High Quality Content by WIKIPEDIA articles! The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1927 it appeared in a second edition with an important Introduction To the Second Edition, an Appendix A that replaced *9 and an all-new Appendix C.PM, as it is often abbreviated (not to be confused with Russell's 1903 Principles of Mathematics), is an attempt to derive all mathematical truths from a well-defined set of axioms and inference rules in symbolic logic. One of the main inspirations and motivations for PM was Frege's earlier work on logic, which had led to paradoxes discovered by Russell. Данное...

ISBN: 978-6-1303-4744-4
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In geometry, a prismatic uniform polyhedron is a uniform polyhedron with dihedral symmetry. They exist in two infinite families, the uniform prisms and the uniform antiprisms. All have their vertices in parallel planes and are therefore prismatoids. The difference between the prismatic and antiprismatic symmetry groups is that Dph has the vertices lined up in both planes, which gives it a reflection plane perpendicular to its p-fold axis (parallel to the {p/q} polygon); while Dpd has the vertices twisted relative to the other plane, which gives it a rotatory reflection. Each has p reflection planes which contain the p-fold axis. Данное издание не является...

ISBN: 978-6-1303-0762-2
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. Thus these joining faces are parallelograms. All cross-sections parallel to the base faces are the same. A prism is a subclass of the prismatoids. A right prism is a prism in which the joining edges and faces are perpendicular to the base faces. This applies if the joining faces are rectangular. If the joining edges and faces are not perpendicular to the base faces, it is called an oblique prism. Данное издание не является оригинальным. Книга печатается по технологии...

ISBN: 978-6-1303-2291-5
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! A prismatic surface is a surface generated by all the lines that are parallel to a given line and intersect a broken line that is not in the same plane as the given line. The broken line is the directrix of the surface; the parallel lines are its generators (or elements). If the broken line is closed (i.e. a closed polygon), then the surface is a closed prismatic surface. With regards to crystallography, a prismatic surface is a single face of a prismatic form, which is an open form consisting of three, four, or six identical faces related by a symmetry operator. Данное издание не является оригинальным. Книга печатается по...

ISBN: 978-6-1311-8773-5
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! Probability theory uses routinely results from other fields of mathematics (mostly, analysis). The opposite cases, collected below, are relatively rare; however, probability theory is used systematically in combinatorics via the probabilistic method. They are particularly used for non-constructive proofs. Normal numbers exist. Moreover, computable normal numbers exist. These non-probabilistic existence theorems follow from probabilistic results: (a) a number chosen at random (uniformly on (0,1)) is normal almost surely (which follows easily from the strong law of large numbers); (b) some probabilistic inequalities behind the strong law. The existence of a normal number follows from (a)...

ISBN: 978-6-1336-7097-6
Издательство: Книга по требованию
Дата выхода: июль 2011

Most tasks require a person or an automated system to reason—to reach conclusions based on available information. The framework of probabilistic graphical models, presented in this book, provides a general approach for this task. The approach is model-based, allowing interpretable models to be constructed and then manipulated by reasoning algorithms. These models can also be learned automatically from data, allowing the approach to be used in cases where manually constructing a model is difficult or even impossible. Because uncertainty is an inescapable aspect of most real-world applications, the book focuses on probabilistic models, which make the uncertainty explicit and provide models that are more faithful to reality....

ISBN: 9780262013192
Дата выхода: август 2009

High Quality Content by WIKIPEDIA articles! In statistics, the term "error" arises in two ways. Firstly, it arises in the context of decision making, where the probability of error may be considered as being the probability of making a wrong decision and which would have a different value for each type of error. Secondly, it arises in the context of statistical modelling (for example regression) where the model's predicted value may be in error regarding the observed outcome and where the term probability of error may refer to the probabilities of various amounts of error occurring. In hypothesis testing in statistics, two types of error are distinguished. Type I errors which consist of rejecting a null hypothesis that is true; this...

ISBN: 978-6-1336-7460-8
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In probability theory, the probability space, or probability triple, is a concept which serves as a rigorous mathematical ground for the conventional idea of randomness. It is a mathematical model of a real-world situation where we recognize that certain things occur “at random”. The model works as following: first, at the outset of the experiment we attempt to envision all possible outcomes which might possibly happen, the set of all such outcomes is called the sample space ?. Second, we recognize that the elementary outcomes could be too little of practical use, and that the more complicated events, consisting possibly of many different elementary outcomes, are of more interest....

