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Probability theory

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Branch of mathematics concerning probability
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Part of a series onstatistics
Probability theory

Probability theory orprobability calculus is the branch ofmathematics concerned withprobability. Although there are several differentprobability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ofaxioms. Typically these axioms formalise probability in terms of aprobability space, which assigns ameasure taking values between 0 and 1, termed theprobability measure, to a set of outcomes called thesample space. Any specified subset of the sample space is called anevent.

Central subjects in probability theory include discrete and continuousrandom variables,probability distributions, andstochastic processes (which provide mathematical abstractions ofnon-deterministic or uncertain processes or measuredquantities 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 thelaw of large numbers and thecentral limit theorem.

As a mathematical foundation forstatistics, probability theory is essential to many human activities that involve quantitative analysis of data.[1] Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as instatistical mechanics orsequential estimation. A great discovery of twentieth-centuryphysics was the probabilistic nature of physical phenomena at atomic scales, described inquantum mechanics.[2]

History of probability

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Main article:History of probability

The modern mathematical theory ofprobability has its roots in attempts to analyzegames of chance byGerolamo Cardano in the sixteenth century, and byPierre de Fermat andBlaise Pascal in the seventeenth century (for example the "problem of points").[3]Christiaan Huygens published a book on the subject in 1657.[4] In the 19th century, what is considered theclassical definition of probability was completed byPierre Laplace.[5]

Initially, probability theory mainly considereddiscrete events, and its methods were mainlycombinatorial. Eventually,analytical considerations compelled the incorporation ofcontinuous variables into the theory.

This culminated in modern probability theory, on foundations laid byAndrey Nikolaevich Kolmogorov. Kolmogorov combined the notion ofsample space, introduced byRichard von Mises, andmeasure theory and presented hisaxiom system for probability theory in 1933. This became the mostly undisputedaxiomatic basis for modern probability theory; but, alternatives exist, such as the adoption of finite rather than countable additivity byBruno de Finetti.[6]

Treatment

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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.

Motivation

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Consider anexperiment that can produce a number of outcomes. The set of all outcomes is called thesample space of the experiment. Thepower set 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 an element of the power set of the sample space of dice rolls. These collections are calledevents. 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.

Probability is away of assigning 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 aprobability distribution, 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.[7]

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 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 thoseelementary events have a number assigned to them. This is done using arandom variable. A random variable is a function that assigns to each elementary event in the sample space areal number. This function is usually denoted by a capital letter.[8] In the case of a die, the assignment of a number to certain elementary events can be done using theidentity function. This does not always work. For example, whenflipping a coin the two possible outcomes are "heads" and "tails". In this example, the random variableX could assign to the outcome "heads" the number "0" (X(heads)=0{\textstyle X({\text{heads}})=0}) and to the outcome "tails" the number "1" (X(tails)=1{\displaystyle X({\text{tails}})=1}).

Discrete probability distributions

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Main article:Discrete probability distribution
ThePoisson distribution, a discrete probability distribution

Discrete probability theory deals with events that occur incountable sample spaces.

Examples: Throwingdice, experiments withdecks of cards,random walk, and tossingcoins.

Classical definition: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: seeClassical definition of probability.

For example, if the event is "occurrence of an even number when a dice is rolled", the probability is given by36=12{\displaystyle {\tfrac {3}{6}}={\tfrac {1}{2}}}, 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 afinite or countable set called thesample space, which relates to the set of allpossible outcomes in classical sense, denoted byΩ{\displaystyle \Omega }. It is then assumed that for each elementx∈Ω{\displaystyle x\in \Omega \,}, an intrinsic "probability" valuef(x){\displaystyle f(x)\,} is attached, which satisfies the following properties:

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

That is, the probability functionf(x) lies between zero and one for every value ofx in the sample spaceΩ, and the sum off(x) over all valuesx in the sample spaceΩ is equal to 1. Anevent is defined as anysubsetE{\displaystyle E\,} of the sample spaceΩ{\displaystyle \Omega \,}. Theprobability of the eventE{\displaystyle E\,} is defined as

P(E)=∑x∈Ef(x).{\displaystyle \mathbb {P} (E)=\sum _{x\in E}f(x)\,.}

So, the probability of the entire sample space is 1, and the probability of the null event is 0.

