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| List Of Stochastic Processes |
List of stochastic processesIn the mathematics of probability, a stochastic process can be thought of as a random function. In practical applications, the domain over which the function is defined is a time interval (time series) or a region of space (random field).
Familiar examples of time series include stock market and exchange rate fluctuations, signals such as speech, audio and video; medical data such as a patient's EKG, EEG, blood pressure or temperature; and random movement such as Brownian motion or random walks.
Examples of random fields include static images, random topographies (landscapes), or composition variations of an inhomogeneous material.
Stochastic processes topics
:This list is currently incomplete.
- Bernoulli process
- Birth-death process
- Branching process
- Brownian motion
- Brownian bridge
- Galton-Watson process
- Gauss-Markov process
- Gaussian process
- Girsanov's theorem
- Itô calculus
- Lévy process
- Loop-erased random walk
- Markov chain
- Continuous-time Markov chain
- Markov process
- Martingale
- Ornstein-Uhlenbeck process
- Poisson process
- Compound Poisson process
- Population process
- Process with independent increments
- Queueing theory
- Queue
- Random field
- Gaussian random field
- Markov random field
- Stationary process
- Stochastic process
- Time series
- Wiener process
Category:Mathematical lists
Mathematics
Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions.
Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries.
The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.
History
:Main article: History of mathematics
The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought.
In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry.
Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse.
Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change.
Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both.
Mathematical discoveries have been made throughout history and continue to be made today.
Inspiration, pure and applied mathematics, and aesthetics
Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts.
As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science.
Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.
Notation, language, and rigor
Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales.
The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary.
Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis).
Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.
Is mathematics a science?
Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)."
If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm]
In any case, mathematics shares much in common with many fields in the physical sciences, notably
the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.
Overview of fields of mathematics
As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics).
The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory.
The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space.
The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics.
Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.
Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science.
An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.
Major themes in mathematics
An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.
Quantity
This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements.
:
:Number – Natural number – Integers – Rational numbers – Real numbers – Complex numbers – Hypercomplex numbers – Quaternions – Octonions – Sedenions – Hyperreal numbers – Surreal numbers – Ordinal numbers – Cardinal numbers – p-adic numbers – Integer sequences – Mathematical constants – Number names – Infinity – Base
Structure
:Pinning down ideas of size, symmetry, and mathematical structure.
:
:Abstract algebra – Number theory – Algebraic geometry – Group theory – Monoids – Analysis – Topology – Linear algebra – Graph theory – Universal algebra – Category theory – Order theory – Measure theory
Space
:A more visual approach to mathematics.
:
:Topology – Geometry – Trigonometry – Algebraic geometry – Differential geometry – Differential topology – Algebraic topology – Linear algebra – Fractal geometry
Change
:Ways to express and handle change in mathematical functions, and changes between numbers.
:
:Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions
Foundations and methods
:Approaches to understanding the nature of mathematics.
:philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols
Discrete mathematics
:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values.
:
:Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory
Applied mathematics
:Applied mathematics uses the full knowledge of mathematics to solve real-world problems.
:Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory
Important theorems
:These theorems have interested mathematicians and non-mathematicians alike.
:See list of theorems for more
:Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.
Important conjectures
See list of conjectures for more
:These are some of the major unsolved problems in mathematics.
:Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.
History and the world of mathematicians
See also list of mathematics history topics
:History of mathematics – Timeline of mathematics – Mathematicians – Fields medal – Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Mathematical abilities and gender issues
Mathematics and other fields
:Mathematics and architecture – Mathematics and education – Mathematics of musical scales
Common misconceptions
Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems.
Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature.
The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne.
Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter.
Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions.
Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences.
Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.
See also
- Mathematical game
- Mathematical problem
- Mathematical puzzle
- Puzzle
Bibliography
- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).
External links
- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
-
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ProbabilityThe word probability derives from the Latin probare (to prove, or to test).
Informally, probable is one of several words applied to uncertain events or knowledge,
being more or less interchangeable with likely, risky, hazardous, uncertain, and doubtful, depending on the context.
