Probability theory is the branch of mathematics concerned with analysis of random phenomena.^[1] The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. Although an individual coin toss or the roll of a die is a random event, if repeated many times the sequence of random events will exhibit certain statistical patterns, which can be studied and predicted. Two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... A numeric sequence is said to be statistically random when it contains no recognizable patterns or regularities; sequences such as the results of an ideal die roll, or the digits of Pi (as far as we can tell) exhibit statistical randomness. ... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... In the mathematics of probability, a stochastic process is a random function. ... In probability theory, an event is a set of outcomes (a subset of the sample space) to which a probability is assigned. ... Determinism is the philosophical proposition that every event, including human cognition and behavior, decision and action, is causally determined by an unbroken chain of prior occurrences. ... // The law of large numbers (LLN) is any of several theorems in probability. ... A central limit theorem is any of a set of weak-convergence results in probability theory. ...

As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to description of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics. This article is about the field of statistics. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ...

History

The mathematical theory of probability has its roots in attempts to analyse games of chance by Gerolamo Cardano in the sixteenth century, and by Pierre de Fermat and Blaise Pascal in the seventeenth century (for example the "problem of points"). Probability is the likelihood that something is the case or will happen. ... A game of chance is a game whose outcome is strongly influenced by some randomizing device, and upon which contestants frequently wager money. ... Gerolamo Cardano. ... Pierre de Fermat Pierre de Fermat IPA: (August 17, 1601 – January 12, 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to modern calculus. ... Blaise Pascal (pronounced ), (June 19, 1623 – August 19, 1662) was a French mathematician, physicist, and religious philosopher. ... The problem of points, also called the problem of division of the stakes, is a classical problem in probability theory. ...

Initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial. Eventually, analytical considerations compelled the incorporation of continuous variables into the theory. This culminated in modern probability theory, the foundations of which were laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of sample space, introduced by Richard von Mises, and measure theory and presented his axiom system for probability theory in 1933. Fairly quickly this became the undisputed axiomatic basis for modern probability theory.^[2] Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ... Analysis has its beginnings in the rigorous formulation of calculus. ... Andrey Nikolaevich Kolmogorov (Андре́й Никола́евич Колмого́ров) (kahl-mah-GAW-raff) (April 25, 1903 in Tambov - October 20, 1987 in Moscow) was a Russian mathematician... In probability theory, the sample space, often denoted S, Ω or U (for universe), of an experiment or random trial is the set of all possible outcomes. ... Richard von Mises. ... In mathematics, a measure is a function that assigns a number, e. ... The probability of some event (denoted ) is defined with respect to a universe or sample space of all possible elementary events in such a way that must satisfy the Kolmogorov axioms. ... In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ...

Treatment

Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The more mathematically advanced measure theory based treatment of probability covers both the discrete, the continuous, any mix of these two and more.

Discrete probability distributions

Discrete probability theory deals with events that occur in countable sample spaces. In mathematics the term countable set is used to describe the size of a set, e. ...

Examples: Throwing dice, experiments with decks of cards, and random walk. Dice (the plural of die, from Old French de, from Latin datum something given or played [1]) are small polyhedral objects, usually cubical, used for generating random numbers or other symbols. ... Category: Possible copyright violations ... Example of eight random walks in one dimension starting at 0. ...

Classical definition: Initially the probability of an event to occur was defined as number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space.

For example, if the event is "occurrence of an even number when a die is rolled", the probability is given by , since 3 faces out of the 6 have even numbers and each face has the same probability of appearing. Dice (the plural of die, from Old French de, from Latin datum something given or played [1]) are small polyhedral objects, usually cubical, used for generating random numbers or other symbols. ...

Modern definition: The modern definition starts with a set called the sample space, which relates to the set of all possible outcomes in classical sense, denoted by . It is then assumed that for each element , an intrinsic "probability" value is attached, which satisfies the following properties: In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In probability theory, the sample space, often denoted S, Ω or U (for universe), of an experiment or random trial is the set of all possible outcomes. ... In mathematics, an element (also called a member) is an object contained in a set (or more generally a class). ...

That is, the probability function f(x) lies between zero and one for every value of x in the sample space Ω, and the sum of f(x) over all values x in the sample space Ω is exactly equal to 1. An event is defined as any subset of the sample space Ω,. The probability of the event defined as In probability theory, an event is a set of outcomes (a subset of the sample space) to which a probability is assigned. ... “Superset” redirects here. ...

