# Probability theory – Wikipedia

arm of mathematics concerning probability
Probability theory is the branch of mathematics concerned with probability. Although there are respective different probability interpretations, probability hypothesis treats the concept in a rigorous numerical manner by expressing it through a set of axioms. typically these axioms formalise probability in terms of a probability outer space, which assigns a measure taking values between 0 and 1, termed the probability standard, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes, which provide numerical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion. Although it is not possible to absolutely bode random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the cardinal terminus ad quem theorem. As a mathematical foundation for statistics, probability hypothesis is essential to many homo activities that involve quantitative analysis of data. [ 1 ] Methods of probability theory besides apply to descriptions of complex systems given only partial derivative cognition of their submit, as in statistical mechanics or consecutive estimate. A capital discovery of twentieth-century physics was the probabilistic nature of physical phenomenon at atomic scales, described in quantum mechanics. [ 2 ] [ unreliable source? ]

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## history of probability

The modern mathematical theory of probability has its roots in attempts to analyze games of chance by Gerolamo Cardano in the sixteenth hundred, and by Pierre de Fermat and Blaise Pascal in the seventeenth century ( for example the “ trouble of points “ ). [ 3 ] Christiaan Huygens published a book on the subject in 1657 [ 4 ] and in the nineteenth hundred, Pierre Laplace completed what is today considered the classical rendition. [ 5 ]

initially, probability theory chiefly considered discrete events, and its methods were chiefly combinative. finally, analytic considerations compelled the incorporation of continuous variables into the theory. This culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of sample space, introduced by Richard von Mises, and measure theory and presented his axiom system for probability theory in 1933. This became the largely undisputed axiomatic footing for mod probability theory ; but, alternatives exist, such as the adoption of finite rather than countable additivity by Bruno de Finetti. [ 6 ]

## treatment

Most introductions to probability theory treat discrete probability distributions and continuous probability distributions individually. The measure theory-based treatment of probability covers the discrete, continuous, a blend of the two, and more .

### motivation

Consider an experiment that can produce a count of outcomes. The hardening of all outcomes is called the sample space of the experiment. The power set of the sample distribution space ( or equivalently, the event space ) is formed by considering all different collections of possible results. For model, rolling an honest die produces one of six possible results. One collection of potential results corresponds to getting an odd number. thus, the subset { 1,3,5 } is an element of the power set of the sample distribution distance of die rolls. These collections are called events. In this encase, { 1,3,5 } is the event that the die falls on some leftover total. If the results that actually occur fall in a given consequence, that event is said to have occurred. probability is a room of assigning every “ event ” a value between zero and one, with the prerequisite that the event made up of all possible results ( in our exemplar, the consequence { 1,2,3,4,5,6 } ) be assigned a value of one. To qualify as a probability distribution, the assignment of values must satisfy the necessity that if you look at a collection of mutually single events ( events that contain no common results, for example, the events { 1,6 }, { 3 }, and { 2,4 } are all mutually exclusive ), the probability that any of these events occurs is given by the union of the probabilities of the events. [ 7 ] The probability that any one of the events { 1,6 }, { 3 }, or { 2,4 } will occur is 5/6. This is the lapp as saying that the probability of consequence { 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 the event { 1,2,3,4,5,6 } has a probability of 1, that is, absolute certainty. When doing calculations using the outcomes of an experiment, it is necessary that all those elementary events have a number assigned to them. This is done using a random variable star. A random variable is a routine that assigns to each elementary consequence in the sample space a real number. This serve is normally denoted by a capital letter. [ 8 ] In the case of a die, the assignment of a number to a certain elementary events can be done using the identity function. This does not always influence. For exemplar, when flipping a mint the two possible outcomes are “ heads ” and “ tails ”. In this case, the random variable X could assign to the consequence “ heads ” the number “ 0 ” ( X ( planck’s constant e a vitamin d mho ) = 0 { \displaystyle X ( heads ) =0 } ) and to the result “ tails ” the count “ 1 ” ( X ( t a iodine fifty south ) = 1 { \displaystyle X ( tails ) =1 } ) .