Chance, odds, and bet are other words expressing similar notions.
As with the theory of mechanics which assigns precise definitions to such everyday terms as work and force,
so the theory of probability attempts to quantify the notion of probable.
Historical remarks
The scientific study of probability is a modern development. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, but exact mathematical descriptions of use in those problems only arose much later.
The doctrine of probabilities dates to the correspondence of Pierre de Fermat and Blaise Pascal (1654). Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject. Jakob Bernoulli's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's Doctrine of Chances (1718) treated the subject as a branch of mathematics.
The theory of errors may be traced back to Roger Cotes's Opera Miscellanea (posthumous, 1722), but a memoir prepared by Thomas Simpson in 1755 (printed 1756) first applied the theory to the discussion of errors of observation. The reprint (1757) of this memoir lays down the axioms that positive and negative errors are equally probable, and that there are certain assignable limits within which all errors may be supposed to fall; continuous errors are discussed and a probability curve is given.
Pierre-Simon Laplace (1774) made the first attempt to deduce a rule for the combination of observations from the principles of the theory of probabilities. He represented the law of probability of errors by a curve , being any error and its probability, and laid down three properties of this curve: (1) It is symmetric as
to the -axis; (2) the -axis is an asymptote, the probability of the error being 0; (3) the area enclosed is 1, it being certain that an error exists. He deduced a formula for the mean of three observations. He also gave (1781) a formula for the law of facility of error (a term due to Lagrange, 1774), but one which led to unmanageable equations. Daniel Bernoulli (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.
The method of least squares is due to Adrien-Marie Legendre (1805), who introduced it in his Nouvelles méthodes pour la détermination des orbites des comètes (New Methods for Determining the Orbits of Comets). In ignorance of Legendre's contribution, an Irish-American writer, Robert Adrain, editor of "The Analyst" (1808), first deduced the law of facility of error,
:
and being constants depending on precision of observation. He gave two proofs, the second being essentially the same as John Herschel's (1850). Gauss gave the first proof which seems to have been known in Europe (the third after Adrain's) in 1809. Further proofs were given by Laplace (1810, 1812), Gauss (1823), James Ivory (1825, 1826), Hagen (1837), Friedrich Bessel (1838), Donkin (1844,
1856), and Morgan Crofton (1870). Other contributors were Ellis (1844), De Morgan (1864), Glaisher (1872), and Giovanni Schiaparelli (1875). Peters's (1856) formula for , the probable error of a single observation, is well known.
In the nineteenth century authors on the general theory included Laplace, Sylvestre Lacroix (1816), Littrow (1833), Adolphe Quetelet (1853), Richard Dedekind (1860), Helmert (1872), Hermann Laurent (1873), Liagre, Didion, and Karl Pearson. Augustus De Morgan and George Boole improved the exposition of the theory.
On the geometric side (see integral geometry) contributors to The Educational Times were influential (Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin).
Concepts
There is essentially one set of mathematical rules for manipulating probability; these rules are listed under "Formalization of probability" below.
(There are other rules for quantifying uncertainty,
such as the Dempster-Shafer theory and possibility theory,
but those are essentially different and not compatible with the laws of probability as they are usually understood.)
However, there is ongoing debate over what, exactly, the rules apply to; this is the topic of probability interpretations.
The general idea of probability is often divided into two related concepts:
- Aleatory probability, which represents the likelihood of future events whose occurrence is governed by some random physical phenomenon. This concept can be further divided into physical phenomena that are predictable, in principle, with sufficient information (see Determinism), and phenomena which are essentially unpredictable. Examples of the first kind include tossing dice or spinning a roulette wheel, and an example of the second kind is radioactive decay.
- Epistemic probability, which represents our uncertainty about propositions when one lacks complete knowledge of causative circumstances. Such propositions may be about past or future events, but need not be. Some examples of epistemic probability are to assign a probability to the proposition that a proposed law of physics is true, and to determine how "probable" it is that a suspect committed a crime, based on the evidence presented.
