Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and 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 ^[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. A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way A stochastic process, or sometimes random process, is the counterpart to a deterministic process (or Deterministic system) in Probability theory. 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 behaviour decision and action is causally determined 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. The law of large numbers (LLN is a theorem in Probability that describes the long-term stability of the mean of a Random variable. The central limit theorem (CLT states that the sum of a sufficiently large number of identically distributed independent Random variables each with finite

As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of large sets of data. Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Methods of probability theory also apply to description of complex systems given only partial knowledge of their state, as in statistical mechanics. Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics A great discovery of twentieth century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons

1 History
2 Treatment
3 Probability distributions
4 Convergence of random variables
5 Law of large numbers
6 Central limit theorem
7 See also
8 References
9 Bibliography

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 or chance 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 Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the Blaise Pascal (blɛz paskal (June 19 1623 &ndash 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. Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects Eventually, analytical considerations compelled the incorporation of continuous variables into the theory. Analysis has its beginnings in the rigorous formulation of Calculus. This culminated in modern probability theory, the foundations of which were laid by Andrey Nikolaevich Kolmogorov. Andrey Nikolaevich Kolmogorov (Андрей Николаевич Колмогоров ( April 25, 1903 - October 20, 1987) was a Soviet 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. In Probability theory, the sample space or universal sample space, often denoted S, Ω or U (for "universe" of an Experiment Richard Edler von Mises ( Lemberg (now Lviv) 19 April 1883 - Boston, 14 July 1953) was a scientist 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 In Probability theory, the Probability P of some event E, denoted P(E is defined in such a way that P satisfies the Fairly quickly this became the undisputed axiomatic basis for modern probability theory. 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 ^[2]

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

Main article: Discrete probability distribution

Discrete probability theory deals with events that occur in countable sample spaces. In Probability theory, a Probability distribution is called discrete if it is characterized by a Probability mass function.

Examples: Throwing dice, experiments with decks of cards, and random walk. For other uses see either Die or Dice (disambiguation. Dice (the Plural of Die, from Old French A playing card is a piece of specially prepared heavy paper thin card or thin plastic figured with distinguishing motifs and used as one of a set for playing Card games A random walk, sometimes denoted RW, is a Mathematical formalization of a trajectory that consists of taking successive Random steps

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 $\tfrac{3}{6}=\tfrac{1}{2}$ , since 3 faces out of the 6 have even numbers and each face has the same probability of appearing. For other uses see either Die or Dice (disambiguation. Dice (the Plural of Die, from Old French

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 $\Omega=\left \{ x_1,x_2,\dots\right \}$ . In Probability theory, the sample space or universal sample space, often denoted S, Ω or U (for "universe" of an Experiment It is then assumed that for each element $x \in \Omega\,$ , an intrinsic "probability" value $f(x)\,$ is attached, which satisfies the following properties:

$f(x)\in[0,1]\mbox{ for all }x\in \Omega\,;$
$\sum_{x\in \Omega} f(x) = 1\,.$

That is, the probability function f(x) lies between zero and one for every value of x in the sample space Ω, and the sum of f(x) over all values x in the sample space Ω is exactly equal to 1. In Mathematics, the elements or members of a set (or more generally a class) are all those objects which when collected together make up the An event is defined as any subset $E\,$ of the sample space $\Omega\,$ . In Probability theory, an event is a set of outcomes (a Subset of the Sample space) to which a probability is assigned The probability of the event $E\,$ defined as

$P(E)=\sum_{x\in E} f(x)\,.$

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

The function $f(x)\,$ mapping a point in the sample space to the "probability" value is called a probability mass function abbreviated as pmf. In Probability theory, a probability mass function (abbreviated pmf) is a function that gives the probability that a discrete Random variable 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

Main article: Continuous probability distribution

Continuous probability theory deals with events that occur in a continuous sample space. In Probability theory, a Probability distribution is called continuous if its Cumulative distribution function is continuous.

Classical definition: The classical definition breaks down when confronted with the continuous case. See Bertrand's paradox. Bertrand's paradox is a problem within the Classical interpretation of Probability theory.

Modern definition: If the sample space is the real numbers ( $\mathbb{R}$ ), then a function called the cumulative distribution function (or cdf) $F\,$ is assumed to exist, which gives $P(X\le x) = F(x)\,$ for a random variable X. In Mathematics, the real numbers may be described informally in several different ways In Probability theory and Statistics, the cumulative distribution function (CDF, also probability distribution function or just distribution function A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way That is, F(x) returns the probability that X will be less than or equal to x.

