Probability theory 23542 224759415 2008-07-10T08:35:27Z Jason Patton 4178568 [[WP:UNDO|Undid]] revision 224759121 by [[Special:Contributions/196.41.181.186|196.41.181.186]] ([[User talk:196.41.181.186|talk]]) and link to dice to avoid confusion '''Probability theory''' is the branch of [[mathematics]] concerned with analysis of [[Statistical randomness|random]] phenomena.<ref>[http://www.britannica.com/ebc/article-9375936 Probability theory, Encyclopaedia Britannica]</ref> The central objects of probability theory are [[random variable]]s, [[stochastic process]]es, and [[event (probability theory)|event]]s: mathematical abstractions of [[determinism|non-deterministic]] events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. Although an individual coin toss or the roll of a [[dice|die]] is a random event, if repeated many times the sequence of random events will exhibit certain statistical patterns, which can be studied and predicted. Two representative mathematical results describing such patterns are the [[law of large numbers]] and the [[central limit theorem]]. As a mathematical foundation for [[statistics]], probability theory is essential to many human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to description of complex systems given only partial knowledge of their state, as in [[statistical mechanics]]. A great discovery of twentieth century [[physics]] was the probabilistic nature of physical phenomena at atomic scales, described in [[quantum mechanics]]. == History == The mathematical theory of [[probability]] has its roots in attempts to analyse [[game of chance|games of chance]] by [[Gerolamo Cardano]] in the sixteenth century, and by [[Pierre de Fermat]] and [[Blaise Pascal]] in the seventeenth century (for example the "[[problem of points]]"). Initially, probability theory mainly considered '''discrete''' events, and its methods were mainly [[combinatorics|combinatorial]]. Eventually, [[mathematical analysis|analytical]] considerations compelled the incorporation of '''continuous''' variables into the theory. This culminated in modern probability theory, the foundations of which were laid by [[Andrey Nikolaevich Kolmogorov]]. Kolmogorov combined the notion of [[sample space]], introduced by [[Richard von Mises]], and '''[[measure theory]]''' and presented his [[Kolmogorov axioms|axiom system]] for probability theory in 1933. Fairly quickly this became the undisputed [[axiom system|axiomatic basis]] for modern probability theory.<ref>[http://www.probabilityandfinance.com/articles/04.pdf "The origins and legacy of Kolmogorov's Grundbegriffe", by Glenn Shafer and Vladimir Vovk]</ref> == Treatment == Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The more mathematically advanced measure theory based treatment of probability covers both the discrete, the continuous, any mix of these two and more. === Discrete probability distributions === {{main|Discrete probability distribution}} '''Discrete probability theory''' deals with events that occur in [[countable]] sample spaces. Examples: Throwing [[dice]], experiments with [[deck of cards|decks of cards]], and [[random walk]]. '''Classical definition:''' Initially the probability of an event to occur was defined as number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space. For example, if the event is "occurrence of an even number when a [[dice|die]] is rolled", the probability is given by <math>\tfrac{3}{6}=\tfrac{1}{2}</math>, since 3 faces out of the 6 have even numbers and each face has the same probability of appearing. '''Modern definition:''' The modern definition starts with a [[set]] called the '''[[sample space]]''', which relates to the set of all ''possible outcomes'' in classical sense, denoted by <math>\Omega=\left \{ x_1,x_2,\dots\right \}</math>. It is then assumed that for each [[Element (mathematics)|element]] <math>x \in \Omega\,</math>, an intrinsic "probability" value <math>f(x)\,</math> is attached, which satisfies the following properties: #<math>f(x)\in[0,1]\mbox{ for all }x\in \Omega\,;</math> #<math>\sum_{x\in \Omega} f(x) = 1\,.</math> That is, the probability function ''f''(''x'') lies between zero and one for every value of ''x'' in the sample space ''Ω'', and the sum of ''f''(''x'') over all values ''x'' in the sample space ''Ω'' is exactly equal to 1. An '''[[Event (probability theory)|event]]''' is defined as any [[subset]] <math>E\,</math> of the sample space <math>\Omega\,</math>. The '''probability''' of the event <math>E\,</math> defined as :<math>P(E)=\sum_{x\in E} f(x)\,.