It is an open question whether aleatory probability is reducible to epistemic probability based on our inability to precisely predict every force that might affect the roll of a die, or whether such uncertainties exist in the nature of reality itself, particularly in quantum phenomena governed by Heisenberg's uncertainty principle. Although the same mathematical rules apply regardless of which interpretation is chosen, the choice has major implications for the way in which probability is used to model the real world.
Formalization of probability
Like other theories, the theory of probability is a representation of probabilistic concepts in formal terms -- that is, in terms that can be considered separately from their meaning.
These formal terms are manipulated by the rules of mathematics and logic, and any results are then interpreted or translated back into the problem domain.
There have been at least two successful attempts to formalize probability, namely the Kolmogorov formulation and the Cox formulation.
In Kolmogorov's formulation,
sets are interpreted as events and probability itself as a measure on a class of sets.
In Cox's formulation,
probability is taken as a primitive (that is, not further analyzed) and the emphasis is on constructing a consistent assignment of probability values to propositions.
In both cases,
the laws of probability are the same, except for technical details:
# a probability is a number between 0 and 1;
# the probability of an event or proposition and its complement must add up to 1; and
# the joint probability of two events or propositions is the product of the probability of one of them and the probability of the second, conditional on the first.
The reader will find an exposition of the Kolmogorov formulation in the probability theory article, and in the Cox's theorem article for Cox's formulation. See also the article on probability axioms.
For an algebraic alternative to Kolmogorov's approach, see algebra of random variables.
Representation and interpretation of probability values
The probability of an event is generally represented as a real number between 0 and 1, inclusive. An impossible event has a probability of exactly 0, and a certain event has a probability of 1, but the converses are not always true: probability 0 events are not always impossible, nor probability 1 events certain.
The rather subtle distinction between "certain" and "probability 1" is treated at greater length in the article on "almost surely".
Most probabilities that occur in practice are numbers between 0 and 1, indicating the event's position on the continuum between impossibility and certainty. The closer an event's probability is to 1, the more likely it is to occur.
For example, if two mutually exclusive events are assumed equally probable, such as a flipped coin landing heads-up or tails-up, we can express the probability of each event as "1 in 2", or, equivalently, "50%" or "1/2".
Probabilities are equivalently expressed as odds, which is the ratio of the probability of one event to the probability of all other events.
The odds of heads-up, for the tossed coin, are (1/2)/(1 - 1/2), which is equal to 1/1. This is expressed as "1 to 1 odds" and often written "1:1".
Odds a:b for some event are equivalent to probability a/(a+b).
For example, 1:1 odds are equivalent to probability 1/2, and 3:2 odds are equivalent to probability 3/5.
There remains the question of exactly what can be assigned probability, and how the numbers so assigned can be used; this is the question of probability interpretations.
There are some who claim that probability can be assigned to any kind of an uncertain logical proposition; this is the Bayesian interpretation.
There are others who argue that probability is properly applied only to random events as outcomes of some specified random experiment, for example sampling from a population; this is the frequentist interpretation.
There are several other interpretations which are variations on one or the other of those, or which have less acceptance at present.
Distributions
A probability distribution is a function that assigns probabilities to events or propositions. For any set of events or propositions there are many ways to assign probabilities, so the choice of one distribution or another is equivalent to making different assumptions about the events or propositions in question.
There are several equivalent ways to specify a probability distribution.
Perhaps the most common is to specify a probability density function.
Then the probability of an event or proposition is obtained by integrating the density function.
The distribution function may also be specified directly.
In one dimension, the distribution function is called the cumulative distribution function.
Probability distributions can also be specified via moments or the characteristic function, or in still other ways.
A distribution is called a discrete distribution if it is defined on a countable, discrete set, such as a subset of the integers.
A distribution is called a continuous distribution if it has a continuous distribution function, such as a polynomial or exponential function.
Most distributions of practical importance are either discrete or continuous, but there are examples of distributions which are neither.
