A First Course In Probability And Markov Chains Pdf
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- A First Course in Probability and Markov Chains
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Mathematical Sciences Research Institute
Provides an introduction to basic structures of probability with a view towards applications in information technology A First Course in Probability and Markov Chains presents an introduction to the basic elements in probability and focuses on two main areas. The first part explores notions and structures in probability, including combinatorics, probability measures, probability distributions, conditional probability, inclusion-exclusion formulas, random variables, dispersion indexes, independent random variables as well as weak and strong laws of large numbers and central limit theorem. This book also looks at making use of measure theory notations that unify all the presentation, in particular avoiding the separate treatment of continuous and discrete distributions. Explores elementary probability with combinatorics, uniform probability, the inclusion-exclusion principle, independence and convergence of random variables. Features applications of Law of Large Numbers.
Probability theory, statistics as well as mathematical physics have increasingly been used in computer science. The goal of this school is to provide a unique opportunity for graduate students and young researchers to developed multi-disciplinary skills in a rapidly evolving area of mathematics. The topics would include spin glasses, constraint satisfiability, randomized algorithms, Monte-Carlo Markov chains and high-dimensional statistics, sparse and random graphs, computational complexity, estimation and approximation algorithms. Those topics will fall into two main categories, on the one hand problems related to spin glasses and on the other hand random algorithms. The part of the summer school dedicated to spin glasses will be split into three parts: an introductory course about traditional spin glasses followed by two more advanced courses where spin glasses meet computer science in addition to a talk on dynamics of spin glasses. The part of the summer school on random algorithms will consist of an introductory course on phase transitions in large random structures, followed by advanced courses on theoretical bounds for computational complexity in reconstruction and inference, and on understanding rare events in random graphs and models of statistical mechanics. The two introductory courses on spin glasses and on random algorithms will be accompanied by three exercises sessions of one hour.
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Markov chains are widely used as models of real-world processes, especially where it is important to incorporate a degree of randomness in the evolution of the system. For instance, when modelling the outbreak of a disease in a population, the current state of the system can be captured reasonably accurately by the number of people presently infected and the number of those susceptible to the disease, but the future course of the epidemic cannot normally be known with any confidence. It may be influenced by a few chance events, especially when the number of infectives is small such as at the start or near the end of the outbreak , which may have a crucial impact on how long it takes before the epidemic is contained and how many people will end up being infected. This course will cover some important aspects of the theory of Markov chains, in discrete and continuous time. We start with the basics, including a discussion of convergence of the time-dependent distribution to equilibrium as time goes to infinity, in the case where the state space has a fixed size.
Jetzt bewerten Jetzt bewerten. Provides an introduction to basic structures of probabilitywith a view towards applications in information technology A First Course in Probability and Markov Chains presentsan introduction to the basic elements in probability and focuses ontwo main areas. The first part explores notions and structures inprobability, including combinatorics, probability measures,probability distributions, conditional probability,inclusion-exclusion formulas, random variables, dispersion indexes,independent random variables as well as weak and strong laws oflarge numbers and central limit theorem. In the second …mehr. DE
A First Course in Probability and Markov Chains
A Markov chain 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. 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 distributions, and have found application in Bayesian statistics , thermodynamics , statistical mechanics , physics , chemistry , economics , finance , signal processing , information theory and artificial intelligence. The adjective Markovian is used to describe something that is related to a Markov process.
This is a text encompassing all of the standard topics in introductory probability theory, together with a significant amount of optional material of emerging importance. The emphasis is on a lucid and accessible writing style, mixed with a large number of interesting examples of a diverse nature. The text will prepare students extremely well for courses in more advanced probability and in statistical theory and for the actuary exam. The book covers combinatorial probability, all the standard univariate discrete and continuous distributions, joint and conditional distributions in the bivariate and the multivariate case, the bivariate normal distribution, moment generating functions, various probability inequalities, the central limit theorem and the laws of large numbers, and the distribution theory of order statistics. In addition, the book gives a complete and accessible treatment of finite Markov chains, and a treatment of modern urn models and statistical genetics. It includes worked out examples and exercises, including a large compendium of supplementary exercises for exam preparation and additional homework.
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