The functionf(x){\displaystyle f(x)\,} mapping a point in the sample space to the "probability" value is called aprobability mass function abbreviated aspmf.

Continuous probability distributions

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Main article:Continuous probability distribution
Thenormal distribution, a 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. SeeBertrand's paradox.

Modern definition:If the sample space of a random variableX is the set ofreal numbers (R{\displaystyle \mathbb {R} }) or a subset thereof, then a function called thecumulative distribution function (CDF)F{\displaystyle F\,} exists, defined byF(x)=P(X≤x){\displaystyle F(x)=\mathbb {P} (X\leq x)\,}. That is,F(x) returns the probability thatX will be less than or equal tox.

The CDF necessarily satisfies the following properties.

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

The random variableX{\displaystyle X} is said to have a continuous probability distribution if the corresponding CDFF{\displaystyle F} is continuous. IfF{\displaystyle F\,} isabsolutely continuous, then its derivative exists almost everywhere and integrating the derivative gives us the CDF back again. In this case, the random variableX is said to have aprobability density function (PDF) or simplydensityf(x)=dF(x)dx.{\displaystyle f(x)={\frac {dF(x)}{dx}}\,.}

For a setE⊆R{\displaystyle E\subseteq \mathbb {R} }, the probability of the random variableX being inE{\displaystyle E\,} is

P(X∈E)=∫x∈EdF(x).{\displaystyle \mathbb {P} (X\in E)=\int _{x\in E}dF(x)\,.}

In case the PDF exists, this can be written as

P(X∈E)=∫x∈Ef(x)dx.{\displaystyle \mathbb {P} (X\in E)=\int _{x\in E}f(x)\,dx\,.}

Whereas thePDF exists only for continuous random variables, theCDF exists for all random variables (including discrete random variables) that take values inR.{\displaystyle \mathbb {R} \,.}

These concepts can be generalized formultidimensional cases onRn{\displaystyle \mathbb {R} ^{n}} and other continuous sample spaces.


Measure-theoretic probability theory

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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.

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(δ[x]+φ(x))/2{\displaystyle (\delta [x]+\varphi (x))/2}, whereδ[x]{\displaystyle \delta [x]} is theDirac delta function.

Other distributions may not even be a mix, for example, theCantor distribution has no positive probability for any single point, neither does it have a density. The modern approach to probability theory solves these problems usingmeasure theory to define theprobability space:

Given any setΩ{\displaystyle \Omega \,} (also calledsample space) and aσ-algebraF{\displaystyle {\mathcal {F}}\,} on it, ameasureP{\displaystyle \mathbb {P} } defined onF{\displaystyle {\mathcal {F}}\,} is called aprobability measure ifP(Ω)=1.{\displaystyle \mathbb {P} (\Omega )=1.\,}

IfF{\displaystyle {\mathcal {F}}\,} is theBorel σ-algebra on the set of real numbers, then there is a unique probability measure onF{\displaystyle {\mathcal {F}}\,} for any CDF, and vice versa. The measure corresponding to a CDF is said to beinduced 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.

Theprobability of a setE{\displaystyle E\,} in the σ-algebraF{\displaystyle {\mathcal {F}}\,} is defined as

P(E)=∫ω∈EμF(dω){\displaystyle \mathbb {P} (E)=\int _{\omega \in E}\mu _{F}(d\omega )\,}

where the integration is with respect to the measureμF{\displaystyle \mu _{F}\,} induced byF.{\displaystyle F\,.}

Along with providing better understanding and unification of discrete and continuous probabilities, measure-theoretic treatment also allows us to work on probabilities outsideRn{\displaystyle \mathbb {R} ^{n}}, as in the theory ofstochastic processes. For example, to studyBrownian motion, probability is defined on a space of functions.