Important discrete distributions include the discrete uniform distribution, the Poisson distribution, the binomial distribution, the negative binomial distribution and the Maxwell-Boltzmann distribution.
Important continuous distributions include the normal distribution, the gamma distribution, the Student's t-distribution, and the exponential distribution.
Probability in mathematics
Probability axioms form the basis for mathematical probability theory. Calculation of probabilities can often be determined using combinatorics or by applying the axioms directly. Probability applications include even more than statistics, which is usually based on the idea of probability distributions and the central limit theorem.
To give a mathematical meaning to probability, consider flipping a "fair" coin. Intuitively, the probability that heads will come up on any given coin toss is "obviously" 50%; but this statement alone lacks mathematical rigor - certainly, while we might expect that flipping such a coin 10 times will yield 5 heads and 5 tails, there is no guarantee that this will occur; it is possible for example to flip 10 heads in a row. What then does the number "50%" mean in this context?
One approach is to use the law of large numbers. In this case, we assume that we can perform any number of coin flips, with each coin flip being independent - that is to say, the outcome of each coin flip is unaffected by previous coin flips. If we perform N trials (coin flips), and let NH be the number of times the coin lands heads, then we can, for any N, consider the ratio NH/N.
As N gets larger and larger, we expect that in our example the ratio NH/N will get closer and closer to 1/2. This allows us to "define" the probability Pr(H) of flipping heads as the limit (mathematics), as N approaches infinity, of this sequence of ratios:
:
In actual practice, of course, we cannot flip a coin an infinite number of times; so in general, this formula most accurately applies to situations in which we have already assigned an a priori probability to a particular outcome (in this case, our assumption that the coin was a "fair" coin). The law of large numbers then says that, given Pr(H), and any arbitrarily small number ε, there exists some number n such that for all N > n,
:
In other words, by saying that "the probability of heads is 1/2", we mean that, if we flip our coin often enough, eventually the number of heads over the number of total flips will become arbitrarily close to 1/2; and will then stay at least as close to 1/2 for as long as we keep performing additional coin flips.
Note that a proper definition requires measure theory which provides means to cancel out those cases where
the above limit does not provide the "right" result or is even undefined by showing that those cases have a measure of zero.
The a priori aspect of this approach to probability is sometimes troubling when applied to real world situations. For example, in the play Rosencrantz and Guildenstern are Dead by Tom Stoppard, a character flips a coin which keeps coming up heads over and over again, a hundred times. He can't decide whether this is just a random event - after all, it is possible (although unlikely) that a fair coin would give this result - or whether his assumption that the coin is fair is at fault.
Remarks on probability calculations
The difficulty of probability calculations lie in determining the number of possible events, counting the occurrences of each event, counting the total number of possible events. Especially difficult is drawing meaningful conclusions from the probabilities calculated. An amusing probability riddle, the Monty Hall problem demonstrates the pitfalls nicely.
To learn more about the basics of probability theory, see the article on probability axioms and the article on Bayes' theorem that explains the use of conditional probabilities in case where the occurrence of two events is related.
Applications of probability theory to everyday life
A major effect of probability theory on everyday life is in risk assessment and in trade on commodity markets. Governments typically apply probability methods in environment regulation where it is called "pathway analysis", and are often measuring well-being using methods that are stochastic in nature, and choosing projects to undertake based on their perceived probable effect on the population as a whole, statistically. It is not correct to say that statistics are involved in the modelling itself, as typically the assessments of risk are one-time and thus require more fundamental probability models, e.g. "the probability of another 9/11". A law of small numbers tends to apply to all such choices and perception of the effect of such choices, which makes probability measures a political matter.
A good example is the effect of the perceived probability of any widespread Middle East conflict on oil prices - which have ripple effects in the economy as a whole. An assessment by a commodity trade that a war is more likely vs. less likely sends prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are not assessed independently nor necessarily very rationally. The theory of behavioral finance emerged to describe the effect of such groupthink on pricing, on policy, and on peace and conflict.
