:
4330:
4292:
4260:
4233:
4220:
4213:
4210:
4179:standard normal
4153:
4149:
4133:
4129:
4120:
4109:
4104:
4102:
4093:
4089:
4087:
4084:
4083:
4056:
4052:
4050:
4047:
4046:
4027:
4024:
4023:
3996:
3992:
3983:
3979:
3977:
3974:
3973:
3955:
3949:
3920:
3909:
3908:
3907:
3905:
3902:
3901:
3871:
3867:
3858:
3857:
3855:
3852:
3851:
3809:
3805:
3796:
3792:
3790:
3787:
3786:
3747:
3744:
3743:
3709:
3703:
3696:
3686:
3685:
3682:
3679:
3678:
3652:
3649:
3648:
3620:
3613:
3603:
3602:
3599:
3596:
3595:
3560:
3554:
3550:
3545:
3543:
3540:
3539:
3523:
3520:
3519:
3498:
3494:
3492:
3489:
3488:
3460:
3456:
3450:
3439:
3434:
3424:
3415:
3405:
3404:
3402:
3399:
3398:
3361:
3355:
3313:
3307:
3300:
3296:
3293:
3290:
3289:
3248:
3244:
3232:
3220:
3217:
3216:
3196:
3193:
3192:
3168:
3164:
3155:
3151:
3149:
3146:
3145:
3110:
3103:
3099:
3096:
3093:
3092:
3042:
3038:
3037:
3033:
3032:
3028:
3013:
3007:
3004:
3003:
2983:
2980:
2979:
2955:
2951:
2942:
2938:
2936:
2933:
2932:
2901:
2895:
2888:
2884:
2881:
2878:
2877:
2847:
2844:
2843:
2826:
2823:
2822:
2805:
2802:
2801:
2777:
2773:
2764:
2760:
2758:
2755:
2754:
2737:
2734:
2733:
2702:
2698:
2689:
2685:
2683:
2680:
2679:
2665:
2659:
2595:
2589:
2561:Brownian motion
2539:
2534:
2533:
2531:
2528:
2527:
2505:
2502:
2501:
2483:
2479:
2477:
2474:
2473:
2440:
2436:
2424:
2420:
2403:
2400:
2399:
2378:
2377:
2375:
2372:
2371:
2354:
2351:
2350:
2321:
2320:
2318:
2315:
2314:
2311:Borel σ-algebra
2292:
2291:
2289:
2286:
2285:
2251:
2248:
2247:
2225:
2224:
2222:
2219:
2218:
2201:
2198:
2197:
2176:
2175:
2173:
2170:
2169:
2145:
2142:
2141:
2094:
2091:
2090:
2071:
2036:
2033:
2032:
2024:
2002:
1997:
1996:
1994:
1991:
1990:
1964:
1962:
1959:
1958:
1901:
1897:
1874:
1871:
1870:
1822:
1818:
1795:
1792:
1791:
1771:
1768:
1767:
1747:
1739:
1736:
1735:
1708:
1691:
1689:
1672:
1669:
1668:
1630:
1627:
1626:
1610:
1607:
1606:
1590:
1587:
1586:
1538:
1532:
1529:
1528:
1479:
1473:
1470:
1469:
1442:
1439:
1438:
1369:
1366:
1365:
1348:
1345:
1344:
1318:
1316:
1313:
1312:
1270:
1264:
1254:abbreviated as
1225:
1222:
1221:
1176:
1155:
1152:
1151:
1131:
1128:
1127:
1106:
1103:
1102:
1085:
1082:
1081:
992:
986:
983:
982:
950:
918:
915:
914:
885:
882:
881:
858:
855:
854:
838:
835:
834:
793:
778:
776:
773:
772:
739:sample spaces.
