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Chebyshev’s Inequality
In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from the mean. Specifically, no more than 1/''k''2 of the distribution's values can be ''k'' or more standard deviations away from the mean (or equivalently, at least 1 − 1/''k''2 of the distribution's values are less than ''k'' standard deviations away from the mean). The rule is often called Chebyshev's theorem, about the range of standard deviations around the mean, in statistics. The inequality has great utility because it can be applied to any probability distribution in which the mean and variance are defined. For example, it can be used to prove the weak law of large numbers. Its practical usage is similar to the 68–95–99.7 rule, which applies only to normal distributions. Chebyshev's inequality is more general, stating t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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 not possible to perfectly p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure Space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that is used for measuring (the measure). One important example of a measure space is a probability space. A measurable space consists of the first two components without a specific measure. Definition A measure space is a triple (X, \mathcal A, \mu), where * X is a set * \mathcal A is a -algebra on the set X * \mu is a measure on (X, \mathcal) In other words, a measure space consists of a measurable space (X, \mathcal) together with a measure on it. Example Set X = \. The \sigma-algebra on finite sets such as the one above is usually the power set, which is the set of all subsets (of a given set) and is denoted by \wp(\cdot). Sticking with this convention, we set \mathcal = \wp(X) In this simple case, the power set can be writte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Mitzenmacher
Michael David Mitzenmacher is an American computer scientist working in algorithms. He is Professor of Computer Science at the Harvard John A. Paulson School of Engineering and Applied Sciences and was area dean of computer science July 2010 to June 2013. He also runs My Biased Coin', a blog about theoretical computer science. Education In 1986, Mitzenmacher attended the Research Science Institute. Mitzenmacher earned his AB at Harvard, where he was on the team that won the 1990 North American Collegiate Bridge Championship. He attended the University of Cambridge on a Churchill Scholarship from 1991–1992. Mitzenmacher received his PhD in computer science at the University of California, Berkeley in 1996 under the supervision of Alistair Sinclair. He joined Harvard University in 1999. Research Mitzenmacher’s research covers the design an analysis of randomised algorithms and processes. With Eli Upfal he is the author of a textbook on randomized algorithms and proba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sankhya (journal)
''Sankhyā: The Indian Journal of Statistics'' is a quarterly peer-reviewed scientific journal on statistics published by the Indian Statistical Institute (ISI). It was established in 1933 by Prasanta Chandra Mahalanobis, founding director of ISI, along the lines of Karl Pearson's ''Biometrika''. Mahalanobis was the founding editor-in-chief. Each volume of ''Sankhya'' consists of four issues, two of them are in Series A, containing articles on theoretical statistics, probability theory, and stochastic processes, whereas the other two issues form Series B, containing articles on applied statistics, i.e. applied probability, applied stochastic processes, econometrics, and statistical computing. ''Sankhya'' is considered as "core journal" of statistics by the Current Index to Statistics. Publication history ''Sankhya'' was first published in June 1933. In 1961, the journal split into two series: Series A which focused on mathematical statistics and Series B which focused on st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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