### Discrete probability distributions

Discrete probability hypothesis deals with events that occur in countable sample spaces. Examples : Throwing die, experiments with decks of cards, random walk, and tossing coins classical music definition : initially the probability of an event to occur was defined as the count of cases favorable for the consequence, over the issue of total outcomes possible in an equiprobable sample distance : see classical music definition of probability. For exemplar, if the event is “ happening of an even numeral when a die is rolled ”, the probability is given by 3 6 = 1 2 { \displaystyle { \tfrac { 3 } { 6 } } = { \tfrac { 1 } { 2 } } } , since 3 faces out of the 6 have even numbers and each face has the lapp probability of appearing. modern definition : The modern definition starts with a finite or countable arrange called the sample space, which relates to the laid of all possible outcomes in classical sense, denoted by Ω { \displaystyle \Omega } . It is then assumed that for each element x ∈ Ω { \displaystyle x\in \Omega \, } , an intrinsic “ probability ” measure degree fahrenheit ( x ) { \displaystyle f ( x ) \, } is attached, which satisfies the follow properties :

1. f ( x ) ∈ [ 0, 1 ] for all x ∈ Ω ; { \displaystyle degree fahrenheit ( x ) \in [ 0,1 ] { \mbox { for all } } x\in \Omega \, ; }
2. ∑ x ∈ Ω farad ( x ) = 1. { \displaystyle \sum _ { x\in \Omega } degree fahrenheit ( x ) =1\ ,. }

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 distribution space Ω is equal to 1. An event is defined as any subset E { \displaystyle E\, } of the sample quad Ω { \displaystyle \Omega \, } . The probability of the event E { \displaystyle E\, } is defined as

P ( E ) = ∑ x ∈ E farad ( adam ). { \displaystyle P ( E ) =\sum _ { x\in e } farad ( x ) \ ,. }

so, the probability of the entire sample space is 1, and the probability of the null consequence is 0. The affair fluorine ( x ) { \displaystyle f ( x ) \, } mapping a point in the sample space to the “ probability ” value is called a probability aggregate function abbreviated as pmf. The mod definition does not try to answer how probability mass functions are obtained ; rather, it builds a theory that assumes their being [ citation needed ] .

### continuous probability distributions

continuous probability hypothesis deals with events that occur in a continuous sample quad. classical definition : The classical definition breaks down when confronted with the continuous case. See Bertrand ‘s paradox. modern definition : If the sample space of a random variable star X is the specify of real numbers ( R { \displaystyle \mathbb { R } } ) or a subset thence, then a serve called the accumulative distribution function ( or cdf ) F { \displaystyle F\, } exists, defined by F ( x ) = P ( X ≤ x ) { \displaystyle F ( x ) =P ( X\leq x ) \, } . That is, F ( x ) returns the probability that X will be less than or equal to x. The cdf necessarily satisfies the following properties .

1. F { \displaystyle F\, }monotonically non-decreasing, right-continuous function;
2. lim x → − ∞ F ( x ) = 0 ; { \displaystyle \lim _ { x\rightarrow -\infty } F ( x ) =0\, ; }
3. lim x → ∞ F ( x ) = 1. { \displaystyle \lim _ { x\rightarrow \infty } F ( x ) =1\ ,. }

If F { \displaystyle F\, } is absolutely continuous, i.e., its derivative instrument exists and integrating the derivative instrument gives us the cdf back again, then the random variable star X is said to have a probability density function or pdf or merely density farad ( x ) = vitamin d F ( x ) d x. { \displaystyle f ( x ) = { \frac { dF ( x ) } { dx } } \ ,. } For a set up E ⊆ R { \displaystyle E\subseteq \mathbb { R } } , the probability of the random variable X being in E { \displaystyle E\, } is

P ( X ∈ E ) = ∫ x ∈ E five hundred F ( x ). { \displaystyle P ( X\in E ) =\int _ { x\in east } dF ( x ) \ ,. }

In case the probability concentration function exists, this can be written as

P ( X ∈ E ) = ∫ x ∈ E fluorine ( x ) five hundred x. { \displaystyle P ( X\in E ) =\int _ { x\in east } degree fahrenheit ( x ) \, dx\ ,. }

Whereas the pdf exists lone for continuous random variables, the cdf exists for all random variables ( including discrete random variables ) that take values in R. { \displaystyle \mathbb { R } \ ,. } These concepts can be generalized for multidimensional cases on R n { \displaystyle \mathbb { R } ^ { nitrogen } } and other continuous sample spaces .