718:
712:
682:
674:
671:
670:
645:
637:
634:
633:
626:flipping a coin
614:random variable
567:
558:
546:axiomatic basis
472:games of chance
464:
458:
365:concerned with
348:
196:Random variable
147:Bernoulli trial
28:
23:
22:
15:
12:
11:
5:
6392:
6382:
6381:
6364:
6363:
6361:
6360:
6347:
6339:
6336:
6335:
6333:
6332:
6327:
6322:
6317:
6316:
6315:
6304:
6302:
6298:
6297:
6295:
6294:
6288:
6286:
6282:
6281:
6279:
6278:
6271:
6266:
6261:
6256:
6250:
6248:
6244:
6243:
6241:
6240:
6235:
6230:
6225:
6220:
6215:
6210:
6204:
6202:
6196:
6195:
6192:
6191:
6189:
6188:
6183:
6178:
6173:
6168:
6163:
6158:
6153:
6148:
6142:
6140:
6131:
6130:
6129:
6128:
6123:
6113:
6112:
6111:
6106:
6096:
6091:
6090:
6089:
6079:
6078:
6077:
6072:
6067:
6062:
6057:
6052:
6047:
6037:
6036:
6035:
6030:
6019:
6017:
6009:
6008:
6006:
6005:
6000:
5995:
5986:
5975:
5973:
5967:
5966:
5964:
5963:
5958:
5957:
5956:
5951:
5946:
5941:
5931:
5930:
5929:
5924:
5919:
5914:
5904:
5903:
5902:
5897:
5892:
5887:
5877:
5876:
5875:
5865:
5864:
5863:
5858:
5848:
5847:
5846:
5844:Control theory
5841:
5831:
5830:
5829:
5824:
5819:
5809:
5808:
5807:
5802:
5797:
5787:
5786:
5785:
5775:
5769:
5767:
5761:
5760:
5758:
5757:
5752:
5747:
5742:
5737:
5731:
5729:
5723:
5722:
5719:
5718:
5716:
5715:
5710:
5705:
5700:
5695:
5690:
5685:
5680:
5675:
5669:
5667:
5658:
5657:
5652:
5647:
5642:
5637:
5636:
5635:
5625:
5620:
5615:
5610:
5605:
5604:
5603:
5598:
5587:
5585:
5579:
5578:
5571:
5570:
5563:
5556:
5548:
5539:
5538:
5536:
5535:
5523:
5511:
5499:
5484:
5481:
5480:
5478:
5477:
5472:
5467:
5462:
5457:
5452:
5451:
5450:
5443:Mathematicians
5439:
5437:
5435:Related topics
5431:
5430:
5428:
5427:
5422:
5417:
5412:
5407:
5402:
5396:
5394:
5388:
5387:
5385:
5384:
5383:
5382:
5377:
5372:
5370:Control theory
5362:
5357:
5352:
5347:
5342:
5337:
5332:
5327:
5322:
5317:
5312:
5306:
5304:
5298:
5297:
5295:
5294:
5289:
5284:
5279:
5274:
5268:
5266:
5260:
5259:
5257:
5256:
5251:
5246:
5241:
5235:
5233:
5227:
5226:
5224:
5223:
5218:
5213:
5208:
5203:
5198:
5193:
5187:
5185:
5179:
5178:
5176:
5175:
5170:
5165:
5159:
5157:
5151:
5150:
5148:
5147:
5145:Measure theory
5142:
5137:
5132:
5127:
5122:
5117:
5112:
5106:
5104:
5098:
5097:
5095:
5094:
5089:
5084:
5079:
5074:
5069:
5064:
5059:
5053:
5051:
5045:
5044:
5042:
5041:
5036:
5031:
5026:
5021:
5016:
5010:
5008:
5002:
5001:
4999:
4998:
4993:
4988:
4987:
4986:
4981:
4970:
4967:
4966:
4957:
4956:
4949:
4942:
4934:
4927:
4926:
4920:
4907:
4901:
4875:
4860:
4859:
4858:
4857:
4851:
4850:
4838:
4821:
4808:
4807:
4806:
4805:
4795:
4794:
4788:
4768:A. Kolmogoroff
4763:
4762:
4761:
4760:
4750:
4749:
4736:
4734:
4731:
4728:
4727:
4710:September 2009
4682:
4680:
4673:
4666:
4665:
4661:David Williams
4653:
4646:
4626:
4601:
4594:
4572:
4559:
4539:
4516:
4505:(3): 234–260.
4485:
4470:
4443:(8): 623–630.
