Probability theory

Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event.

Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion.

Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem.

As a mathematical foundation for statistics, 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 in statistical mechanics. A great discovery of twentieth-century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics.[2]

History of probability

The mathematical theory of probability has its roots in attempts to analyze 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"). Christiaan Huygens published a book on the subject in 1657[3] and in the 19th century, Pierre Laplace completed what is today considered the classic interpretation.[4]

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, on foundations laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of sample space, introduced by Richard von Mises, and measure theory and presented his axiom system for probability theory in 1933. This became the mostly undisputed axiomatic basis for modern probability theory; but, alternatives exist, such as the adoption of finite rather than countable additivity by Bruno de Finetti.[5]

Treatment

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

Consider an experiment that can produce a number of outcomes. The set of all outcomes is called the sample space of the experiment. The power 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 die rolls. These collections are called events. 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 a way 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 a probability 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.[6]

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 those elementary events have a number assigned to them. This is done using a random variable. A random variable is a function that assigns to each elementary event in the sample space a real number. This function is usually denoted by a capital letter.[7] In the case of a die, the assignment of a number to a certain elementary events can be done using the identity function. This does not always work. For example, when flipping a coin the two possible outcomes are "heads" and "tails". In this example, the random variable X could assign to the outcome "heads" the number "0" () and to the outcome "tails" the number "1" ().

Discrete probability distributions

NYW-DK-Poisson(5)
The Poisson distribution, a discrete probability distribution.

Discrete probability theory deals with events that occur in countable sample spaces.

Examples: Throwing dice, experiments with decks of cards, random walk, and tossing coins

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: see Classical definition of probability.

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.

Modern definition: The modern definition starts with a finite or countable 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:

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 equal to 1. An event is defined as any subset of the sample space . The probability of the event is defined as

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.

Continuous probability distributions

Gaussian distribution 2
The normal 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. See Bertrand's paradox.

Modern definition: If the outcome space of a random variable X is the set of real numbers () or a subset thereof, then a function called the cumulative distribution function (or cdf) exists, defined by . That is, F(x) returns the probability that X will be less than or equal to x.

The cdf necessarily satisfies the following properties.

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

If is absolutely continuous, i.e., its derivative exists and integrating the derivative gives us the cdf back again, then the random variable X is said to have a probability density function or pdf or simply density

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

In case the probability density function exists, this 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 in

These concepts can be generalized for multidimensional cases on and other continuous sample spaces.

Measure-theoretic probability theory

The raison d'être of the measure-theoretic treatment of probability is that it unifies the discrete and the continuous 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 , where is the Dirac delta function.

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

Given any set (also called sample space) and a σ-algebra on it, a measure defined on is called a probability measure if

If is the Borel σ-algebra on the set of real numbers, 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.

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.

When it's convenient to work with a dominating measure, the Radon-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 a counting measure over the set of all possible outcomes. Densities for absolutely continuous distributions are usually defined as this derivative with respect to the Lebesgue 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

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.

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 of convergence in the list implies convergence according to all of the preceding notions.

Weak convergence
A sequence of random variables converges weakly to the random variable if their respective cumulative distribution functions converge to the cumulative distribution function of , wherever is continuous. Weak convergence is also called convergence in distribution.
Most common shorthand notation:
Convergence in probability
The sequence of random variables is said to converge towards the random variable in probability if for every ε > 0.
Most common shorthand 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 shorthand notation:

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

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 theory provides a formal version of this intuitive idea, known as the law 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.[8]

The law of large numbers (LLN) states that the sample average

of a sequence of independent and identically distributed random variables converges towards their common expectation , provided that the expectation of is finite.

It is in the different forms of convergence of random variables that separates the weak and the strong law of large numbers

Weak law: for
Strong law: for

It follows from the 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.

For example, if are independent Bernoulli random variables taking values 1 with probability p and 0 with probability 1-p, then for all i, so that converges to p almost surely.