</math> So, the probability of the entire sample space is 1, and the probability of the null event is 0. The function <math>f(x)\,</math> mapping a point in the sample space to the "probability" value is called a '''[[probability mass function]]''' abbreviated as '''pmf'''. The modern definition does not try to answer how probability mass functions are obtained; instead it builds a theory that assumes their existence. === Continuous probability distributions === {{main|Continuous probability distribution}} '''Continuous probability theory''' deals with events that occur in a continuous sample space. '''Classical definition:''' The classical definition breaks down when confronted with the continuous case. See [[Bertrand's paradox (probability)|Bertrand's paradox]]. '''Modern definition:''' If the sample space is the [[real numbers]] (<math>\mathbb{R}</math>), then a function called the '''[[cumulative distribution function]]''' (or '''cdf''') <math>F\,</math> is assumed to exist, which gives <math>P(X\le x) = F(x)\,</math> for a [[random variable]] ''X''. That is, ''F''(''x'') returns the probability that ''X'' will be less than or equal to ''x''. The cdf must satisfy the following properties. #<math>F\,</math> is a [[Monotonic function|monotonically non-decreasing]], [[right-continuous]] function; #<math>\lim_{x\rightarrow -\infty} F(x)=0\,;</math> #<math>\lim_{x\rightarrow \infty} F(x)=1\,.</math> If <math>F\,</math> is [[differentiable]], then the random variable ''X'' is said to have a '''[[probability density function]]''' or '''pdf''' or simply '''density''' <math>f(x)=\frac{dF(x)}{dx}\,.</math> For a set <math>E \subseteq \mathbb{R}</math>, the probability of the random variable ''X'' being in <math>E\,</math> is defined as :<math>P(X\in E) = \int_{x\in E} dF(x)\,.</math> In case the probability density function exists, this can be written as :<math>P(X\in E) = \int_{x\in E} f(x)\,dx\,.</math> Whereas the ''pdf'' exists only for continuous random variables, the ''cdf'' exists for all random variables (including discrete random variables) that take values on <math>\mathbb{R}\,.</math> These concepts can be generalized for [[Dimension|multidimensional]] cases on <math>\mathbb{R}^n</math> and other continuous sample spaces. === Measure-theoretic probability theory === The ''raison d'être'' of the measure-theoretic treatment of probability is that it unifies the discrete and the continuous, 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. An example of such distributions could be a mix of discrete and continuous distributions, for example, a random variable which is 0 with probability 1/2, and takes a value from random normal distribution with probability 1/2. It can still be studied to some extent by considering it to have a pdf of <math>(\delta[x] + \varphi(x))/2</math>, where <math>\delta[x]</math> is the [[Kronecker delta]] function. Other distributions may not even be a mix, for example, the [[Cantor distribution]] has no positive probability for any single point, neither does it have a density. The modern approach to probability theory solves these problems using [[measure theory]] to define the [[probability space]]: Given any set <math>\Omega</math>, (also called '''sample space''') and a [[sigma-algebra|σ-algebra]] <math>\mathcal{F}\,</math> on it, a [[measure (mathematics)|measure]] <math>P</math> is called a '''probability measure''' if #<math>P\,</math> is non-negative; #<math>P(\Omega)=1\,.</math> If <math>\mathcal{F}\,</math> is a [[Borel algebra|Borel σ-algebra]] then there is a unique probability measure on <math>\mathcal{F}\,</math> for any cdf, and vice versa. The measure corresponding to a cdf is said to be '''induced''' 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. The ''probability'' of a set <math>E\,</math> in the σ-algebra <math>\mathcal{F}\,</math> is defined as <!--the correct formulation; X has nothing to do with it--> :<math>P(E) = \int_{\omega\in E} \mu_F(d\omega)\,.</math> where the integration is with respect to the measure <math>\mu_F\,</math> induced by <math>F\,.