ISBN: 978-6-1303-4306-4
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! A randomized algorithm or probabilistic algorithm is an algorithm which employs a degree of randomness as part of its logic. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random bits. Formally, the algorithm's performance will be a random variable determined by the random bits; thus either the running time, or the output (or both) are random variables.In both cases, random performance and random output, the term "algorithm" for a procedure is somewhat questionable, as it is no longer formally effective. However, in some cases, probabilistic algorithms...

ISBN: 978-6-1312-3934-2
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In probability theory, a probability density function (abbreviated as pdf, or just density) of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point in the observation space. The probability of a random variable falling within a given set is given by the integral of its density over the set.On rare occasions the term “probability distribution function” is used to denote the probability density function. However special care should be taken around this term, since in other sources the “probability distribution function” may refer to either the probability distribution function, or the cumulative distribution...

ISBN: 978-6-1303-4792-5
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In analysis of algorithms, probabilistic analysis of algorithms is an approach to estimate the computational complexity of an algorithm or a computational problem. It starts from an assumption about a probabilistic distribution of the set of all possible inputs. This assumption is then used to design an efficient algorithm or to derive the complexity of a known algorithm. This approach is not the same as that of probabilistic algorithms, but the two may be combined. For non-probabilistic, more specifically, for deterministic algorithms, the most common types of complexity estimates are: the average-case complexity (expected time complexity), in which given an input distribution, the expected...

ISBN: 978-6-1336-7363-2
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The probability derivations for starting hands making four of a kind, a full house, three of a kind, two pair, one pair and no pair in Omaha hold 'em are separate for each of the starting hand rank types. The derivations require identifying the individual cases that yield each possible hand and are sometimes rather detailed, so it is useful to use a notation to indicate the shape of the board for each case. The rank type of the hand is shown using upper case letters to indicate ranks. The ranks on the board are indicated using upper case letters for matches with the starting hand and lower case letters to indicate ranks that don't match the starting hand. So the rank type XXYZ is any hand with...

ISBN: 978-6-1336-7267-3
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In probability theory, the probability-generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. Probability-generating functions are often employed for their succinct description of the sequence of probabilities Pr(X = i) in the probability mass function for a random variable X, and to make available the well-developed theory of power series with non-negative coefficients. Probability-generating functions obey all the rules of power series with non-negative coefficients. In particular, G(1–) = 1, where G(1–) = limz->1G(z) from below, since the probabilities must sum to one. So the...

ISBN: 978-6-1336-7117-1
Издательство: Книга по требованию
Дата выхода: июль 2011

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. Although an individual coin toss or the roll of a die is a random event, if repeated many times the sequence of random events will exhibit certain statistical patterns, which can be studied and predicted. Two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem. Данное издание не является...

ISBN: 978-6-1303-0285-6
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The aim of a probabilistic logic (or probability logic) is to combine the capacity of probability theory to handle uncertainty with the capacity of deductive logic to exploit structure. The result is a richer and more expressive formalism with a broad range of possible application areas. Probabilistic logic is a natural extension of traditional logic truth tables: the results they define are derived through probabilistic expressions instead. The difficulty with probabilistic logics is that they tend to multiply the computational complexities of their probabilistic and logical components. There are numerous proposals for probabilistic logics: The term "probabilistic logic" was first used in a...

ISBN: 978-6-1336-7379-3
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In probability theory, a probability density function of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point in the observation space. The probability of a random variable falling within a given set is given by the integral of its density over the set. On rare occasions the term “probability distribution function” is used to denote the probability density function. However special care should be taken around this term, since in other sources the “probability distribution function” may refer to either the probability distribution function, or the cumulative distribution function, or may be a probability mass...

ISBN: 978-6-1303-4296-8
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In statistics, the probability integral transform or transformation relates to the result that data values that are modelled as being random variables from any given continuous distribution can be converted to random variables having a uniform distribution. This holds exactly provided that the distribution being used is the true distribution of the random variables; if the distribution is one fitted to the data the result will hold approximately in large samples. Suppose that a random variable X has a continuous distribution for which the cumulative distribution function is F. One use for the probability integral transform in statistical data analysis is to provide the basis for testing whether...