4423:
4405:
4393:
4392:
4390:
4387:
4385:
4382:
4381:
4380:
4375:
4370:
4365:
4358:
4355:
4354:
4353:
4347:
4342:
4333:
4324:
4318:
4312:
4306:
4300:
4295:
4286:
4280:
4275:
4269:
4263:
4254:
4248:
4242:
4239:Expected value
4236:
4226:
4225:
4209:
4206:
4175:
4174:
4157:
4152:
4147:
4144:
4141:
4136:
4132:
4128:
4123:
4118:
4115:
4112:
4108:
4101:
4096:
4092:
4067:
4064:
4059:
4055:
4031:
4007:
4004:
3999:
3995:
3991:
3986:
3982:
3951:Main article:
3948:
3945:
3923:
3916:
3913:
3885:
3882:
3879:
3874:
3870:
3866:
3826:
3823:
3820:
3817:
3812:
3808:
3804:
3799:
3795:
3772:
3771:
3760:
3757:
3754:
3751:
3730:
3722:
3719:
3715:
3712:
3707:
3699:
3693:
3690:
3674:
3673:
3662:
3659:
3656:
3635:
3628:
3624:
3616:
3610:
3607:
3563:
3557:
3553:
3548:
3527:
3501:
3497:
3478:
3477:
3463:
3459:
3453:
3448:
3445:
3442:
3438:
3431:
3428:
3423:
3418:
3412:
3409:
3393:
3386:
3357:Main article:
3354:
3351:
3346:
3345:
3333:
3325:
3322:
3319:
3316:
3311:
3303:
3299:
3285:
3284:
3282:
3268:
3265:
3262:
3259:
3256:
3251:
3247:
3241:
3238:
3235:
3231:
3227:
3224:
3214:
3200:
3179:
3176:
3171:
3167:
3163:
3158:
3154:
3142:
3138:
3137:
3125:
3118:
3114:
3106:
3102:
3088:
3087:
3075:
3072:
3068:
3064:
3061:
3057:
3053:
3050:
3045:
3041:
3036:
3031:
3027:
3022:
3019:
3016:
3012:
3001:
3000:in probability
2987:
2966:
2963:
2958:
2954:
2950:
2945:
2941:
2929:
2925:
2924:
2912:
2904:
2899:
2891:
2887:
2873:
2872:
2870:
2851:
2830:
2809:
2788:
2785:
2780:
2776:
2772:
2767:
2763:
2741:
2731:
2716:
2713:
2710:
2705:
2701:
2697:
2692:
2688:
2676:
2661:Main article:
2658:
2655:
2591:Main article:
2588:
2585:
2542:
2537:
2513:
2509:
2486:
2482:
2470:
2469:
2457:
2454:
2451:
2448:
2443:
2439:
2433:
2430:
2427:
2423:
2419:
2416:
2413:
2410:
2407:
2395:is defined as
2381:
2358:
2341:
2324:
2295:
2270:
2267:
2264:
2261:
2258:
2255:
2245:
2228:
2205:
2179:
2164:
2149:
2140:Given any set
2133:to define the
2131:measure theory
2107:
2104:
2101:
2098:
2078:
2074:
2070:
2067:
2064:
2061:
2058:
2055:
2052:
2049:
2046:
2043:
2040:
2023:
2020:
2005:
2000:
1972:
1967:
1947:
1946:
1935:
1931:
1928:
1924:
1921:
1918:
1915:
1910:
1907:
1904:
1900:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1864:
1863:
1852:
1848:
1845:
1842:
1839:
1836:
1831:
1828:
1825:
1821:
1817:
1814:
1811:
1808:
1805:
1802:
1799:
1775:
1750:
1746:
1743:
1721:
1714:
1711:
1706:
1703:
1700:
1697:
1694:
1688:
1685:
1682:
1679:
1676:
1667:
1663:
1659:
1634:
1614:
1594:
1583:
1582:
1571:
1567:
1564:
1561:
1558:
1555:
1552:
1547:
1544:
1541:
1537:
1526:
1515:
1511:
1508:
1505:
1502:
1499:
1496:
1491:
1488:
1485:
1482:
1478:
1467:
1446:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1352:
1342:
1338:
1321:
1307:is the set of
1302:
1292:
1286:
1266:Main article:
1263:
1260:
1257:
1253:
1238:
1235:
1232:
1229:
1215:
1214:
1203:
1199:
1196:
1193:
1190:
1185:
1182:
1179:
1175:
1171:
1168:
1165:
1162:
1159:
1147:is defined as
1135:
1125:
1110:
1089:
1076:
1037:
1036:
1025:
1021:
1018:
1015:
1012:
1009:
1006:
1001:
998:
995:
991:
980:
969:
965:
962:
959:
949:
946:
943:
940:
937:
934:
931:
928:
925:
922:
898:
895:
892:
889:
868:
865:
862:
842:
820:
801:
798:
792:
786:
783:
763:
754:, and tossing
748:decks of cards
734:
714:Main article:
711:
708:
695:
692:
689:
681:
678:
658:
655:
652:
644:
641:
566:
563:
557:
554:
538:measure theory
520:
513:. Eventually,
508:
500:Pierre Laplace
460:Main article:
457:
454:
350:
349:
347:
346:
339:
332:
324:
321:
320:
319:
318:
313:
305:
304:
303:
302:
297:
295:Bayes' theorem
292:
287:
282:
277:
269:
268:
267:
266:
261:
256:
251:
243:
242:
241:
240:
239:
238:
233:
228:
226:Observed value
223:
218:
213:
211:Expected value
208:
203:
193:
188:
187:
186:
181:
176:
171:
166:
161:
151:
150:
149:
139:
138:
137:
132:
127:
122:
117:
107:
102:
94:
93:
92:
91:
86:
81:
80:
79:
69:
68:
67:
54:
53:
45:
44:
38:
37:
26:
9:
6:
4:
3:
2:
6391:
6380:
6377:
6376:
6374:
6359:
6355:
6351:
6348:
6346:
6345:
6341:
6340:
6337:
6331:
6328:
6326:
6323:
6321:
6318:
6314:
6311:
6310:
6309:
6306:
6305:
6303:
6301:Organizations
6299:
6293:
6290:
6289:
6287:
6283:
6276:
6272:
6270:
6267:
6265:
6262:
6260:
6257:
6255:
6252:
6251:
6249:
6245:
6239:
6236:
6234:
6231:
6229:
6226:
6224:
6221:
6219:
6216:
6214:
6211:
6209:
6206:
6205:
6203:
6201:
6197:
6187:
6184:
6182:
6179:
6177:
6174:
6172:
6169:
6167:
6164:
6162:
6161:Quantum group
6159:
6157:
6154:
6152:
6149:
6147:
6144:
6143:
6141:
6139:
6135:
6127:
6124:
6122:
6119:
6118:
6117:
6116:Supersymmetry
6114:
6110:
6107:
6105:
6102:
6101:
6100:
6099:String theory
6097:
6095:
6092:
6088:
6085:
6084:
6083:
6080:
6076:
6073:
6071:
6068:
6066:
6063:
6061:
6058:
6056:
6053:
6051:
6048:
6046:
6043:
6042:
6041:
6038:
6034:
6031:
6029:
6026:
6025:
6024:
6021:
6020:
6018:
6016:
6010:
6004:
6001:
5999:
5998:Path integral
5996:
5994:
5990:
5987:
5984:
5980:
5979:Distributions
5977:
5976:
5974:
5972:
5968:
5962:
5959:
5955:
5952:
5950:
5947:
5945:
5942:
5940:
5937:
5936:
5935:
5932:
5928:
5925:
5923:
5920:
5918:
5915:
5913:
5910:
5909:
5908:
5905:
5901:
5898:
5896:
5893:
5891:
5888:
5886:
5883:
5882:
5881:
5878:
5874:
5871:
5870:
5869:
5866:
5862:
5859:
5857:
5854:
5853:
5852:
5849:
5845:
5842:
5840:
5837:
5836:
5835:
5832:
5828:
5825:
5823:
5820:
5818:
5815:
5814:
5813:
5810:
5806:
5803:
5801:
5798:
5796:
5793:
5792:
5791:
5788:
5784:
5781:
5780:
5779:
5776:
5774:
5771:
5770:
5768:
5766:
5762:
5756:
5753:
5751:
5748:
5746:
5745:Combinatorics
5743:
5741:
5738:
5736:
5733:
5732:
5730:
5728:
5724:
5714:
5711:
5709:
5706:
5704:
5701:
5699:
5696:
5694:
5691:
5689:
5686:
5684:
5681:
5679:
5676:
5674:
5671:
5670:
5668:
5666:
5662:
5656:
5653:
5651:
5648:
5646:
5643:
5641:
5638:
5634:
5631:
5630:
5629:
5626:
5624:
5621:
5619:
5618:Coding theory
5616:
5614:
5611:
5609:
5606:
5602:
5599:
5597:
5594:
5593:
5592:
5589:
5588:
5586:
5584:
5583:Computational
5580:
5576:
5569:
5564:
5562:
5557:
5555:
5550:
5549:
5546:
5534:
5533:
5524:
5522:
5521:
5512:
5510:
5509:
5500:
5498:
5497:
5492:
5486:
5485:
5482:
5476:
5473:
5471:
5468:
5466:
5463:
5461:
5458:
5456:
5453:
5449:
5446:
5445:
5444:
5441:
5440:
5438:
5436:
5432:
5426:
5423:
5421:
5418:
5416:
5413:
5411:
5408:
5406:
5403:
5401:
5398:
5397:
5395:
5393:
5392:Computational
5389:
5381:
5378:
5376:
5373:
5371:
5368:
5367:
5366:
5363:
5361:
5358:
5356:
5353:
5351:
5348:
5346:
5343:
5341:
5338:
5336:
5333:
5331:
5328:
5326:
5323:
5321:
5318:
5316:
5313:
5311:
5308:
5307:
5305:
5303:
5299:
5293:
5290:
5288:
5285:
5283:
5280:
5278:
5275:
5273:
5270:
5269:
5267:
5265:
5261:
5255:
5252:
5250:
5247:
5245:
5242:
5240:
5237:
5236:
5234:
5232:
5231:Number theory
5228:
5222:
5219:
5217:
5214:
5212:
5209:
5207:
5204:
5202:
5199:
5197:
5194:
5192:
5189:
5188:
5186:
5184:
5180:
5174:
5171:
5169:
5166:
5164:
5163:Combinatorics
5161:
5160:
5158:
5156:
5152:
5146:
5143:
5141:
5138:
5136:
5133:
5131:
5128:
5126:
5123:
5121:
5118:
5116:
5115:Real analysis
5113:
5111:
5108:
5107:
5105:
5103:
5099:
5093:
5090:
5088:
5085:
5083:
5080:
5078:
5075:
5073:
5070:
5068:
5065:
5063:
5060:
5058:
5055:
5054:
5052:
5050:
5046:
5040:
5037:
5035:
5032:
5030:
5027:
5025:
5022:
5020:
5017:
5015:
5012:
5011:
5009:
5007:
5003:
4997:
4994:
4992:
4989:
4985:
4982:
4980:
4977:
4976:
4975:
4972:
4971:
4968:
4963:
4955:
4950:
4948:
4943:
4941:
4936:
4935:
4932:
4923:
4921:0-387-22833-0
4917:
4913:
4908:
4904:
4902:9781108473682
4898:
4894:
4890:
4886:
4885:
4880:
4879:Durrett, Rick
4876:
4874:
4873:0-387-25115-4
4870:
4866:
4862:
4861:
4855:
4854:
4853:
4852:
4847:
4843:
4839:
4837:
4836:0-387-95313-2
4833:
4829:
4825:
4822:
4818:
4814:
4810:
4809:
4803:
4799:
4798:
4797:
4796:
4791:
4785:
4781:
4777:
4773:
4769:
4765:
4764:
4758:
4754:
4753:
4752:
4751:
4746:
4742:
4738:
4737:
4724:
4721:
4713:
4703:
4699:
4693:
4692:
4686:
4681:
4672:
4671:
4662:
4657:
4649:
4643:
4639:
4638:
4630:
4616:on 2014-01-26
4615:
4611:
4605:
4597:
4591:
4587:
4583:
4576:
4562:
4556:
4552:
4551:
4543:
4526:
4520:
4512:
4508:
4504:
4500:
4496:
4489:
4481:
4474:
4466:
4462:
4458:
4454:
4450:
4446:
4442:
4438:
4434:
4427:
4419:
4415:
4409:
4403:
4398:
4394:
4379:
4376:
4374:
4371:
4369:
4366:
4364:
4361:
4360:
4351:
4348:
4346:
4343:
4337:
4334:
4328:
4325:
4322:
4319:
4316:
4313:
4310:
4307:
4304:
4301:
4299:
4296:
4290:
4287:
4284:
4281:
4279:
4276:
4273:
4270:
4267:
4264:
4258:
4255:
4252:
4249:
4246:
4243:
4240:
4237:
4231:
4228:
4227:
4223:
4217:
4212:
4205:
4203:
4199:
4195:
4191:
4187:
4182:
4180:
4155:
4150:
4142:
4139:
4134:
4130:
4121:
4116:
4113:
4110:
4106:
4099:
4094:
4090:
4082:
4081:
