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Self-information
In information theory, the information content, self-information, surprisal, or Shannon information is a basic quantity derived from the probability of a particular event occurring from a random variable. It can be thought of as an alternative way of expressing probability, much like odds or log-odds, but which has particular mathematical advantages in the setting of information theory. The Shannon information can be interpreted as quantifying the level of "surprise" of a particular outcome. As it is such a basic quantity, it also appears in several other settings, such as the length of a message needed to transmit the event given an optimal source coding of the random variable. The Shannon information is closely related to ''entropy'', which is the expected value of the self-information of a random variable, quantifying how surprising the random variable is "on average". This is the average amount of self-information an observer would expect to gain about a random variable whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Information Theory
Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. The field is at the intersection of probability theory, statistics, computer science, statistical mechanics, information engineering, and electrical engineering. A key measure in information theory is entropy. Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a fair coin flip (with two equally likely outcomes) provides less information (lower entropy) than specifying the outcome from a roll of a die (with six equally likely outcomes). Some other important measures in information theory are mutual information, channel capacity, error exponents, and relative entropy. Important sub-fields of information theory include s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a die. A probability space consists of three elements:Stroock, D. W. (1999). Probability theory: an analytic view. Cambridge University Press. # A sample space, \Omega, which is the set of all possible outcomes. # An event space, which is a set of events \mathcal, an event being a set of outcomes in the sample space. # A probability function, which assigns each event in the event space a probability, which is a number between 0 and 1. In order to provide a sensible model of probability, these elements must satisfy a number of axioms, detailed in this article. In the example of the throw of a standard die, we would take the sample space to be \. For the event space, we could simply use the set of all subsets of the sam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joint Probability Mass Function
Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered for any given number of random variables. The joint distribution encodes the marginal distributions, i.e. the distributions of each of the individual random variables. It also encodes the conditional probability distributions, which deal with how the outputs of one random variable are distributed when given information on the outputs of the other random variable(s). In the formal mathematical setup of measure theory, the joint distribution is given by the pushforward measure, by the map obtained by pairing together the given random variables, of the sample space's probability measure. In the case of real-valued random variables, the joint distribution, as a particular multivariate distribution, may be expressed by a multivariate cumulati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independent Random Variables
Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independents (Oporto artist group), a Portuguese artist group historically linked to abstract art and to Fernando Lanhas, the central figure of Portuguese abstractionism Music Groups, labels, and genres * Independent music, a number of genres associated with independent labels * Independent record label, a record label not associated with a major label * Independent Albums, American albums chart Albums * ''Independent'' (Ai album), 2012 * ''Independent'' (Faze album), 2006 * ''Independent'' (Sacred Reich album), 1993 Songs * "Independent" (song), a 2007 song by Webbie * "Independent", a 2002 song by Ayumi Hamasaki from '' H'' News and media organizations * ''The Independent'', a British online newspaper. * '' The Malta Independent'', a Malte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure (mathematics)
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sigma Additivity
In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of ''k'' disjoint sets (where ''k'' is a finite number) equals the sum of its values on the sets. Therefore, an additive set function is also called a finitely-additive set function (the terms are equivalent). However, a finitely-additive set function might not have the additivity property for a union of an ''infinite'' number of sets. A σ-additive set function is a function that has the additivity property even for countably infinite many sets, that is, \mu\left(\bigcup_^\infty A_n\right) = \sum_^\infty \mu(A_n). Additivity and sigma-additivity are particularly important properties of measures. They are abs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Map
In algebra, an additive map, Z-linear map or additive function is a function f that preserves the addition operation: f(x + y) = f(x) + f(y) for every pair of elements x and y in the domain of f. For example, any linear map is additive. When the domain is the real numbers, this is Cauchy's functional equation. For a specific case of this definition, see additive polynomial. More formally, an additive map is a \Z- module homomorphism. Since an abelian group is a \Z- module, it may be defined as a group homomorphism between abelian groups. A map V \times W \to X that is additive in each of two arguments separately is called a bi-additive map or a \Z-bilinear map. Examples Typical examples include maps between rings, vector spaces, or modules that preserve the additive group. An additive map does not necessarily preserve any other structure of the object; for example, the product operation of a ring. If f and g are additive maps, then the map f + g (defined pointwise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independent Events
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds. Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other. When dealing with collections of more than two events, two notions of independence need to be distinguished. The events are called pairwise independent if any two events in the collection are independent of each other, while mutual independence (or collective independence) of events means, informally speaking, that each event is independent of any combination of other events in the collection. A similar notion exists for collections of random variables. Mutual independence implies pairwise independenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by \sigma^2, s^2, \operatorname(X), V(X), or \mathbb(X). An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lottery Mathematics
Lottery mathematics is used to calculate probabilities of winning or losing a lottery game. It is based primarily on combinatorics, particularly the twelvefold way and combinations without replacement. Choosing 6 from 49 In a typical 6/49 game, each player chooses six distinct numbers from a range of 1-49. If the six numbers on a ticket match the numbers drawn by the lottery, the ticket holder is a jackpot winner— regardless of the order of the numbers. The probability of this happening is 1 in 13,983,816. The chance of winning can be demonstrated as follows: The first number drawn has a 1 in 49 chance of matching. When the draw comes to the second number, there are now only 48 balls left in the bag, because the balls are drawn without replacement. So there is now a 1 in 48 chance of predicting this number. Thus for each of the 49 ways of choosing the first number there are 48 different ways of choosing the second. This means that the probability of correctly predicting 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Winning The Lottery
A lottery is a form of gambling that involves the drawing of numbers at random for a prize. Some governments outlaw lotteries, while others endorse it to the extent of organizing a national or state lottery. It is common to find some degree of regulation of lottery by governments. The most common regulation is prohibition of sale to minors, and vendors must be licensed to sell lottery tickets. Although lotteries were common in the United States and some other countries during the 19th century, by the beginning of the 20th century, most forms of gambling, including lotteries and sweepstakes, were illegal in the U.S. and most of Europe as well as many other countries. This remained so until well after World War II. In the 1960s, casinos and lotteries began to re-appear throughout the world as a means for governments to raise revenue without raising taxes. Lotteries come in many formats. For example, the prize can be a fixed amount of cash or goods. In this format, there is risk t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lottery
A lottery is a form of gambling that involves the drawing of numbers at random for a prize. Some governments outlaw lotteries, while others endorse it to the extent of organizing a national or state lottery. It is common to find some degree of regulation of lottery by governments. The most common regulation is prohibition of sale to minors, and vendors must be licensed to sell lottery tickets. Although lotteries were common in the United States and some other countries during the 19th century, by the beginning of the 20th century, most forms of gambling, including lotteries and sweepstakes, were illegal in the U.S. and most of Europe as well as many other countries. This remained so until well after World War II. In the 1960s, casinos and lotteries began to re-appear throughout the world as a means for governments to raise revenue without raising taxes. Lotteries come in many formats. For example, the prize can be a fixed amount of cash or goods. In this format, there is ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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