|
Boole's Inequality
In probability theory, Boole's inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events. This inequality provides an upper bound on the probability of occurrence of at least one of a countable number of events in terms of the individual probabilities of the events. Boole's inequality is named for its discoverer, George Boole. Formally, for a countable set of events ''A''1, ''A''2, ''A''3, ..., we have :\left(\bigcup_^ A_i \right) \le \sum_^ (A_i). In measure-theoretic terms, Boole's inequality follows from the fact that a measure (and certainly any probability measure) is ''σ''- sub-additive. Thus Boole's inequality holds not only for probability measures , but more generally when is replaced by any finite measure. Proof Proof using induction Boole's inequality may be proved for finite collections of n e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory or probability calculus 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 of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Unions
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A refers to a union of zero () sets and it is by definition equal to the empty set. For explanation of the symbols used in this article, refer to the table of mathematical symbols. Binary union The union of two sets ''A'' and ''B'' is the set of elements which are in ''A'', in ''B'', or in both ''A'' and ''B''. In set-builder notation, : A \cup B = \. For example, if ''A'' = and ''B'' = then ''A'' ∪ ''B'' = . A more elaborate example (involving two infinite sets) is: : ''A'' = : ''B'' = : A \cup B = \ As another example, the number 9 is ''not'' contained in the union of the set of prime numbers and the set of even numbers , because 9 is neither prime nor even. Sets cannot have duplicate elements, so the union of the sets and is . Multiple occurr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pairwise Independence
In probability theory, a pairwise independent collection of random variables is a set of random variables any two of which are statistical independence, independent. Any collection of Mutual independence, mutually independent random variables is pairwise independent, but some pairwise independent collections are not mutually independent. Pairwise independent random variables with finite variance are uncorrelated. A pair of random variables ''X'' and ''Y'' are independent if and only if the random vector (''X'', ''Y'') with joint distribution, joint cumulative distribution function (CDF) F_(x,y) satisfies :F_(x,y) = F_X(x) F_Y(y), or equivalently, their joint density f_(x,y) satisfies :f_(x,y) = f_X(x) f_Y(y). That is, the joint distribution is equal to the product of the marginal distributions. Unless it is not clear in context, in practice the modifier "mutual" is usually dropped so that independence means mutual independence. A statement such as " ''X'', ''Y'', ''Z'' are in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fréchet Inequalities
In probabilistic logic, the Fréchet inequalities, also known as the Boole–Fréchet inequalities, are rules implicit in the work of George BooleBoole, G. (1854). ''An Investigation of the Laws of Thought, On Which Are Founded the Mathematical Theories of Logic and Probability.'' Walton and Maberly, London. See Boole's "major" and "minor" limits of a conjunction on page 299.Hailperin, T. (1986). ''Boole's Logic and Probability''. North-Holland, Amsterdam. and explicitly derived by Maurice FréchetFréchet, M. (1935). Généralisations du théorème des probabilités totales. ''Fundamenta Mathematicae'' 25: 379–387.Fréchet, M. (1951). Sur les tableaux de corrélation dont les marges sont données. ''Annales de l'Université de Lyon. Section A: Sciences mathématiques et astronomie'' 9: 53–77. that govern the combination of probabilities about logical propositions or events logically linked together in conjunctions ( AND operations) or disjunctions ( OR operations) as in Boolea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schuette–Nesbitt Formula
In mathematics, the Schuette–Nesbitt formula is a generalization of the inclusion–exclusion principle. It is named after Donald R. Schuette and Cecil J. Nesbitt. The probabilistic version of the Schuette–Nesbitt formula has practical applications in actuarial science, where it is used to calculate the net single premium for life annuities and life insurances based on the general symmetric status. Combinatorial versions Consider a set and subsets . Let denote the number of subsets to which belongs, where we use the indicator functions of the sets . Furthermore, for each , let denote the number of intersections of exactly sets out of , to which belongs, where the intersection over the empty index set is defined as , hence . Let denote a vector space over a field such as the real or complex numbers (or more generally a module over a ring with multiplicative identity). Then, for every choice of , where denotes the indicator function of the set of all with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diluted Inclusion–exclusion Principle