When it is convenient to work with a dominating measure, theRadon–Nikodym theorem 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 acounting measure over the set of all possible outcomes. Densities forabsolutely continuous distributions are usually defined as this derivative with respect to theLebesgue measure. 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.

Classical probability distributions

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Main article:Probability distributions

Certain random variables occur very often in probability theory because they well describe many natural or physical processes. Their distributions, therefore, have gainedspecial importance in probability theory. Some fundamentaldiscrete distributions are thediscrete uniform,Bernoulli,binomial,negative binomial,Poisson andgeometric distributions. Importantcontinuous distributions include thecontinuous uniform,normal,exponential,gamma andbeta distributions.

Convergence of random variables

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Main article:Convergence of random variables

In probability theory, there are several notions of convergence forrandom variables. 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.

Weak convergence
A sequence of random variablesX1,X2,…,{\displaystyle X_{1},X_{2},\dots ,\,} convergesweakly to the random variableX{\displaystyle X\,} if their respective CDF convergesF1,F2,…{\displaystyle F_{1},F_{2},\dots \,} converges to the CDFF{\displaystyle F\,} ofX{\displaystyle X\,}, whereverF{\displaystyle F\,} iscontinuous. Weak convergence is also calledconvergence in distribution.
Most common shorthand notation:Xn→DX{\displaystyle \displaystyle X_{n}\,{\xrightarrow {\mathcal {D}}}\,X}
Convergence in probability
The sequence of random variablesX1,X2,…{\displaystyle X_{1},X_{2},\dots \,} is said to converge towards the random variableX{\displaystyle X\,}in probability iflimn→∞P(|Xn−X|≥ε)=0{\displaystyle \lim _{n\rightarrow \infty }\mathbb {P} \left(\left|X_{n}-X\right|\geq \varepsilon \right)=0} for every ε > 0.
Most common shorthand notation:Xn→PX{\displaystyle \displaystyle X_{n}\,\xrightarrow {\mathbb {P} } \,X}
Strong convergence
The sequence of random variablesX1,X2,…{\displaystyle X_{1},X_{2},\dots \,} is said to converge towards the random variableX{\displaystyle X\,}strongly ifP(limn→∞Xn=X)=1{\displaystyle \mathbb {P} (\lim _{n\rightarrow \infty }X_{n}=X)=1}. Strong convergence is also known asalmost sure convergence.
Most common shorthand notation:Xn→a.s.X{\displaystyle \displaystyle X_{n}\,{\xrightarrow {\mathrm {a.s.} }}\,X}

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.

Law of large numbers

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Main article:Law of large numbers

Common intuition suggests that if a fair coin is tossed many times, thenroughly half of the time it will turn upheads, and the other half it will turn uptails. Furthermore, the more often the coin is tossed, the more likely it should be that the ratio of the number ofheads to the number oftails will approach unity. Modern probability theory provides a formal version of this intuitive idea, known as thelaw of large numbers. 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.[9]

Thelaw of large numbers (LLN) states that the sample average

X¯n=1n∑k=1nXk{\displaystyle {\overline {X}}_{n}={\frac {1}{n}}{\sum _{k=1}^{n}X_{k}}}

of asequence ofindependent and identically distributed random variablesXk{\displaystyle X_{k}} converges towards their commonexpectation (expected value)μ{\displaystyle \mu }, provided that the expectation of|Xk|{\displaystyle |X_{k}|} is finite.

It is in the different forms ofconvergence of random variables that separates theweak and thestrong law of large numbers[10]

Weak law:X¯n→Pμ{\displaystyle \displaystyle {\overline {X}}_{n}\,\xrightarrow {\mathbb {P} } \,\mu } forn→∞{\displaystyle n\to \infty }
Strong law:X¯n→a.s.μ{\displaystyle \displaystyle {\overline {X}}_{n}\,{\xrightarrow {\mathrm {a.\,s.} }}\,\mu } forn→∞.{\displaystyle n\to \infty .}

It follows from the LLN that if an event of probabilityp is observed repeatedly during independent experiments, the ratio of the observed frequency of that event to the total number of repetitions converges towardsp.