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

The function mapping a point in the sample space to the "probability" value is called a probability mass function abbreviated as pmf. The modern definition does not try to answer how probability mass functions are obtained; instead it builds a theory that assumes their existence. In probability theory, a probability mass function (abbreviated pmf) gives the probability that a discrete random variable is exactly equal to some value. ...

Continuous probability distributions

Continuous probability theory deals with events that occur in a continuous sample space.

Classical definition: The classical definition breaks down when confronted with the continuous case. See Bertrand's paradox. Bertrands paradox is a problem in probability theory. ...

Modern definition: If the sample space is the real numbers (), then a function called the cumulative distribution function (or cdf) is assumed to exist, which gives for a random variable X. That is, F(x) returns the probability that X will be less than or equal to x. Please refer to Real vs. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ...

The cdf must satisfy the following properties.

1. is a monotonically non-decreasing, right-continuous function

If is differentiable, then the random variable X is said to have a probability density function or pdf or simply density . A monotonically increasing function (it is strictly increasing on the left and just non-decreasing on the right). ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ...

For a set , the probability of the random variable X being in is defined as

In case the probability density function exists, then it can be written as

Whereas the pdf exists only for continuous random variables, the cdf exists for all random variables (including discrete random variables) that take values on .

These concepts can be generalized for multidimensional cases on and other continuous sample spaces. 2-dimensional renderings (ie. ...

Measure theoretic probability theory

The raison d'être of the measure theoretic treatment of probability is that it unifies the discrete and the continuous, and makes the difference a question of which measure is used. Furthermore, it covers distributions that are neither discrete nor continuous. An example of such distributions could be a mix of discrete and continuous distributions, e.g., a sum of a discrete and a continuous random variable will neither have a pmf nor a pdf. Other distributions may not even be a mix: For example, the Cantor distribution has no point mass and no density. The modern approach to probability theory solves these problems using measure theory to define the probability space: The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematics, the definition of the probability space is the foundation of probability theory. ...

Given any set Ω, (also called sample space) and a σ-algebra on it, a measure P is called a probability measure if In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ... In mathematics the concept of a measure generalizes notions such as length, area, and volume (but not all of its applications have to do with physical sizes). ...

1. is non-negative

If is a Borel σ-algebra then there is a unique probability measure on for any cdf, and vice versa. The measure corresponding to a cdf is said to be induced by the cdf. This measure coincides with the pmf for discrete variables, and pdf for continuous variables, making the measure theoretic approach free of fallacies. In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space X is a σ-algebra of subsets of X associated to the topology of X. In the mathematics literature, there are at least two inequivalent definitions of this σ-algebra: The minimal σ-algebra containing the open sets. ...

The probability of a set in the σ-algebra is defined as

where the integration is with respect to the measure induced by .

Along with providing better understanding and unification of discrete and continuous probabilities, measure theoretic treatment also allows us to work on probabilities outside , as in the theory of stochastic processes. For example to study Brownian motion, probability is defined on a space of functions. In the mathematics of probability, a stochastic process is a random function. ... Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors. ...

Probability distributions

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 gained special importance in probability theory. Some fundamental discrete distributions are the discrete uniform, Bernoulli, binomial, negative binomial, Poisson and geometric distributions. Important continuous distributions include the continuous uniform, normal, exponential, gamma and beta distributions. In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In probability theory and statistics, the discrete uniform distribution is a discrete probability distribution that can be characterized by saying that all values of a finite set of possible values are equally probable. ... In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability . ... In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. ... In probability and statistics the negative binomial distribution is a discrete probability distribution. ... In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate, and are independent of the time since the last event. ... In probability theory and statistics, the geometric distribution is either of two discrete probability distributions: the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}, or the probability distribution of the number Y = X âˆ’ 1 of failures before... In probability theory and statistics, the continuous uniform distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distributions support are equally probable. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. ... In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function (pdf) defined on the interval [0, 1]: where α and β are parameters that must be greater than zero and B is the beta function. ...

Convergence of random variables

Main article: Convergence of random variables

In probability theory, there are several notions of convergence for random variables. They are listed below in the order of strength, i.e., any subsequent notion convergence in the list implies convergence according to all of the preceding notions. In probability theory, there exist several different notions of convergence of random variables. ... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ...

Convergence in distribution: As the name implies, a sequence of random variables converges to the random variable in distribution if their respective cumulative distribution functions converge to the cumulative distribution function of , wherever is continuous.