</math> Along with providing better understanding and unification of discrete and continuous probabilities, measure-theoretic treatment also allows us to work on probabilities outside <math>\mathbb{R}^n</math>, as in the theory of [[stochastic process]]es. For example to study [[Brownian motion]], probability is defined on a space of functions. == Probability distributions == {{main|Probability distributions}} Certain random variables occur very often in probability theory because they well describe many natural or physical processes. Their distributions therefore have gained ''special importance'' in probability theory. Some fundamental ''discrete distributions'' are the [[uniform distribution (discrete)|discrete uniform]], [[Bernoulli distribution|Bernoulli]], [[binomial distribution|binomial]], [[negative binomial distribution|negative binomial]], [[Poisson distribution|Poisson]] and [[geometric distribution|geometric]] distributions. Important ''continuous distributions'' include the [[uniform distribution (continuous)|continuous uniform]], [[normal distribution|normal]], [[exponential distribution|exponential]], [[gamma distribution|gamma]] and [[beta distribution|beta]] distributions. == Convergence of random variables == {{main|Convergence of random variables}} In probability theory, there are several notions of convergence for [[random variable]]s. 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. :'''Convergence in distribution:''' As the name implies, a sequence of random variables <math>X_1,X_2,\dots,\,</math> converges to the random variable <math>X\,</math> '''in distribution''' if their respective cumulative ''distribution functions'' <math>F_1,F_2,\dots\,</math> converge to the cumulative distribution function <math>F\,</math> of <math>X\,</math>, wherever <math>F\,</math> is [[continuous]]. ::''Most common short hand notation:'' <math>X_n \, \xrightarrow{\mathcal D} \, X\,.</math> :'''Weak convergence:''' The sequence of random variables <math>X_1,X_2,\dots\,</math> is said to converge towards the random variable <math>X\,</math> '''weakly''' if <math>\lim_{n\rightarrow\infty}P\left(\left|X_n-X\right|\geq\varepsilon\right)=0</math> for every &epsilon; > 0. Weak convergence is also called '''convergence in probability'''. ::''Most common short hand notation:'' <math>X_n \, \xrightarrow{P} \, X\,.</math> :'''Strong convergence:''' The sequence of random variables <math>X_1,X_2,\dots\,</math> is said to converge towards the random variable <math>X\,</math> '''strongly''' if <math>P(\lim_{n\rightarrow\infty} X_n=X)=1</math>. Strong convergence is also known as '''almost sure convergence'''. ::''Most common short hand notation:'' <math>X_n \, \xrightarrow{\mathrm{a.s.}} \, X\,.</math> Intuitively, ''strong'' convergence is a stronger version of the ''weak'' convergence, and in both cases the random variables <math>X_1,X_2,\dots\,</math> show an increasing correlation with <math>X\,</math>. However, in case of ''convergence in distribution'', the realized values of the random variables do not need to converge, and any possible correlation among them is immaterial. == Law of large numbers == {{main|Law of large numbers}} Common intuition suggests that if a fair coin is tossed many times, then ''roughly'' half of the time it will turn up ''heads'', and the other half it will turn up ''tails''. Furthermore, the more often the coin is tossed, the more likely it should be that the ratio of the number of ''heads'' to the number of ''tails'' will approach unity. Modern probability provides a formal version of this intuitive idea, known as the '''law of large numbers'''. This law is remarkable because it is nowhere assumed in the foundations of probability theory, but instead emerges out of 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.[http://www.leithner.com.au/circulars/circular17.htm] <!-- Note to editors: Please provide better citation for the historical importance of LLN if you have it --> The '''law of large numbers''' (LLN) states that the sample average <math>\overline{X}_n=\tfrac1n{\sum X_n}</math> of <math>X_1,X_2,\dots\,</math> (independent and identically distributed random variables with finite expectation <math>\mu</math>) converges towards the theoretical expectation <math>\mu.</math> It is in the different forms of [[convergence of random variables]] that separates the ''weak'' and the ''strong'' law of large numbers <math> \begin{array}{lll} \text{Weak law:} & \overline{X}_n \, \xrightarrow{P} \, \mu & \text{for } n \to \infty \\ \text{Strong law:} & \overline{X}_n \, \xrightarrow{\mathrm{a.\,s.