It can reasonably be said that the discovery of rigorous methods to assess and combine probability assessments has had a profound effect on modern society. A good example is the application of game theory, itself based strictly on probability, to the Cold War and the mutual assured destruction doctrine. Accordingly, it may be of some importance to most citizens to understand how odds and probability assessments are made, and how they contribute to reputations and to decisions, especially in a democracy.
Another significant application of probability theory in everyday life is reliability. Many consumer products, such as automobiles and consumer electronics, utilize reliability theory in the design of the product in order to reduce the probability of failure. The probability of failure is also closely associated with the product's warranty.
See also
- Bayesian probability
- Bernoulli process
- Cox's theorem
- Decision theory
- Fuzzy measure theory
- Game of chance
- Game theory
- Information theory
- Law of averages
- Law of large numbers
- Measure theory
- Normal distribution
- Random fields
- Random variable
- Statistics
- List of statistical topics
- Stochastic process
- Wiener process
- Important publications in probability
External links
- [http://www.cut-the-knot.org/probability.shtml A Collection of articles on Probability, many of which are accompanied by Java simulations] at cut-the-knot
- Edwin Thompson Jaynes. Probability Theory: The Logic of Science. Preprint: Washington University, (1996). -- [http://omega.albany.edu:8008/JaynesBook.html HTML] and [http://bayes.wustl.edu/etj/prob/book.pdf PDF]
- [http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html An online probability textbook which uses computer programming as a teaching aid]
- "[http://www.npr.org/display_pages/features/feature_1697475.html The Not So Random Coin Toss], Mathematicians Say Slight but Real Bias Toward Heads". NPR.
- [http://www.benbest.com/science/theodds.html Figuring the Odds (Probability Puzzles)]
- [http://etext.lib.virginia.edu/cgi-local/DHI/dhi.cgi?id=dv1-43 Dictionary of the History of Ideas:] Certainty in Seventeenth-Century Thought
- [http://etext.lib.virginia.edu/cgi-local/DHI/dhi.cgi?id=dv1-44 Dictionary of the History of Ideas:] Certainty since the Seventeenth Century
Quotations
- Damon Runyon, "It may be that the race is not always to the swift, nor the battle to the strong - but that is the way to bet."
- Pierre-Simon Laplace "It is remarkable that a science which began with the consideration of games of chance should have become the most important object of human knowledge." Théorie Analytique des Probabilités, 1812.
- Richard von Mises "The unlimited extension of the validity of the exact sciences was a characteristic feature of the exaggerated rationalism of the eighteenth century" (in reference to Laplace). Probability, Statistics, and Truth, p 9. Dover edition, 1981 (republication of second English edition, 1957).
Category:Applied mathematics
Category:Decision theory
Category:Probability theory
ja:確率
simple:Probability
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Stochastic processIn the mathematics of probability, a stochastic process is a random function. In the most common applications, the domain over which the function is defined is a time interval (a stochastic process of this kind is called a time series in applications) or a region of space (a stochastic process being called a random field).
Familiar examples of time series include stock market and exchange rate fluctuations, signals such as speech, audio and video; medical data such as a patient's EKG, EEG, blood pressure or temperature; and random movement such as Brownian motion or random walks. Examples of random fields include static images, random topographies (landscapes), or composition variations of an inhomogeneous material.
Definition
A stochastic process is a random function, that is a random variable X defined on a probability space (Ω , Pr) with values in a space of functions F. The space F in turn consists of functions I → D. Thus a stochastic process can also be regarded as
an indexed collection of random variables , where the index i ranges through an index set I, defined on the probability space (Ω, Pr) and taking values on the same codomain D (often the real numbers R). This view of a stochastic process as an indexed collection of random variables is the most common one.
A notable special case is where the index set is a discrete set I, often the nonnegative integers .
In a continuous stochastic process the index set is continuous (usually space or time), resulting in a uncountably infinite number of random variables.
Each point in the sample space Ω corresponds to a particular value for each of the random variables and the resulting function (mapping a point in the index set to the value of the random variable attached to it) is known as a realisation of the stochastic process. In the case the index family is a real (finite or infinite) interval, the resulting function is called a sample path.