Central limit theorem

"The central limit theorem (CLT) is one of the great results of mathematics." (Chapter 18 in[9]) It explains the ubiquitous occurrence of the normal distribution in nature.

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 mean and variance Then the sequence of random variables

converges in distribution to a standard normal random variable.

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

See also

Notes

  1. ^ Inferring From Data
  2. ^ "Why is quantum mechanics based on probability theory?". StackExchange. July 1, 2014.
  3. ^ Grinstead, Charles Miller; James Laurie Snell. "Introduction". Introduction to Probability. pp. vii.
  4. ^ Hájek, Alan. "Interpretations of Probability". Retrieved 2012-06-20.
  5. ^ ""The origins and legacy of Kolmogorov's Grundbegriffe", by Glenn Shafer and Vladimir Vovk" (PDF). Retrieved 2012-02-12.
  6. ^ Ross, Sheldon (2010). A First Course in Probability (8th ed.). Pearson Prentice Hall. pp. 26–27. ISBN 978-0-13-603313-4. Retrieved 2016-02-28.
  7. ^ 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.
  8. ^ "Leithner & Co Pty Ltd - Value Investing, Risk and Risk Management - Part I". Leithner.com.au. 2000-09-15. Archived from the original on 2014-01-26. Retrieved 2012-02-12.
  9. ^ David Williams, "Probability with martingales", Cambridge 1991/2008

References

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

External links

Almost surely

In probability theory, one says that an event happens almost surely (sometimes abbreviated as a.s.) if it happens with probability one. In other words, the set of possible exceptions may be non-empty, but it has probability zero. The concept is precisely the same as the concept of "almost everywhere" in measure theory.

In probability experiments on a finite sample space, there is often no difference between almost surely and surely. However, the distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability zero.

Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, and the continuity of the paths of Brownian motion.

The terms almost certainly (a.c.) and almost always (a.a.) are also used. Almost never describes the opposite of almost surely: an event that happens with probability zero happens almost never.

Bayesian probability

Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief.The Bayesian interpretation of probability can be seen as an extension of propositional logic that enables reasoning with hypotheses, i.e., the propositions whose truth or falsity is uncertain. In the Bayesian view, a probability is assigned to a hypothesis, whereas under frequentist inference, a hypothesis is typically tested without being assigned a probability.

Bayesian probability belongs to the category of evidential probabilities; to evaluate the probability of a hypothesis, the Bayesian probabilist specifies some prior probability, which is then updated to a posterior probability in the light of new, relevant data (evidence). The Bayesian interpretation provides a standard set of procedures and formulae to perform this calculation.

The term Bayesian derives from the 18th century mathematician and theologian Thomas Bayes, who provided the first mathematical treatment of a non-trivial problem of statistical data analysis using what is now known as Bayesian inference. Mathematician Pierre-Simon Laplace pioneered and popularised what is now called Bayesian probability.

Characteristic function (probability theory)

In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables.

In addition to univariate distributions, characteristic functions can be defined for vector or matrix-valued random variables, and can also be extended to more generic cases.

The characteristic function always exists when treated as a function of a real-valued argument, unlike the moment-generating function. There are relations between the behavior of the characteristic function of a distribution and properties of the distribution, such as the existence of moments and the existence of a density function.

Event (probability theory)

In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. A single outcome may be an element of many different events, and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. An event defines a complementary event, namely the complementary set (the event not occurring), and together these define a Bernoulli trial: did the event occur or not?

Typically, when the sample space is finite, any subset of the sample space is an event (i.e. all elements of the power set of the sample space are defined as events). However, this approach does not work well in cases where the sample space is uncountably infinite. So, when defining a probability space it is possible, and often necessary, to exclude certain subsets of the sample space from being events (see Events in probabiliity spaces, below).

Filtration (probability theory)

In the theory of stochastic processes, a subdiscipline of probability theory, filtrations are used to model the information that is available at a given point and therefore play an important role in the formalization of random processes.