ISBN: 978-6-1336-7299-4
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The probabilistic method is a nonconstructive method, primarily used in combinatorics and pioneered by Paul Erd?s, for proving the existence of a prescribed kind of mathematical object. It works by showing that if one randomly chooses objects from a specified class, the probability that the result is of the prescribed kind is more than zero. Although the proof uses probability, the final conclusion is determined for certain, without any possible error. This method has now been applied to other areas of mathematics such as number theory, linear algebra, and real analysis. Данное издание не является оригинальным. Книга печатается по...

ISBN: 978-6-1303-4293-7
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! A probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity. The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign 1 to the entire probability space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events, e.g. the value assigned to "Heads or Tails" in a coin toss should be the sum of the values assigned to Heads and Tails. Probability measures have...

ISBN: 978-6-1336-7132-4
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In computability theory, a probabilistic Turing machine is a non-deterministic Turing machine which randomly chooses between the available transitions at each point according to some probability distribution. In the case of equal probabilities for the transitions, it can be defined as a deterministic Turing machine having an additional "write" instruction where the value of the write is uniformly distributed in the Turing Machine's alphabet (generally, an equal likelihood of writing a '1' or a '0' on to the tape.) Another common reformulation is simply a deterministic Turing machine with an added tape full of random bits called the random tape. As a consequence, a probabilistic Turing machine...

ISBN: 978-6-1336-7085-3
Издательство: Книга по требованию
Дата выхода: июль 2011

In this book, I strove to propose a both precise and handy probabilistic modeling in some areas of finance and biology. Both fields are made of complex random systems, described by huge and noisy data, and call for expertise in probability theory and statistics. My work is a contribution to modeling interactions in such systems, numerical analysis of the models, and statistical analysis of experimental data. In finance, I first focus on analysis of weak error of the density of the Euler scheme for stochastic differential equations, deriving new expansions in Gaussian-like functional spaces. Then I study ergodic properties of stochastic volatility models, with extended numerical experiments. In biology, I work on cellular aging,...

ISBN: 978-3-8383-1446-4
Издательство: Книга по требованию
Дата выхода: июль 2011

The word probability has been used in a variety of ways since it was first coined in relation to games of chance. Does probability measure the real, physical tendency of something to occur, or is it just a measure of how strongly one believes it will occur? In answering such questions, we interpret the probability values of probability theory. There are two broad categories of probability interpretations which can be called 'physical' and 'evidential' probabilities. Physical probabilities, which are also called objective or frequency probabilities, are associated with random physical systems such as roulette wheels, rolling dice and radioactive atoms. In such systems, a given type of event (such as the dice yielding a six) tends to occur...

ISBN: 978-6-1303-0288-7
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In probability theory, the probability space, or probability triple, is a concept which serves as a rigorous mathematical ground for the conventional idea of randomness. It is a mathematical model of a real-world situation (or “experiment”) where we recognize that certain things occur “at random”. The model works as following: first, at the outset of the experiment we attempt to envision all possible outcomes which might possibly happen, the set of all such outcomes is called the sample space ?. Second, we recognize that the elementary outcomes could be too little of practical use, and that the more complicated events, consisting possibly of many different elementary outcomes, are of...

ISBN: 978-6-1303-4802-1
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The term probable error has different meanings in different contexts. In statistics, the probable error of a quantity is a value describing the probability distribution of that quantity. It defines the half-range of an interval about a cental point for the distribution, such that half of the values from the distribution will lie within the interval and half outside. Thus it is equivalent to half the interquartile range However the term also has an older meaning that is still currently repeated as the only meaning, but that has been deprecated for some time. This specifies the probable error as being fixed a multiple of the standard deviation, , where the multiplying factor derives from the...

ISBN: 978-6-1336-7503-2
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! Probabilistic number theory is a subfield of number theory, which explicitly uses probability to answer questions of number theory. One basic idea underlying it is that different prime numbers are, in some serious sense, like independent random variables. This however is not an idea that has a unique useful formal expression. The founders of the theory were Paul Erdos, Aurel Wintner and Mark Kac during the 1930s, one of the most intense periods of investigation in analytic number theory. The Erdos-Wintner theorem and the Erdos-Kac theorem on additive functions were foundational results. Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and...

ISBN: 978-6-1336-7401-1
Издательство: Книга по требованию
Дата выхода: июль 2011