A particular stochastic process is determined by specifying the joint probability distributions of the various random variables.
Stochastic processes may be defined in higher dimensions by attaching a multivariate random variable to each point in the index set, which is equivalent to using a multidimensional index set. Indeed a multivariate random variable can itself be viewed as a stochastic process with index set .
Examples
The paradigm continuous stochastic process is that of the Wiener process. In its original form the problem was concerned with a particle floating on a liquid surface, receiving "kicks" from the molecules of the liquid. The particle is then viewed as being subject to a random force which, since the molecules are very small and very close together, is treated as being continuous and, since the particle is constrained to the surface of the liquid by surface tension, is at each point in time a vector parallel to the surface. Thus the random force is described by a two component stochastic process; two real-valued random variables are associated to each point in the index set, time, (note that since the liquid is viewed as being homogeneous the force is independent of the spatial coordinates) with the domain of the two random variables being R, giving the x and y components of the force. A treatment of Brownian motion generally also includes the effect of viscosity, resulting in an equation of motion known as the Langevin equation.
If the index set of the process N the natural numbers, and the range is R (the real numbers), there are some natural questions to ask about the sample sequences of a process i ∈ N, where a sample sequence is
i ∈ N.
# What is the probability that each sample sequence is bounded?
# What is the probability that each sample sequence is monotonic?
# What is the probability that each sample sequence has a limit as as the index approaches ∞?
# What is the probability that the series obtained from a sample sequence from converges?
# What is the probability distribution of the sum?
Similarly, if the index space I is a finite or infinite interval, we can ask about the sample paths t ∈ I
# What is the probability that it is bounded/integrable/continuous/differentiable...?
# What is the probability that it has a limit at ∞
# What is the probability distribution of the integral?
Interesting special cases
- Homogeneous processes: processes where the domain has some symmetry and the finite-dimensional probability distributions also have that symmetry. Special cases include stationary processes, also called time-homogeneous.
- Bernoulli processes: discrete-time processes with two possible states.
- Bernoulli schemes: discrete-time processes with N possible states; every stationary process in N outcomes is a Bernoulli scheme, and vice-versa.
- Processes with independent increments: processes where the domain is at least partially ordered and, if , all the variables are independent. Markov chains are a special case.
- See also continuous-time Markov chain.
- Markov processes are those in which the future is conditionally independent of the past given the present.
- Point processes: random arrangements of points in a space . They can be modelled as stochastic processes where the domain is a sufficiently large family of subsets of , ordered by inclusion; the range is the set of natural numbers; and, if A is a subset of B, with probability 1.
- Gaussian processes: processes where all linear combinations of coordinates are normally distributed random variables.
- Poisson processes
- Gauss-Markov processes: processes that are both Gaussian and Markov
- Martingales -- processes with constraints on the expectation
- Galton-Watson processes
- Elevator paradox
- Branching processes
- Many stochastic processes are Lévy processes.
Constructing stochastic processes
In the ordinary axiomatization of probability theory by means of measure theory, the problem is to construct a sigma-algebra of measurable subsets of the space of all functions, and then put a finite measure on it. For this purpose one traditionally uses a method called Kolmogorov extension.
There is at least one alternative axiomatization of probability theory by means of expectations on C-star algebras of random variables. In this case the method goes by the name of Gelfand-Naimark-Segal construction.
This is analogous to the two approaches to measure and integration, where one has the choice to construct measures of sets first and define integrals later, or construct integrals first and define set measures as integrals of characteristic functions.
The Kolmogorov extension
The Kolmogorov extension proceeds along the following lines: assuming that a probability measure on the space of all functions exists, then it can be used to specify the probability distribution of finite-dimensional random variables . Now, from this n-dimensional probability distribution we can deduce an (n − 1)-dimensional marginal probability distribution for . There is an obvious compatibility condition, namely, that this marginal probability distribution be the same as the one derived from the full-blown stochastic process. When this condition is expressed in terms of probability densities, the result is called the Chapman-Kolmogorov equation.