Most common short hand notation:

Weak convergence: The sequence of random variables is said to converge towards the random variable weakly if for every ε > 0. Weak convergence is also called convergence in probability.

Most common short hand notation:

Strong convergence: The sequence of random variables is said to converge towards the random variable strongly if Strong convergence is also known as almost sure convergence.

Most common short hand notation:

Intuitively, strong convergence is a stronger version of the weak convergence, and in both cases the random variables show an increasing correlation with . However, in case of convergence in distribution, the realized values of the random variables do not need to converge, and any possible correlation among them is immaterial. In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...

Law of large numbers

Main article: Law of large numbers

Common intuition suggests that if a fair coin is tossed many times, then roughly half of the time it will turn up heads, and the other half it will turn up tails. Furthermore, the more often the coin is tossed, the more likely it should be that the ratio of the number of heads to the number of tails will approach unity. Modern probability provides a formal version of this intuitive idea, known as the law of large numbers. This law is remarkable because it is nowhere assumed in the foundations of probability theory, but instead emerges out of these foundations as a theorem. Since it links theoretically-derived probabilities to their actual frequency of occurrence in the real world, the law of large numbers is considered as a pillar in the history of statistical theory.[1] // The law of large numbers (LLN) is any of several theorems in probability. ...

The law of large numbers (LLN) states that the sample average of (independent and identically distributed random variables with finite expectation μ) converges towads the theoretical expectation μ.

It is in the different forms of convergence of random variables that separates the weak and the strong law of large numbers In probability theory, there exist several different notions of convergence of random variables. ...

Weak law:

Strong law:

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

Putting this in terms of random variables and LLN we have are independent Bernoulli random variables taking values 1 with probability p and 0 with probability 1-p. E(Y_i) = p for all i and it follows from LLN that converges to p almost surely. In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability . ... In probability theory, an event happens almost surely (a. ...

Central limit theorem

Main article: Central limit theorem

The central limit theorem is the reason for the ubiquitous occurrence of the normal distribution in nature; it is one of the most celebrated theorems in probability and statistics.^{[citation needed]} A central limit theorem is any of a set of weak-convergence results in probability theory. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ...

The theorem states that the average of many independent and identically distributed random variables with finite variance tends towards a normal distribution irrespective of the distribution followed by the original random variables. Formally, let be independent random variables with means , and variances Then the sequence of random variables In mathematics, an average or central tendency of a set (list) of data refers to a measure of the middle of the data set. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... In statistics, mean has two related meanings: the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ... This article is about mathematics. ...

converges in distribution to a standard normal random variable. The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields, especially in physics and engineering. ...

See also

In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... This article is about mathematics. ... Fuzzy logic is derived from fuzzy set theory dealing with reasoning that is approximate rather than precisely deduced from classical predicate logic. ... Fuzzy measure theory considers a number of special classes of measures, each of which is characterized by a special property. ... Terms in statistics and probability theory : Concerned fields Probability theory Algebra of random variables (linear algebra) Statistics Measure theory Estimation theory Probability interpretations: Bayesian probability (or personal probability) Frequency probability Eclectic probability Glossary Atomic event : another name for elementary event. ... Look up likelihood in Wiktionary, the free dictionary. ... This is a list of probability topics, by Wikipedia page. ... // Probability The Doctrine of Chances Author: Abraham de Moivre Publication data: 1738 (2nd ed. ... Please add any Wikipedia articles related to statistics that are not already on this list. ... In probability theory and statistics, some special forms of mathematical notation are of interest : Random variables (for example, the height of students) are written in upper case. ... Predictive modelling is the process by which a model is created or chosen to try and best predict the probability of an outcome. ... The aim of a probabilistic logic (or probability logic) is to combine the capacity of probability theory to handle uncertainty with the capacity of deductive logic to exploit structure. ... The word probability has been used in a variety of ways since it was first coined in relation to games of chance. ... In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs. ...

Bibliography

• 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.

• Andrei Nikolajevich Kolmogorov (1950). Foundations of the Theory of Probability. 

The modern measure-theoretic foundation of probability theory; the original German version (Grundbegriffe der Wahrscheinlichkeitrechnung) appeared in 1933.

• Patrick Billingsley (1979). Probability and Measure. New York, Toronto, London: John Wiley and Sons. 

• Henk Tijms (2004). Understanding Probability. Cambridge Univ. Press. 

A lively introduction to probability theory for the beginner.

• Gut, Allan (2005). Probability: A Graduate Course. Springer-Verlag. ISBN 0387228330. 

References