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Kemeny's Constant
In probability theory, Kemeny’s constant is the expected number of time steps required for a Markov chain to transition from a starting state ''i'' to a random destination state sampled from the Markov chain's stationary distribution. Surprisingly, this quantity does not depend on which starting state ''i'' is chosen. It is in that sense a constant, although it is different for different Markov chains. When first published by John Kemeny in 1960 a prize was offered for an intuitive explanation as to why the quantity was constant. (Corollary 4.3.6) Definition For a finite ergodic Markov chain with transition matrix ''P'' and invariant distribution ''π'', write ''m''''ij'' for the mean first passage time from state ''i'' to state ''j'' (denoting the mean recurrence time for the case ''i'' = ''j''). Then :K = \sum_ \pi_j m_ is a constant and not dependent on ''i''. Prize Kemeny wrote, (for ''i'' the starting state of the Markov chain) “A prize is offered for the fir ... [...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] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as or \mathbb. History The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes ''in a fair way'' between two players, who have to e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Chain
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs ''now''." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). It is named after the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes, such as studying cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, currency exchange rates and animal population dynamics. Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Kemeny (computer Scientist)
John George Kemeny (born Kemény János György; May 31, 1926 – December 26, 1992) was a Hungarian-born American mathematician, computer scientist, and educator best known for co-developing the BASIC programming language in 1964 with Thomas E. Kurtz and Mary Kenneth Keller. Kemeny served as the 13th President of Dartmouth College from 1970 to 1981 and pioneered the use of computers in college education. Kemeny chaired the presidential commission that investigated the Three Mile Island accident in 1979. According to György Marx he was one of The Martians (scientists), The Martians. Early life Born in Budapest, Kingdom of Hungary, Hungary, into a History of the Jews in Hungary, Jewish family, Kemeny attended the Rácz private primary school in Budapest and was a classmate of Nándor Balázs. In 1938 his father left for the United States alone. In 1940, he took the whole Kemeny family to the United States when the adoption of the second anti-Jewish law in Hungary became imminent. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Matrix
In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It is also called a probability matrix, transition matrix, substitution matrix, or Markov matrix. The stochastic matrix was first developed by Andrey Markov at the beginning of the 20th century, and has found use throughout a wide variety of scientific fields, including probability theory, statistics, mathematical finance and linear algebra, as well as computer science and population genetics. There are several different definitions and types of stochastic matrices: :A right stochastic matrix is a real square matrix, with each row summing to 1. :A left stochastic matrix is a real square matrix, with each column summing to 1. :A doubly stochastic matrix is a square matrix of nonnegative real numbers with each row and column summing to 1. In the same vein, one may define a stochastic vector (als ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Principle
In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. In the simplest case, consider a function of two variables such that :\frac+\frac=0. The weak maximum principle, in this setting, says that for any open precompact subset of the domain of , the maximum of on the closure of is achieved on the boundary of . The strong maximum principle says that, unless is a constant function, the maximum cannot also be achieved anywhere on itself. Such statements give a striking qualitative picture of solutions of the given differential equation. Such a qualitative picture can be extended to many kinds of differential equations. In many situations, one can also use such maximum principles to draw precise quantitative conclusions about solutions of differential equations, such as control over the siz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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