}} \, \mu & \text{for } n \to \infty . \end{array} </math> It follows from LLN that if an event of probability ''p'' is observed repeatedly during independent experiments, the ratio of the observed frequency of that event to the total number of repetitions converges towards ''p''. Putting this in terms of random variables and LLN we have <math>Y_1,Y_2,...\,</math> are independent [[Bernoulli distribution|Bernoulli random variables]] taking values 1 with probability ''p'' and 0 with probability 1-''p''. <math>\textrm{E}(Y_i)=p</math> for all ''i'' and it follows from LLN that <math>\frac{\sum Y_n}{n}\,</math> converges to ''p'' [[almost surely]]. ==Central limit theorem== {{main|Central limit theorem}} The '''central limit theorem''' explains the ubiquitous occurrence of the [[normal distribution]] in nature; it is one of the most celebrated theorems in probability and statistics.{{Fact|date=July 2007}} <!-- Note to editors: Request citations and historical importance of this theorem here if you have a source (and enough time) ~~~~ --> The theorem states that the [[average]] of many independent and identically distributed random variables with finite variance tends towards a [[normal distribution]] ''irrespective'' of the distribution followed by the original random variables. Formally, let <math>X_1,X_2,\dots\,</math> be independent random variables with [[mean]] <math>\mu_\,</math> and [[variance]] <math>\sigma^2 > 0.\,</math> Then the sequence of random variables :<math>Z_n=\frac{\sum_{i=1}^n (X_i - \mu)}{\sigma\sqrt{n}}\,</math> converges in distribution to a [[standard normal]] random variable. ==See also== {{multicol}} *[[Expected value]] and [[Variance]] *[[Fuzzy logic]] and [[Fuzzy measure theory]] *[[Glossary of probability and statistics]] *[[Likelihood function]] *[[List of probability topics]] *[[List of publications in statistics]] *[[List of statistical topics]] {{multicol-break}} *[[Notation in probability]] *[[Predictive modelling]] *[[Probabilistic logic]] - A combination of probability theory and logic *[[Probability interpretations]] *[[Statistical independence]] *[[Subjective logic]] {{multicol-end}} ==References== {{reflist}} {{refimprove|date=January 2008}} == Bibliography == *{{cite book | author = Pierre Simon de Laplace | year = 1812 | title = Analytical Theory of Probability}} :: The first major treatise blending calculus with probability theory, originally in French: ''Théorie Analytique des Probabilités''. *{{cite book | author = Andrei Nikolajevich Kolmogorov | year = 1950 | title = Foundations of the Theory of Probability}} :: The modern measure-theoretic foundation of probability theory; the original German version (''Grundbegriffe der Wahrscheinlichkeitrechnung'') appeared in 1933. *{{cite book | author = Patrick Billingsley | title = Probability and Measure | publisher = John Wiley and Sons | location = New York, Toronto, London | year = 1979}} *{{cite book | author = Henk Tijms | year = 2004 | publisher = Cambridge Univ. Press | title = Understanding Probability}} :: A lively introduction to probability theory for the beginner. *{{cite book | last = Gut | first = Allan | title = Probability: A Graduate Course | publisher = Springer-Verlag | year = 2005 | id = ISBN 0387228330 }} {{Mathematics-footer}} [[Category:Probability theory| ]] [[Category:Mathematical analysis]] {{Link FA|ka}} [[af:Waarskynlikheidsleer]] [[ar:نظرية الاحتمالات]] [[bg:Теория на вероятностите]] [[ca:Teoria de la probabilitat]] [[cs:Teorie pravděpodobnosti]] [[da:Sandsynlighedsregning]] [[de:Wahrscheinlichkeitstheorie]] [[et:Tõenäosusteooria]] [[el:Πιθανότητα]] [[es:Teoría de la probabilidad]] [[eo:Probablokalkulo]] [[eu:Probabilitate teoria]] [[fa:نظریه احتمالات]] [[fr:Théorie des probabilités]] [[gl:Teoría da probabilidade]] [[ko:확률론]] [[id:Probabilitas]] [[is:Líkindafræði]] [[it:Teoria della probabilità]] [[he:תורת ההסתברות]] [[ka:ალბათობის თეორია]] [[lv:Varbūtību teorija]] [[lt:Tikimybių teorija]] [[mk:Теорија на веројатност]] [[ms:Teori kemungkinan]] [[nl:Kansrekening]] [[ja:確率論]] [[no:Sannsynlighetsteori]] [[pl:Teoria prawdopodobieństwa]] [[pt:Teoria das probabilidades]] [[ro:Teoria probabilităţilor]] [[ru:Теория вероятностей]] [[sr:Теорија вероватноће]] [[fi:Todennäköisyysteoria]] [[sv:Sannolikhetsteori]] [[ta:நிகழ்தகவுக் கோட்பாடு]] [[th:ทฤษฎีความน่าจะเป็น]] [[tk:Ahtimallyk Teoriyasy]] [[tr:Olasılık kuramı]] [[vi:Lý thuyết xác suất]] [[uk:Теорія ймовірностей]] [[zh:概率论]]