Independence (probability theory)

In probability theory, two events are independent, statistically independent, or stochastically independent if the occurrence of one does not affect the probability of occurrence of the other. Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other.

The concept of independence extends to dealing with collections of more than two events or random variables, in which case the events are pairwise independent if each pair are independent of each other, and the events are mutually independent if each event is independent of each other combination of events.

Independent and identically distributed random variables

In probability theory and statistics, a sequence or collection of random variables is independent and identically distributed (i.i.d. or iid or IID) if each random variable has the same probability distribution as the others and all are mutually independent. Identically distributed, on its own, is often abbreviated ID. For uniformity, as both are discussed—and in widespread use—this article uses the visually cleaner IID in preference to the more prevalent convention i.i.d.

The annotation IID is particularly common in statistics, where observations in a sample are often assumed to be effectively IID for the purposes of statistical inference. The assumption (or requirement) that observations be IID tends to simplify the underlying mathematics of many statistical methods (see mathematical statistics and statistical theory). However, in practical applications of statistical modeling the assumption may or may not be realistic. To test how realistic the assumption is on a given data set, the autocorrelation can be computed, lag plots drawn or turning point test performed.

The generalization of exchangeable random variables is often sufficient and more easily met.

The assumption is important in the classical form of the central limit theorem, which states that the probability distribution of the sum (or average) of IID variables with finite variance approaches a normal distribution.

Often the IID assumption arises in the context of sequences of random variables. Then "independent and identically distributed" in part implies that an element in the sequence is independent of the random variables that came before it. In this way, an IID sequence is different from a Markov sequence, where the probability distribution for the nth random variable is a function of the previous random variable in the sequence (for a first order Markov sequence). An IID sequence does not imply the probabilities for all elements of the sample space or event space must be the same. For example, repeated throws of loaded dice will produce a sequence that is IID, despite the outcomes being biased.

Martingale (probability theory)

In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence, given all prior values, is equal to the present value.

Outline of probability

Probability is a measure of the likeliness that an event will occur. Probability is used to quantify an attitude of mind towards some proposition of whose truth we are not certain. The proposition of interest is usually of the form "A specific event will occur." The attitude of mind is of the form "How certain are we that the event will occur?" The certainty we adopt can be described in terms of a numerical measure and this number, between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty), we call probability. Probability theory is used extensively in statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.

Poly-Weibull distribution

In probability theory and statistics, the poly-Weibull distribution is a continuous probability distribution. The distribution is defined to be that of a random variable defined to be the smallest of a number of statistically independent random variables having non-identical Weibull distributions.

Probability

Probability is the measure of the likelihood that an event will occur. See glossary of probability and statistics. Probability quantifies as a number between 0 and 1, where, loosely speaking, 0 indicates impossibility and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%).

These concepts have been given an axiomatic mathematical formalization in probability theory, which is used widely in such areas of study as mathematics, statistics, finance, gambling, science (in particular physics), artificial intelligence/machine learning, computer science, game theory, and philosophy to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of complex systems.

Probability distribution

In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events. For instance, if the random variable X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 for X = heads, and 0.5 for X = tails (assuming the coin is fair). Examples of random phenomena can include the results of an experiment or survey.

A probability distribution is specified in terms of an underlying sample space, which is the set of all possible outcomes of the random phenomenon being observed. The sample space may be the set of real numbers or a set of vectors, or it may be a list of non-numerical values; for example, the sample space of a coin flip would be {heads, tails} .

Probability distributions are generally divided into two classes. A discrete probability distribution (applicable to the scenarios where the set of possible outcomes is discrete, such as a coin toss or a roll of dice) can be encoded by a discrete list of the probabilities of the outcomes, known as a probability mass function. On the other hand, a continuous probability distribution (applicable to the scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day) is typically described by probability density functions (with the probability of any individual outcome actually being 0). The normal distribution is a commonly encountered continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.