The Kolmogorov extension theorem guarantees the existence of a stochastic process with a given family of finite-dimensional probability distributions satisfying the Chapman-Kolmogorov compatibility condition.
Separability, or what the Kolmogorov extension does not provide
Recall that, in the Kolmogorov axiomatization, measurable sets are the sets which have a probability or, in other words, the sets corresponding to yes/no questions that have a probabilistic answer.
The Kolmogorov extension starts by declaring to be measurable all sets of functions where finitely many coordinates are restricted to lie in measurable subsets of . In other words, if a yes/no question about f can be answered by looking at the values of at most finitely many coordinates, then it has a probabilistic answer.
In measure theory, if we have a countably infinite collection of measurable sets, then the union and intersection of all of them is a measurable set. For our purposes, this means that yes/no questions that depend on countably many coordinates have a probabilistic answer.
The good news is that the Kolmogorov extension makes it possible to construct stochastic processes with fairly arbitrary finite-dimensional distributions. Also, every question that one could ask about a sequence has a probabilistic answer when asked of a random sequence. The bad news is that certain questions about functions on a continuous domain don't have a probabilistic answer. One might hope that the questions that depend on uncountably many values of a function be of little interest, but the really bad news is that virtually all concepts of calculus are of this sort. For example:
#boundedness
#continuity
#differentiability
all require knowledge of uncountably many values of the function.
One solution to this problem is to require that the stochastic process be separable. In other words, that there be some countable set of coordinates whose values determine the whole random function f.
See also
- List of stochastic processes topics
References
-
Function (mathematics)In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). The concept of a function is fundamental to virtually every branch of mathematics and every quantitative science.
The terms function, mapping, map and transformation are usually used synonymously. The term operation is frequently used for binary functions; functions whose domain is a set of functions, or a vector space, are often called operators (see also operator (programming)).
Intuitive introduction
Essentially, a function is a "rule" or procedure that assigns an "output" value to each given "input" value. The following are examples of functions:
- In a group of people, each person has a favorite colour—from the set of red, orange, yellow, green, cyan, blue, indigo, or violet. Here, the input is the person, and the output is one of the 8 colours. The favorite colour is a function of the person. For example, John has favorite colour red, while Kim has favorite colour violet. Note that more than one person may be associated with a given colour (e.g., John and Kim may both like red), but one person cannot have more or less than one favorite color.
- A stone is dropped from different stories of a tall building. The dropped stone may take 2 seconds to fall from the second story, and 4 seconds to fall from the 8th story. Here, the input is the story, and the output is the number of seconds. The relevant function describes the relationship between the time it takes the stone to reach the ground and the story. (See acceleration)
The "rule" defining a function can be specified by a formula, a relationship, or simply a table listing the outputs against inputs. The most important feature of a function is that it is consistent, or deterministic, always producing the same output from a given input. In this way, a function may be thought of as a mechanism or "machine" (a "black box") consistently converting a given valid input into its unique associated output. In certain technical contexts, the input is often called the argument of the function, and the output the value of the function.
A very common type of function occurs when the argument (input) and the value (output) are both numbers, the functional relationship is expressed by a formula, and the value (output) of the function is obtained by direct substitution of the argument into the formula. Consider for example
:
which for any number x, assigns to x the associated value the square of x.
A straightforward generalization is to allow functions depending on several arguments. For instance,
:
is a function which takes the input, two expressions x and y, and assigns to it its product (output), xy. It might seem that this is not really a function as we described above, because this "rule" depends on two inputs. However, if we think of the two inputs together as a single pair (x, y), then we can interpret g as a function -- the argument (unified single input) is the ordered pair (x, y), and the function value (output) is xy.
Such functions whose input consists of ordered pairs are called "binary" or "2-ary".
In the sciences, we often encounter functions that are not given by (known) formulas. Consider for instance the temperature distribution on earth over time: this is a function which takes location and time as arguments and gives as output value the temperature at the indicated location at the indicated moment in time.