### Measure-theoretic probability hypothesis

The raison d’être of the measure-theoretic treatment of probability is that it unifies the discrete and the continuous cases, and makes the dispute a interview of which measuring stick is used. furthermore, it covers distributions that are neither discrete nor continuous nor mixtures of the two. An model of such distributions could be a mix of discrete and continuous distributions—for example, a random varying that is 0 with probability 1/2, and takes a random measure from a normal distribution with probability 1/2. It can hush be studied to some extent by considering it to have a pdf of ( δ [ x ] + φ ( x ) ) / 2 { \displaystyle ( \delta [ x ] +\varphi ( x ) ) /2 } , where δ [ x ] { \displaystyle \delta [ x ] } is the Dirac delta affair. other distributions may not even be a mix, for case, the Cantor distribution has no incontrovertible probability for any single point, neither does it have a concentration. The advanced approach to probability theory solves these problems using measuring stick hypothesis to define the probability space : Given any set Ω { \displaystyle \Omega \, } ( besides called sample distance ) and a σ-algebra F { \displaystyle { \mathcal { F } } \, } on it, a measure P { \displaystyle P\, } defined on F { \displaystyle { \mathcal { F } } \, } is called a probability bill if P ( Ω ) = 1. { \displaystyle P ( \Omega ) =1.\, } If F { \displaystyle { \mathcal { F } } \, } is the Borel σ-algebra on the set of real numbers, then there is a unique probability quantify on F { \displaystyle { \mathcal { F } } \, } for any cdf, and vice versa. The bill corresponding to a cdf is said to be induced by the cdf. This measurement coincides with the pmf for discrete variables and pdf for continuous variables, making the measure-theoretic approach release of fallacies. The probability of a set E { \displaystyle E\, } in the σ-algebra F { \displaystyle { \mathcal { F } } \, } is defined as

P ( E ) = ∫ ω ∈ E μ F ( five hundred ω ) { \displaystyle P ( E ) =\int _ { \omega \in E } \mu _ { F } ( d\omega ) \, }

where the integration is with esteem to the measure μ F { \displaystyle \mu _ { F } \, } induced by F. { \displaystyle F\ ,. } Along with providing better understand and union of discrete and continuous probabilities, measure-theoretic treatment besides allows us to work on probabilities outside R n { \displaystyle \mathbb { R } ^ { n } }, as in the hypothesis of stochastic processes. For exercise, to study Brownian motion, probability is defined on a space of functions. When it ‘s commodious to work with a ascendant measure, the Radon-Nikodym theorem is used to define a concentration as the Radon-Nikodym derived function of the probability distribution of interest with obedience to this predominate measure. Discrete densities are normally defined as this derived function with deference to a count measure over the set of all possible outcomes. Densities for absolutely continuous distributions are normally defined as this derivative instrument with obedience to the Lebesgue measurement. If a theorem can be proved in this general typeset, it holds for both discrete and continuous distributions vitamin a well as others ; divide proofs are not required for discrete and continuous distributions.

## classical probability distributions

Certain random variables occur very frequently in probability theory because they well identify many natural or physical processes. Their distributions, therefore, have gained special importance in probability theory. Some fundamental discrete distributions are the discrete undifferentiated, Bernoulli, binomial, negative binomial, Poisson and geometric distributions. Important continuous distributions include the continuous uniform, convention, exponential, gamma and beta distributions .

## overlap of random variables

In probability hypothesis, there are several notions of convergence for random variables. They are listed below in the rate of military capability, i.e., any subsequent impression of convergence in the list implies convergence according to all of the preceding notions .

Weak convergence
A sequence of random variables x 1, X 2, …, { \displaystyle X_ { 1 }, X_ { 2 }, \dots, \, } adam { \displaystyle X\, }distribution functions F 1, F 2, … { \displaystyle F_ { 1 }, F_ { 2 }, \dots \, } F { \displaystyle F\, } x { \displaystyle X\, } F { \displaystyle F\, }continuous. Weak convergence is also called convergence in distribution.
Most common shorthand notation: adam n → D X { \displaystyle \displaystyle X_ { nitrogen } \, { \xrightarrow { \mathcal { D } } } \, X }
Convergence in probability
The sequence of random variables ten 1, X 2, … { \displaystyle X_ { 1 }, X_ { 2 }, \dots \, } adam { \displaystyle X\, } lim n → ∞ P ( | X n − X | ≥ ε ) = 0 { \displaystyle \lim _ { n\rightarrow \infty } P\left ( \left|X_ { normality } -X\right|\geq \varepsilon \right ) =0 }
Most common shorthand notation: x n → P X { \displaystyle \displaystyle X_ { north } \, { \xrightarrow { P } } \, X }
Strong convergence
The sequence of random variables adam 1, X 2, … { \displaystyle X_ { 1 }, X_ { 2 }, \dots \, } adam { \displaystyle X\, } P ( lim north → ∞ X n = X ) = 1 { \displaystyle P ( \lim _ { n\rightarrow \infty } X_ { n } =X ) =1 }
Most common shorthand notation: ten n → a. s. X { \displaystyle \displaystyle X_ { nitrogen } \, { \xrightarrow { \mathrm { a.s. } } } \, X }

As the names indicate, weak convergence is weaker than strong convergence. In fact, impregnable overlap implies convergence in probability, and convergence in probability implies decrepit convergence. The invert statements are not constantly true .

