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Lindeberg's Condition
In probability theory, Lindeberg's condition is a Necessary and sufficient condition, sufficient condition (and under certain conditions also a necessary condition) for the central limit theorem (CLT) to hold for a sequence of independent random variables. Unlike the classical CLT, which requires that the random variables in question have finite variance and be both Independent and identically-distributed random variables, independent and identically distributed, Central limit theorem#Lindeberg CLT, Lindeberg's CLT only requires that they have finite variance, satisfy Lindeberg's condition, and be Statistical independence, independent. It is named after the Finnish mathematician Jarl Waldemar Lindeberg. Statement Let (\Omega, \mathcal, \mathbb) be a probability space, and X_k : \Omega \to \mathbb,\,\, k \in \mathbb, be ''independent'' random variables defined on that space. Assume the expected values \mathbb\,[X_k] = \mu_k and variances \mathrm\,[X_k] = \sigma_k^2 exist and are fin ... [...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] |
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 . 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 of a random process under consideration. # An event space, \mathcal, which is a set of events, where an event is a subset of outcomes in the sample space. # A '' probability function'', P, which assigns, to each event in the event space, a probability, which is a number between 0 and 1 (inclusive). In order to provide a model of probability, these elements must satisfy probability axioms. In the example of the throw of a standard die, # The sample space \Omega is typically the set \ where each element in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Condition
In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample mean converges to a standard normal distribution. This holds even if the original variables themselves are not normally distributed. There are several versions of the CLT, each applying in the context of different conditions. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions. This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern form it was only precisely stated as late as 1920. In statistics, the CLT can be stated as: let X_1, X_2, \dots, X_n denote a statistical sample of size n from a population with expected value (average) \mu and finite positive varia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Combination
In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be any expression of the form ''ax'' + ''by'', where ''a'' and ''b'' are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field (mathematics), field, with some generalizations given at the end of the article. Definition Let ''V'' be a vector space over the field ''K''. As usual, we call elements of ''V'' ''vector space, vectors'' and call elements of ''K'' ''scalar (mathematics), scalars''. If v1,...,v''n'' are vectors and ''a''1,...,''a''''n'' are scalars, then the ''linear combination of those vectors with those scalars as coefficients'' is :a_1 \mathbf v_1 + a_2 \mathbf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof By Contradiction
In logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally valid. More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved. In this general sense, proof by contradiction is also known as indirect proof, proof by assuming the opposite, and ''reductio ad impossibile''. A mathematical proof employing proof by contradiction usually proceeds as follows: #The proposition to be proved is ''P''. #We assume ''P'' to be false, i.e., we assume ''¬P''. #It is then shown that ''¬P'' implies falsehood. This is typically accomplished by deriving two mutually co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Distribution
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac e^\,. The parameter is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma^2 is the variance. The standard deviation of the distribution is (sigma). A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergence In Distribution
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of convergence capture different properties about the sequence, with some notions of convergence being stronger than others. For example, convergence in distribution tells us about the limit distribution of a sequence of random variables. This is a weaker notion than convergence in probability, which tells us about the value a random variable will take, rather than just the distribution. The concept is important in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that certain properties of a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indicator Function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator function of is the function \mathbf_A defined by \mathbf_\!(x) = 1 if x \in A, and \mathbf_\!(x) = 0 otherwise. Other common notations are and \chi_A. The indicator function of is the Iverson bracket of the property of belonging to ; that is, \mathbf_(x) = \left x\in A\ \right For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers. Definition Given an arbitrary set , the indicator function of a subset of is the function \mathbf_A \colon X \mapsto \ defined by \operatorname\mathbf_A\!( x ) = \begin 1 & \text x \in A \\ 0 & \text x \notin A \,. \end The Iverson bracket provides the equivalent notation \left x\in A\ \right/math> or that can be used instead of \mathbf_\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Zeitschrift
''Mathematische Zeitschrift'' ( German for ''Mathematical Journal'') is a mathematical journal for pure and applied mathematics published by Springer Verlag. History The journal was founded in 1917, with its first issue appearing in 1918. It was initially edited by Leon Lichtenstein together with Konrad Knopp, Erhard Schmidt, and Issai Schur. Because Lichtenstein was Jewish, he was forced to step down as editor in 1933 under the Nazi rule of Germany; he fled to Poland and died soon after. The editorship was offered to Helmut Hasse Helmut Hasse (; 25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory and ..., but he refused, Translated by Bärbel Deninger from the 1982 German original. and Konrad Knopp took it over. Other past editors include Erich Kamke, Friedrich Karl Schmidt, Rolf Nevanlinna, Hel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Necessary And Sufficient Condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of is guaranteed by the truth of . (Equivalently, it is impossible to have without , or the falsity of ensures the falsity of .) Similarly, is sufficient for , because being true always implies that is true, but not being true does not always imply that is not true. In general, a necessary condition is one (possibly one of several conditions) that must be present in order for another condition to occur, while a sufficient condition is one that produces the said condition. The assertion that a statement is a "necessary ''and'' sufficient" condition of another means that the former statement is true if and only if the latter is true. That is, the two statements must be either simultaneously true, or simultaneously false. In ordinary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jarl Waldemar Lindeberg
Jarl Waldemar Lindeberg (4 August 1876, Helsinki – 24 December 1932, Helsinki) was a Finnish mathematician known for work on the central limit theorem. Life and work Lindeberg was son of a teacher at the Helsinki Polytechnical Institute and at an early age showed mathematical talent and interest. The family was well off and later Jarl Waldemar would prefer to be a reader than a full professor. Lindeberg's career centred on the University of Helsinki. His early interests were in partial differential equations and the calculus of variations but from 1920 he worked in probability and statistics. In 1920 he published his first paper on the central limit theorem. His result was similar to that obtained earlier by Lyapunov whose work he did not then know. However, their approaches were quite different; Lindeberg's was based on a convolution argument while Lyapunov used the characteristic function. Two years later Lindeberg used his method to obtain a stronger result: the so-called ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