A probability distribution whose sample space is the set of real numbers is called univariate, while a distribution whose sample space is a vector space is called multivariate. A univariate distribution gives the probabilities of a single random variable taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.

Probability measure

In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity. The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire probability space.

Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events, e.g. the value assigned to "1 or 2" in a throw of a die should be the sum of the values assigned to "1" and "2".

Probability measures have applications in diverse fields, from physics to finance and biology.

Probability space

In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process (or “experiment”) consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind. One proposes that each time a situation of that kind arises, the set of possible outcomes is the same and the probabilities are also the same.

A probability space consists of three parts:

An outcome is the result of a single execution of the model. Since individual outcomes might be of little practical use, more complex events are used to characterize groups of outcomes. The collection of all such events is a σ-algebra . Finally, there is a need to specify each event's likelihood of happening. This is done using the probability measure function, .

Once the probability space is established, it is assumed that “nature” makes its move and selects a single outcome, , from the sample space . All the events in that contain the selected outcome (recall that each event is a subset of ) are said to “have occurred”. The selection performed by nature is done in such a way that if the experiment were to be repeated an infinite number of times, the relative frequencies of occurrence of each of the events would coincide with the probabilities prescribed by the function .

The Russian mathematician Andrey Kolmogorov introduced the notion of probability space, together with other axioms of probability, in the 1930s. Nowadays alternative approaches for axiomatization of probability theory exist; see “Algebra of random variables”, for example.

This article is concerned with the mathematics of manipulating probabilities. The article probability interpretations outlines several alternative views of what “probability” means and how it should be interpreted. In addition, there have been attempts to construct theories for quantities that are notionally similar to probabilities but do not obey all their rules; see, for example, free probability, fuzzy logic, possibility theory, negative probability and quantum probability.

Sample space

In probability theory, the sample space of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually denoted using set notation, and the possible ordered outcomes are listed as elements in the set. It is common to refer to a sample space by the labels S, Ω, or U (for "universal set").

For example, if the experiment is tossing a coin, the sample space is typically the set {head, tail}. For tossing two coins, the corresponding sample space would be {(head,head), (head,tail), (tail,head), (tail,tail)}, commonly written {HH, HT, TH, TT}. If the sample space is unordered, it becomes {{head,head}, {head,tail}, {tail,tail}}.

For tossing a single six-sided die, the typical sample space is {1, 2, 3, 4, 5, 6} (in which the result of interest is the number of pips facing up).A well-defined sample space is one of three basic elements in a probabilistic model (a probability space); the other two are a well-defined set of possible events (a sigma-algebra) and a probability assigned to each event (a probability measure function).

Stochastic process

In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a collection of random variables. Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. They have applications in many disciplines including sciences such as biology, chemistry, ecology, neuroscience, and physics as well as technology and engineering fields such as image processing, signal processing, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process, used by Louis Bachelier to study price changes on the Paris Bourse, and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time. These two stochastic processes are considered the most important and central in the theory of stochastic processes, and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.The term random function is also used to refer to a stochastic or random process, because a stochastic process can also be interpreted as a random element in a function space. The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables. But often these two terms are used when the random variables are indexed by the integers or an interval of the real line. If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead. The values of a stochastic process are not always numbers and can be vectors or other mathematical objects.Based on their mathematical properties, stochastic processes can be divided into various categories, which include random walks, martingales, Markov processes, Lévy processes, Gaussian processes, random fields, renewal processes, and branching processes. The study of stochastic processes uses mathematical knowledge and techniques from probability, calculus, linear algebra, set theory, and topology as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and functional analysis. The theory of stochastic processes is considered to be an important contribution to mathematics and it continues to be an active topic of research for both theoretical reasons and applications.

Type-1 Gumbel distribution

In probability theory, the Type-1 Gumbel density function is

for

The distribution is mainly used in the analysis of extreme values and in survival analysis (also known as duration analysis or event-history modelling).

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