Dilution may refer to: * Reducing the concentration of a chemical * Serial dilution, stepwise * Homeopathic dilution * Dilution (equation), an equation to calculate the rate a gas dilutes *Trademark dilution, weakening of a trademark by unauthorised use *Stock dilution, issuing of new company shares *Dilution gene, lightening animal coat color * Dilution ratio * Hemodynamics#Hemodilution, of blood * Dilution refrigerator A 3He/4He dilution refrigerator is a cryogenics, cryogenic device that provides continuous cooling to temperatures as low as 2 Kelvin, mK, with no moving parts in the low-temperature region. The cooling power is provided by the heat o ..., cryogenic device See also * Delusion (other) * Dilation (other) {{disambig zh:稀释 nl:Verdunning ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when identifying the system, or when evaluating its performance, status, condition, etc. ''Parameter'' has more specific meanings within various disciplines, including mathematics, computer programming, engineering, statistics, logic, linguistics, and electronic musical composition. In addition to its technical uses, there are also extended uses, especially in non-scientific contexts, where it is used to mean defining characteristics or boundaries, as in the phrases 'test parameters' or 'game play parameters'. Modelization When a system theory, system is modeled by equations, the values that describe the system are called ''parameters''. For example, in mechanics, the masses, the dimensions and shapes (for solid bodies), the densities and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal's Rule
In mathematics, Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients. The binomial coefficients are the numbers that appear in Pascal's triangle. Pascal's rule states that for positive integers ''n'' and ''k'', + = , where \tbinom is the binomial coefficient, namely the coefficient of the term in the expansion of . There is no restriction on the relative sizes of and ; in particular, the above identity remains valid when since \tbinom = 0 whenever . Together with the boundary conditions \tbinom = \tbinom= 1 for all nonnegative integers ''n'', Pascal's rule determines that \binom = \frac, for all integers . In this sense, Pascal's rule is the recurrence relation that defines the binomial coefficients. Pascal's rule can also be generalized to apply to multinomial coefficients. Combinatorial proof Pascal's rule has an intuitive combinatorial meaning, that is clearly expressed in this counting proof. ''Proof''. Recall that \tbinom e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sample Space
In probability theory, the sample space (also called sample description space, possibility space, or outcome 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, or sample points, 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"). The elements of a sample space may be numbers, words, letters, or symbols. They can also be finite, countably infinite, or uncountably infinite. A subset of the sample space is an event, denoted by E. If the outcome of an experiment is included in E, then event E has occurred. For example, if the experiment is tossing a single coin, the sample space is the set \, where the outcome H means that the coin is heads and the outcome T means that the coin is tails. The possible events are E=\, E=\, E = \, and E = \. For tossing two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Content (measure Theory)
In mathematics, in particular in measure theory, a content \mu is a real-valued function defined on a collection of subsets \mathcal such that # \mu(A)\in\ , \infty\text A \in \mathcal. # \mu(\varnothing) = 0. # \mu\Bigl(\bigcup_^n A_i\Bigr) = \sum_^n \mu(A_i) \text A_1, \dots, A_n, \bigcup_^n A_i \in \mathcal \text A_i \cap A_j = \varnothing \text i \neq j. That is, a content is a generalization of a measure: while the latter must be countably additive, the former must only be finitely additive. In many important applications the \mathcal is chosen to be a ring of sets or to be at least a semiring of sets in which case some additional properties can be deduced which are described below. For this reason some authors prefer to define contents only for the case of semirings or even rings. If a content is additionally ''σ''-additive it is called a pre-measure and if furthermore \mathcal is a ''σ''-algebra, the content is called a measure. Therefore, every (real-valued) measur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