The cdf must satisfy the following properties.

$F\,$ is a monotonically non-decreasing, right-continuous function;
$\lim_{x\rightarrow -\infty} F(x)=0\,;$
$\lim_{x\rightarrow \infty} F(x)=1\,.$

If $F\,$ is differentiable, then the random variable X is said to have a probability density function or pdf or simply density $f(x)=\frac{dF(x)}{dx}\,.$

For a set $E \subseteq \mathbb{R}$ , the probability of the random variable X being in $E\,$ is defined as

$P(X\in E) = \int_{x\in E} dF(x)\,.$

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

$P(X\in E) = \int_{x\in E} f(x)\,dx\,.$

Whereas the pdf exists only for continuous random variables, the cdf exists for all random variables (including discrete random variables) that take values on $\mathbb{R}\,.$

These concepts can be generalized for multidimensional cases on $\mathbb{R}^n$ and other continuous sample spaces. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematics, a probability density function (pdf is a function that represents a Probability distribution in terms of Integrals Formally a probability In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it

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 nor mixtures of the two.

An example of such distributions could be a mix of discrete and continuous distributions, for example, a random variable which is 0 with probability 1/2, and takes a value from random normal distribution with probability 1/2. It can still be studied to some extent by considering it to have a pdf of $(\delta[x] + \varphi(x))/2$ , where $δ[x]$ is the Kronecker delta function. In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two

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 Cantor distribution is the Probability distribution whose Cumulative distribution function is the Cantor function. The modern approach to probability theory solves these problems using measure theory to define the probability space:

Given any set $Ω$ , (also called sample space) and a σ-algebra $\mathcal{F}\,$ on it, a measure $P$ is called a probability measure if

$P\,$ is non-negative;
$P(\Omega)=1\,.$

If $\mathcal{F}\,$ is a Borel σ-algebra then there is a unique probability measure on $\mathcal{F}\,$ for any cdf, and vice versa. 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 A probability space, in Probability theory, is the conventional Mathematical model of Randomness. In Mathematics, a σ-algebra (or sigma-algebra) ( Sigma is a Greek letter upper case Σ lower case σ over a set X is a nonempty 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 In Mathematics, the Borel algebra (or Borel &sigma-algebra) on a Topological space X is a &sigma-algebra of Subsets of 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 $E\,$ in the σ-algebra $\mathcal{F}\,$ is defined as

$P(E) = \int_{\omega\in E} \mu_F(d\omega)\,.$

where the integration is with respect to the measure $\mu_F\,$ induced by $F\,.$

Along with providing better understanding and unification of discrete and continuous probabilities, measure-theoretic treatment also allows us to work on probabilities outside $\mathbb{R}^n$ , as in the theory of stochastic processes. A stochastic process, or sometimes random process, is the counterpart to a deterministic process (or Deterministic system) in Probability theory. For example to study Brownian motion, probability is defined on a space of functions. This article is about the physical phenomenon for the stochastic process see Wiener process.

Probability distributions

Main article: Probability distributions

Certain random variables occur very often in probability theory because they well describe many natural or physical processes. In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable 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. WikipediaWikiProject Probability#Standards for a discussion of standards used for probability distribution articles such as this one ---> In Probability In Probability theory and Statistics, the Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete Probability WikipediaWikiProject Probability#Standards for a discussion of standards used for probability distribution articles such as this one 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 In Probability theory and Statistics, the geometric distribution is either of two Discrete probability distributions the probability Important continuous distributions include the continuous uniform, normal, exponential, gamma and beta distributions. In Probability theory and Statistics, the continuous uniform distribution is a family of Probability distributions such that for each member of the The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields WikipediaWikiProject Probability#Standards for a discussionof standards used for probability distribution articles such as this one In Probability theory and Statistics, the gamma distribution is a two-parameter family of continuous Probability distributions It has a Scale parameter In Probability theory and Statistics, the beta distribution is a family of continuous Probability distributions defined on the interval 1 parameterized

Convergence of random variables

Main article: Convergence of random variables

In probability theory, there are several notions of convergence for random variables. In Probability theory, there exist several different notions of Convergence of Random variables The convergence (in one of the senses presented below of Sequences A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way 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.