### law of large numbers

common intuition suggests that if a bonny 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 frequently 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 oneness. modern probability hypothesis provides a ball interpretation of this intuitive mind, known as the law of big numbers. This law is noteworthy because it is not assumed in the foundations of probability theory, but alternatively emerges from these foundations as a theorem. Since it links theoretically derived probabilities to their actual frequency of happening in the real universe, the law of large numbers is considered as a column in the history of statistical theory and has had far-flung influence. [ 9 ] The law of boastfully numbers ( LLN ) states that the sample modal

x ¯ n = 1 north ∑ kelvin = 1 newton X k { \displaystyle { \overline { X } } _ { n } = { \frac { 1 } { normality } } { \sum _ { k=1 } ^ { north } X_ { thousand } } }

of a sequence of independent and identically distribute random variables X k { \displaystyle X_ { kelvin } } converges towards their coarse expectation μ { \displaystyle \mu } , provided that the arithmetic mean of | X k | { \displaystyle |X_ { thousand } | } is finite. It is in the different forms of convergence of random variables that separates the weak and the strong law of big numbers [ 10 ]

Weak law: ten ¯ n → P μ { \displaystyle \displaystyle { \overline { X } } _ { n } \, { \xrightarrow { P } } \, \mu } north → ∞ { \displaystyle n\to \infty }
Strong law: adam ¯ north → a. sulfur. μ { \displaystyle \displaystyle { \overline { X } } _ { normality } \, { \xrightarrow { \mathrm { a.\, mho. } } } \, \mu } newton → ∞. { \displaystyle n\to \infty. }

It follows from the LLN that if an event of probability p is observed repeatedly during independent experiments, the proportion of the observe frequency of that event to the full issue of repetitions converges towards p. For exemplar, if Y 1, Y 2 ,. .. { \displaystyle Y_ { 1 }, Y_ { 2 }, … \, } are mugwump Bernoulli random variables taking values 1 with probability p and 0 with probability 1- p, then E ( Y iodine ) = phosphorus { \displaystyle { \textrm { e } } ( Y_ { i } ) =p } for all i, so that Y ¯ nitrogen { \displaystyle { \bar { Y } } _ { nitrogen } } converges to p about surely .

### central specify theorem

“ The cardinal limit theorem ( CLT ) is one of the bang-up results of mathematics. ” ( Chapter 18 in [ 11 ] ) It explains the omnipresent occurrence of the normal distribution in nature. The theorem states that the average of many mugwump and identically distribute random variables with finite discrepancy tends towards a normal distribution irrespective of the distribution followed by the original random variables. formally, let X 1, X 2, … { \displaystyle X_ { 1 }, X_ { 2 }, \dots \, } be independent random variables with mean μ { \displaystyle \mu } and variance σ 2 > 0. { \displaystyle \sigma ^ { 2 } > 0.\, } Z n = ∑ one = 1 north ( X i − μ ) σ normality { \displaystyle Z_ { normality } = { \frac { \sum _ { i=1 } ^ { north } ( X_ { one } -\mu ) } { \sigma { \sqrt { north } } } } \, }

converges in distribution to a standard normal random variable star. For some classes of random variables the classic central terminus ad quem theorem works preferably fast ( see Berry–Esseen theorem ), for example the distributions with finite first base, second base, and third gear moment from the exponential class ; on the other hand, for some random variables of the big tail and fat fag end variety, it works very lento or may not work at all : in such cases one may use the Generalized Central Limit Theorem ( GCLT ) .

## References

• Pierre Simon de Laplace (1812). Analytical Theory of Probability.
The first major treatise blending calculus with probability theory, originally in French: Théorie Analytique des Probabilités.
An English translation by Nathan Morrison appeared under the title Foundations of the Theory of Probability (Chelsea, New York) in 1950, with a second edition in 1956.
• Patrick Billingsley (1979). Probability and Measure. New York, Toronto, London: John Wiley and Sons.
• Olav Kallenberg; Foundations of Modern Probability, 2nd ed. Springer Series in Statistics. (2002). 650 pp. ISBN 0-387-95313-2
• Henk Tijms (2004). Understanding Probability. Cambridge Univ. Press.
A lively introduction to probability theory for the beginner.
• Olav Kallenberg; Probabilistic Symmetries and Invariance Principles. Springer -Verlag, New York (2005). 510 pp. ISBN 0-387-25115-4
• Gut, Allan (2005). Probability: A Graduate Course. Springer-Verlag. ISBN 0-387-22833-0.
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