4080:
4065:
4062:
4057:
4053:
4045:
4029:
4022:
4005:
4002:
3997:
3993:
3989:
3984:
3980:
3971:
3967:
3962:
3960:
3954:
3944:
3942:
3941:almost surely
3939:
3936:converges to
3921:
3911:
3899:
3883:
3880:
3872:
3868:
3849:
3845:
3841:
3824:
3821:
3818:
3815:
3810:
3806:
3802:
3797:
3793:
3783:
3781:
3777:
3758:
3749:
3728:
3720:
3713:
3705:
3697:
3688:
3676:
3675:
3654:
3633:
3626:
3622:
3614:
3605:
3593:
3592:
3591:
3589:
3585:
3581:
3576:
3555:
3551:
3525:
3517:
3499:
3495:
3487:
3483:
3461:
3457:
3451:
3446:
3443:
3440:
3436:
3429:
3426:
3421:
3416:
3407:
3397:
3396:
3395:
3391:
3388:
3384:
3382:
3378:
3374:
3370:
3366:
3360:
3350:
3331:
3323:
3317:
3309:
3301:
3297:
3287:
3286:
3280:
3266:
3263:
3257:
3254:
3249:
3245:
3233:
3222:
3212:
3198:
3177:
3174:
3169:
3165:
3161:
3156:
3152:
3143:
3140:
3139:
3123:
3116:
3112:
3104:
3100:
3090:
3089:
3073:
3070:
3066:
3062:
3059:
3055:
3051:
3048:
3043:
3039:
3034:
3029:
3025:
3014:
2999:
2985:
2964:
2961:
2956:
2952:
2948:
2943:
2939:
2930:
2927:
2926:
2910:
2897:
2889:
2885:
2875:
2874:
2868:
2866:
2849:
2828:
2807:
2786:
2783:
2778:
2774:
2770:
2765:
2761:
2739:
2729:
2714:
2711:
2708:
2703:
2699:
2695:
2690:
2686:
2677:
2674:
2673:
2672:
2670:
2664:
2654:
2652:
2648:
2644:
2640:
2636:
2632:
2628:
2624:
2620:
2616:
2612:
2608:
2604:
2600:
2594:
2584:
2582:
2578:
2574:
2569:
2564:
2562:
2558:
2540:
2524:
2511:
2507:
2484:
2480:
2452:
2449:
2441:
2437:
2431:
2428:
2425:
2421:
2417:
2411:
2405:
2398:
2397:
2396:
2356:
2348:
2343:
2339:
2312:
2282:
2268:
2265:
2253:
2243:
2203:
2196:
2168:
2162:
2161:(also called
2138:
2136:
2132:
2128:
2123:
2121:
2102:
2096:
2076:
2072:
2062:
2056:
2053:
2047:
2041:
2028:
2019:
2003:
1988:
1983:
1970:
1956:
1952:
1933:
1929:
1926:
1919:
1913:
1908:
1905:
1902:
1898:
1894:
1888:
1885:
1882:
1876:
1869:
1868:
1867:
1850:
1843:
1837:
1834:
1829:
1826:
1823:
1819:
1815:
1809:
1806:
1803:
1797:
1790:
1789:
1788:
1773:
1765:
1744:
1741:
1732:
1719:
1712:
1709:
1701:
1695:
1692:
1686:
1680:
1674:
1665:
1661:
1658:
1655:
1653:
1649:
1632:
1612:
1592:
1569:
1565:
1562:
1556:
1550:
1539:
1527:
1513:
1509:
1506:
1500:
1494:
1486:
1480:
1468:
1465:
1461:
1444:
1437:
1436:
1435:
1432:
1430:
1426:
1422:
1418:
1398:
1395:
1392:
1386:
1383:
1377:
1371:
1350:
1340:
1337:
1334:
1310:
1306:
1300:
1298:
1296:
1290:
1288:
1284:
1279:
1274:
1269:
1259:
1255:
1251:
1233:
1227:
1220:The function
1218:
1201:
1194:
1188:
1183:
1180:
1177:
1173:
1169:
1163:
1157:
1150:
1149:
1148:
1133:
1126:of the event
1123:
1087:
1080:
1075:
1072:
1070:
1066:
1062:
1058:
1054:
1050:
1046:
1042:
1023:
1019:
1016:
1010:
1004:
996:
993:
989:
981:
967:
960:
957:
944:
941:
938:
932:
926:
920:
913:
912:
911:
893:
887:
863:
860:
832:
828:
824:
818:
816:
799:
796:
790:
784:
781:
769:
767:
761:
759:
757:
753:
749:
745:
740:
738:
732:
727:
722:
717:
707:
693:
690:
676:
656:
653:
639:
631:
627:
623:
619:
615:
611:
606:
602:
600:
596:
591:
589:
584:
583:
578:
577:
572:
562:
553:
551:
547:
543:
539:
535:
531:
527:
522:
518:
516:
512:
511:combinatorial
506:
503:
501:
497:
493:
489:
485:
484:Blaise Pascal
481:
477:
473:
469:
463:
453:
451:
447:
443:
439:
435:
430:
428:
424:
419:
415:
411:
407:
403:
398:
396:
392:
388:
384:
380:
376:
372:
368:
364:
360:
356:
345:
340:
338:
333:
331:
326:
325:
323:
322:
317:
314:
312:
309:
308:
307:
306:
301:
298:
296:
293:
291:
288:
286:
283:
281:
278:
276:
273:
272:
271:
270:
265:
262:
260:
257:
255:
252:
250:
247:
246:
245:
244:
237:
234:
232:
229:
227:
224:
222:
219:
217:
214:
212:
209:
207:
204:
202:
199:
198:
197:
194:
192:
189:
185:
182:
180:
177:
175:
172:
170:
167:
165:
162:
160:
157:
156:
155:
152:
148:
145:
144:
143:
140:
136:
133:
131:
128:
126:
123:
121:
118:
116:
113:
112:
111:
108:
106:
103:
101:
98:
97:
96:
95:
90:
87:
85:
84:Indeterminism
82:
78:
75:
74:
73:
70:
66:
63:
62:
61:
58:
57:
56:
55:
51:
47:
46:
43:
40:
39:
36:
32:
31:
19:
6356: /
6352: /
6342:
6218:Optimization
6181:Superalgebra
6040:Field theory
6013:Mathematical
5991: /
5970:
5839:Chaos theory
5822:Gauge theory
5750:Graph theory
5645:Cryptography
5530:
5518:
5506:
5487:
5420:Optimization
5354:
5282:Differential
5206:Differential
5173:Order theory
5168:Graph theory
5072:Group theory
4911:
4883:
4864:
4845:
4827:
4816:
4801:
4771:
4756:
4744:
4716:
4707:
4688:
4656:
4636:
4629:
4618:. Retrieved
4614:the original
4604:
4581:
4575:
4564:. Retrieved
4549:
4542:
4531:. Retrieved
4519:
4502:
4498:
4488:
4479:
4473:
4440:
4436:
4426:
4417:
4408:
4397:
4183:
4176:
3970:irrespective
3969:
3963:
3956:
3937:
3897:
3847:
3843:
3784:
3779:
3775:
3773:
3677:Strong law:
3587:
3583:
3577:
3479:
3389:
3380:
3376:
3372:
3368:
3364:
3362:
3347:
2666:
2633:include the
2630:
2629:. Important
2602:
2598:
2596:
2565:
2525:
2471:
2346:
2344:
2283:
2242:is called a
2163:sample space
2139:
2124:
2029:
2025:
1984:
1954:
1950:
1949:Whereas the
1948:
1865:
1763:
1733:
1664:) or simply
1651:
1584:
1433:
1428:
1424:
1420:
1416:
1309:real numbers
1304:
1299:
1289:
1283:
1219:
1216:
1068:
1064:
1060:
1056:
1052:
1048:
1044:
1040:
1038:
830:
827:sample space
817:
770:
760:
741:
731:
629:
607:
603:
592:
587:
580:
576:sample space
574:
569:Consider an
568:
559:
542:axiom system
530:sample space
523:
504:
465:
431:
399:
391:sample space
358:
354:
353:
316:Tree diagram
311:Venn diagram
275:Independence
221:Markov chain
105:Sample space
41:
6358:topics list
6292:Mathematics
6208:Game theory
6109:Topological
6075:Topological
6070:Statistical
6033:Hamiltonian
5532:WikiProject
5375:Game theory
5355:Probability
5092:Homological
5082:Multilinear
5062:Commutative
5039:Type theory
5006:Foundations
4962:mathematics
4702:introducing
4251:Fuzzy logic
3575:is finite.
