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A first look at rigorous probability theory by Jeffrey S. Rosenthal

By Jeffrey S. Rosenthal

Книга дает строгое изложение всех базовых концепций теории вероятностей на основе теории меры, в то же время не перегружая читателя дополнительными сведениями. В книге даются строгие доказательства закона больших чисел, центральной предельной теоремы, леммы Фату, формулируется лемма Ито. В тексте и математическом приложении содержатся все необходимые сведения, так что книга доступна для понимания любому выпускнику школы.This textbook is an creation to chance thought utilizing degree idea. it truly is designed for graduate scholars in quite a few fields (mathematics, information, economics, administration, finance, computing device technological know-how, and engineering) who require a operating wisdom of likelihood thought that's mathematically targeted, yet with no over the top technicalities. The textual content offers entire proofs of the entire crucial introductory effects. however, the remedy is concentrated and available, with the degree concept and mathematical information offered by way of intuitive probabilistic options, instead of as separate, enforcing topics. during this new version, many workouts and small extra themes were extra and present ones improved. The textual content moves a suitable stability, carefully constructing chance conception whereas keeping off pointless detail.

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Sample text

Randomness remains an enigmatic concept. (A very readable discussion of it has been written for a lay audience by Beltrami [1999]), Its usefulness in many contexts, especially the physical and social sciences, is beyond question. A variety of tests of randomness, in addition to the compressibility test, have been developed and are applied in specific instances, but they tend to be indications and not proofs of randomness, and, at best, they represent necessary but not sufficient conditions for considering something to be random.

We should note, however, that in principle the approach is not unlike the practice of repeating a failed experiment until it yields the desired statistical result. Every so often, there is a report in the news media of a neighborhood or town that has become alarmed because there appears to be a higher-than-chance incidence of some disease (usually cancer) among its residents. From the perspective of a resident, this is an understandable cause for concern and for an effort to determine whether there is a causal explanation for the unusually high rate of the condition.

There are, however, certain concepts that one encounters often in discussions of randomness and that characterize properties that a random set or sequence is expected, at least by some observers, to have. Among the more common of these properties are equal representation, irregularity or unpredictability, and incompressibility. Some Conceptions During the 1870s, William Shanks published the value of π to 707 places, a prodigious feat, given that the computation was done entirely by hand. Three quarters of a century passed before someone produced, with the help of computing machinery, an approximation with a larger number of digits.

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