We have seen that the intuitive notion of function is not limited to computations using single numbers and not even limited to computations; the mathematical notion of function is still more general and is not limited to situations involving numbers. Rather, a function links a "domain" (set of inputs) to a "codomain" (set of possible outputs) in such a way that every element of the domain is associated to precisely one element of the codomain. Functions are abstractly defined as certain relations, as will be seen below. Because of this generality, the function concept is fundamental to virtually every branch of mathematics and the quantitative sciences.
History
As a mathematical term, "function" was coined by Leibniz in 1694, to describe a quantity related to a curve, such as a curve's slope or a specific point of a curve. The functions Leibniz considered are today called differentiable functions, and they are the type of function most frequently encountered by nonmathematicians. For this type of function, one can talk about limits and derivatives; both are measurements of the change of output values associated to a change of input values, and these measurements are the basis of calculus.
The word function was later used by Euler during the mid-18th century to describe an expression or formula involving various arguments, e.g. f(x) = sin(x) + x3.
During the 19th century, mathematicians started to formalize all the different branches of mathematics. Weierstrass advocated building calculus on arithmetic rather than on geometry, which favoured Euler's definition over Leibniz's (see arithmetization of analysis).
By broadening the definition of functions, mathematicians were then able to study "strange" mathematical objects such as continuous functions that are nowhere differentiable. These functions were first thought to be only theoretical curiosities, and they were collectively called "monsters" as late as the turn of the 20th century. However, powerful techniques from functional analysis have shown that these functions are in some sense "more common" than differentiable functions. Such functions have since been applied to the modeling of physical phenomena such as Brownian motion.
Towards the end of the 19th century, mathematicians started trying to formalize all of mathematics using set theory, and they sought to define every mathematical object as a set. Dirichlet and Lobachevsky independently and almost simultaneously gave the modern "formal" definition of function (see formal definition below).
In this definition, a function is a special case of a relation. In most cases of practical interest, however, the differences between the modern definition and Euler's definition are negligible.
The notion of function as a rule for computing, rather than a special kind of relation, has been formalized in mathematical logic and theoretical computer science by means of several systems, including the lambda calculus, the theory of recursive functions and the Turing machine.
Formal definition
Formally a function f from a set X to a set Y, written f : X → Y, is an ordered triple (X, Y, G(f)), where G(f) is a subset of the cartesian product X × Y, such that for each x in X, there is a unique y in Y such that the ordered pair (x, y) is in G(f). X is called the domain of f, Y is called the codomain of F, and G(f) is called the graph of f. For each "input value" x in the domain, the corresponding unique "output value" y in the codomain is denoted by f(x).
Equivalently a function f can be defined as a relation between X and Y which satisfies:
# f is total, or entire: for all x in X, there exists a y in Y such that x f y (x is f-related to y), i.e. for each input value, there is at least one output value in Y.
# f is many-to-one, or functional: if x f y and x f z, then y = z. i.e., many input values can be related to one output value, but one input value cannot be related to many output values.
A relation between X and Y that satisfies condition (1) is a multivalued function. Every function is a multivalued function, but not every multivalued function is a function. A relation between X and Y that satisfies condition (2) is a partial function. Every function is a partial function, but not every partial function is a function. In this encyclopedia, the term "function" will mean a relation satisfying both conditions (1) and (2), unless otherwise stated.
Consider the following three examples:
| image:notMap1.png | This relation is total but not many-to-one; the element 3 in X is related to two elements b and c in Y. Therefore, this is a multivalued function, but not a function. |
| image:notMap2.png | This relation is many-to-one but not total; the element 1 in X is not related to any element of Y. Therefore, this is a partial function, but not a function. | |
| image:mathmap2.png | This relation is both total and many-to-one, and so it is a function from X to Y. Note that the emphasis is on "-to-one" as "many" may actually mean "one". The function can be given explicitly by specifying its graph G(f) = or as
: | | | |