Convergence in distribution: As the name implies, a sequence of random variables $X_1,X_2,\dots,\,$ converges to the random variable $X\,$ in distribution if their respective cumulative distribution functions $F_1,F_2,\dots\,$ converge to the cumulative distribution function $F\,$ of $X\,$ , wherever $F\,$ is continuous. Continuity may refer to In mathematics: Continuous probability distribution or random variable in probability and statistics For

Most common short hand notation: $X_n \, \xrightarrow{\mathcal D} \, X\,.$

Weak convergence: The sequence of random variables $X_1,X_2,\dots\,$ is said to converge towards the random variable $X\,$ weakly if $\lim_{n\rightarrow\infty}P\left(\left|X_n-X\right|\geq\varepsilon\right)=0$ for every ε > 0. Weak convergence is also called convergence in probability.

Most common short hand notation: $X_n \, \xrightarrow{P} \, X\,.$

Strong convergence: The sequence of random variables $X_1,X_2,\dots\,$ is said to converge towards the random variable $X\,$ strongly if $P(\lim_{n\rightarrow\infty} X_n=X)=1$ . Strong convergence is also known as almost sure convergence.

Most common short hand notation: $X_n \, \xrightarrow{\mathrm{a.s.}} \, X\,.$

Intuitively, strong convergence is a stronger version of the weak convergence, and in both cases the random variables $X_1,X_2,\dots\,$ show an increasing correlation with $X\,$ . 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.

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. The law of large numbers (LLN is a theorem in Probability that describes the long-term stability of the mean of a Random variable. 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) states that the sample average $\overline{X}_n=\tfrac1n{\sum X_n}$ of $X_1,X_2,\dots\,$ (independent and identically distributed random variables with finite expectation $μ$ ) converges towards 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

$\begin{array}{lll} \text{Weak law:} & \overline{X}_n \, \xrightarrow{P} \, \mu & \text{for } n \to \infty \\ \text{Strong law:} & \overline{X}_n \, \xrightarrow{\mathrm{a.\,s.}} \, \mu & \text{for } n \to \infty . \end{array}$

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. In Probability theory, there exist several different notions of Convergence of Random variables The convergence (in one of the senses presented below of Sequences

Putting this in terms of random variables and LLN we have $Y_1,Y_2,...\,$ are independent Bernoulli random variables taking values 1 with probability p and 0 with probability 1-p. In Probability theory and Statistics, the Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete Probability $E(Y i) = p$ for all i and it follows from LLN that $\frac{\sum Y_n}{n}\,$ converges to p almost surely. In Probability theory, one says that an event happens almost surely (a

Central limit theorem

Main article: Central limit theorem

The central limit theorem explains the ubiquitous occurrence of the normal distribution in nature; it is one of the most celebrated theorems in probability and statistics. The central limit theorem (CLT states that the sum of a sufficiently large number of identically distributed independent Random variables each with finite 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. In Mathematics, an average, or central tendency of a Data set refers to a measure of the "middle" or " expected " value of The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields Formally, let $X_1,X_2,\dots\,$ be independent random variables with mean $\mu_\,$ and variance $\sigma^2 > 0.\,$ Then the sequence of random variables

$Z_n=\frac{\sum_{i=1}^n (X_i - \mu)}{\sigma\sqrt{n}}\,$

converges in distribution to a standard normal random variable. In Statistics, mean has two related meanings the Arithmetic mean (and is distinguished from the Geometric mean or Harmonic mean In Probability theory and Statistics, the variance of a Random variable, Probability distribution, or sample is one measure of The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields

References

Bibliography

Pierre Simon de Laplace (1812). In Probability theory and Statistics, the variance of a Random variable, Probability distribution, or sample is one measure of Fuzzy logic is a form of Multi-valued logic derived from Fuzzy set theory to deal with Reasoning that is approximate rather than precise Fuzzy measure theory considers a number of special classes of measures each of which is characterized by a special property The following is a Glossary of terms It is not intended to be all-inclusive In Statistics, the likelihood function (often simply the likelihood) is a function of the Parameters of a Statistical model that plays a key role This is a list of probability topics, by Wikipedia page It overlaps with the (alphabetical List of statistical topics. Please add any Wikipedia articles related to Statistics that are not already on this list Probability theory and Statistics has some commonly-used conventions of its own in addition to standard Mathematical notation and mathematical symbols. Predictive modelling is the process by which a model is created or chosen to try to 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 See also Philosophy of probability 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 Subjective logic is a type of Probabilistic logic that explicitly takes uncertainty and belief ownership into account 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.

Dictionary

-noun

(mathematics) The mathematical study of probability (the likelihood of occurrence of random events in order to predict the behavior of defined systems).

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