3516:expectation
2842:, wherever
2643:exponential
2500:induced by
2347:probability
2217:defined on
1415:. That is,
1124:probability
825:called the
752:random walk
618:real number
468:probability
367:probability
363:mathematics
231:Random walk
72:Determinism
60:Probability
6264:Psychology
6228:Statistics
6028:Lagrangian
5655:Statistics
5591:Algorithms
5360:Statistics
5239:Arithmetic
5201:Arithmetic
5067:Elementary
5034:Set theory
4842:Henk Tijms
4685:references
4620:2012-02-12
4566:2016-02-28
4533:2012-02-12
4384:References
4194:heavy tail
3900:, so that
3594:Weak law:
2865:continuous
2728:converges
1734:For a set
571:experiment
565:Motivation
519:continuous
515:analytical
434:statistics
418:quantities
142:Experiment
89:Randomness
35:statistics
6269:Sociology
6259:Chemistry
6055:Effective
6050:Conformal
6045:Classical
5917:Geometric
5890:Geometric
5287:Geometric
5277:Algebraic
5216:Euclidean
5191:Algebraic
5087:Universal
4457:0025-5769
4389:Citations
4151:σ
4143:μ
4140:−
4107:∑
4054:σ
4030:μ
4006:…
3915:¯
3756:∞
3753:→
3729:μ
3692:¯
3661:∞
3658:→
3634:μ
3609:¯
3526:μ
3437:∑
3411:¯
3240:∞
3237:→
3178:…
3063:ε
3060:≥
3049:−
3021:∞
3018:→
2965:…
2787:…
2712:…
2611:Bernoulli
2481:μ
2453:ω
2438:μ
2429:∈
2426:ω
2422:∫
2349:of a set
2260:Ω
2193:on it, a
2167:σ-algebra
2148:Ω
2097:δ
2057:φ
2042:δ
1989:cases on
1906:∈
1899:∫
1886:∈
1827:∈
1820:∫
1807:∈
1766:being in
1745:⊆
1546:∞
1543:→
1490:∞
1487:−
1484:→
1466:function;
1396:≤
1181:∈
1174:∑
1109:Ω
1000:Ω
997:∈
990:∑
964:Ω
961:∈
933:∈
867:Ω
864:∈
841:Ω
737:countable
582:power set
556:Treatment
135:Singleton
6373:Category
6344:Category
5993:analysis
5912:Exterior
5885:Exterior
5765:Analysis
5727:Discrete
5601:analysis
5508:Category
5264:Topology
5211:Discrete
5196:Analytic
5183:Geometry
5155:Discrete
5110:Calculus
5102:Analysis
5057:Abstract
4996:Glossary
4979:Timeline
4881:(2019).
4844:(2004).
4815:(1979).
4770:(1933).
4743:(1812).
4465:27967334
4245:Variance
4208:See also
4204:(GCLT).
4198:fat tail
4044:variance
3896:for all
3706:→
3623:→
3586:and the
3482:sequence
3310:→
3213:strongly
3113:→
2898:→
2615:binomial
2605:are the
2165:) and a
2089:, where
507:discrete
425:and the
216:Variance
6354:outline
6285:Related
6254:Biology
6104:Bosonic
6065:Quantum
6015:physics
5981: (
5713:Solvers
5520:Commons
5302:Applied
5272:General
5049:Algebra
4974:History
4733:Sources
4698:improve
3966:average
3850:, then
3365:roughly
2623:Poisson
2340:induced
2309:is the
2195:measure
2118:is the
1666:density
446:physics
383:measure
130:Outcome
5927:Vector
5922:Tensor
5900:Vector
5895:Tensor
5596:design
5221:Finite
5077:Linear
4984:Future
4960:Major
4918:
4899:
4871:
4834:
4786:
4687:, but
4644:
4592:
4557:
4463:
4455:
3588:strong
2730:weakly
2639:normal
1122:. The
1079:subset
588:events
536:, and
408:, and
375:axioms
77:System
65:Axioms
6060:Gauge
5448:lists
4991:Lists
4964:areas
4528:(PDF)
4461:JSTOR
4357:Lists
3480:of a
3381:tails
3377:heads
3373:tails
3369:heads
2647:gamma
1458:is a
1074:event
756:coins
684:tails
647:heads
395:event
110:Event
4916:ISBN
4897:ISBN
4869:ISBN
4832:ISBN
4784:ISBN
4642:ISBN
4590:ISBN
4555:ISBN
4453:ISSN
4196:and
4063:>
4042:and
4021:mean
3742:for
3647:for
3584:weak
3390:The
2649:and
2625:and
2345:The
1276:The
744:dice
724:The
490:").
482:and
4776:doi
4507:doi
4445:doi
3484:of
3230:lim
3215:if
3011:lim
3002:if
2863:is
2821:of
2284:If
2246:if
1955:CDF
1951:PDF
1787:is
1662:PDF
1646:is
1536:lim
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1258:.
1256:pmf
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474:by
452:.
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4889:UK
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