For example, ifY1,Y2,...{\displaystyle Y_{1},Y_{2},...\,} are independentBernoulli random variables taking values 1 with probabilityp and 0 with probability 1-p, thenE(Yi)=p{\displaystyle {\textrm {E}}(Y_{i})=p} for alli, so thatY¯n{\displaystyle {\bar {Y}}_{n}} converges topalmost surely.

Central limit theorem

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Main article:Central limit theorem

The central limit theorem (CLT) explains the ubiquitous occurrence of thenormal distribution in nature, and this theorem, according to David Williams, "is one of the great results of mathematics."[11]

The theorem states that theaverage of many independent and identically distributed random variables with finite variance tends towards a normal distributionirrespective of the distribution followed by the original random variables. Formally, letX1,X2,…{\displaystyle X_{1},X_{2},\dots \,} be independent random variables withmeanμ{\displaystyle \mu } andvarianceσ2>0.{\displaystyle \sigma ^{2}>0.\,} Then the sequence of random variables

Zn=∑i=1n(Xi−μ)σn{\displaystyle Z_{n}={\frac {\sum _{i=1}^{n}(X_{i}-\mu )}{\sigma {\sqrt {n}}}}\,}

converges in distribution to astandard normal random variable.

For some classes of random variables, the classic central limit theorem works rather fast, as illustrated in theBerry–Esseen theorem. For example, the distributions with finite first, second, and third moment from theexponential family; on the other hand, for some random variables of theheavy tail andfat tail variety, it works very slowly or may not work at all: in such cases one may use theGeneralized Central Limit Theorem (GCLT).

See also

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Lists

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References

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Citations

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  1. ^Inferring From Data
  2. ^"Quantum Logic and Probability Theory".The Stanford Encyclopedia of Philosophy. 10 August 2021.
  3. ^LIGHTNER, JAMES E. (1991)."A Brief Look at the History of Probability and Statistics".The Mathematics Teacher.84 (8):623–630.doi:10.5951/MT.84.8.0623.ISSN 0025-5769.JSTOR 27967334.
  4. ^Grinstead, Charles Miller; James Laurie Snell. "Introduction".Introduction to Probability. pp. vii.
  5. ^Daston, Lorraine J. (1980)."Probabilistic Expectation and Rationality in Classical Probability Theory".Historia Mathematica.7 (3):234–260.doi:10.1016/0315-0860(80)90025-7.
  6. ^""The origins and legacy of Kolmogorov's Grundbegriffe", by Glenn Shafer and Vladimir Vovk"(PDF). Retrieved2012-02-12.
  7. ^Ross, Sheldon (2010).A First Course in Probability (8th ed.). Pearson Prentice Hall. pp. 26–27.ISBN 978-0-13-603313-4. Retrieved2016-02-28.
  8. ^Bain, Lee J.; Engelhardt, Max (1992).Introduction to Probability and Mathematical Statistics (2nd ed.).Belmont, California: Brooks/Cole. p. 53.ISBN 978-0-534-38020-5.
  9. ^"Leithner & Co Pty Ltd - Value Investing, Risk and Risk Management - Part I". Leithner.com.au. 2000-09-15. Archived fromthe original on 2014-01-26. Retrieved2012-02-12.
  10. ^Dekking, Michel (2005). "Chapter 13: The law of large numbers".A modern introduction to probability and statistics : understanding why and how. Library Genesis. London : Springer. pp. 180–194.ISBN 978-1-85233-896-1.{{cite book}}: CS1 maint: publisher location (link)
  11. ^David Williams, "Probability with martingales", Cambridge 1991/2008

Sources

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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 titleFoundations of the Theory of Probability (Chelsea, New York) in 1950, with a second edition in 1956.
A lively introduction to probability theory for the beginner.
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