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An Introduction to Probabilistic Modeling Pierre Bremaud

An Introduction to Probabilistic Modeling By Pierre Bremaud

An Introduction to Probabilistic Modeling by Pierre Bremaud

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Summary

Introduction to the basic concepts of probability theory: independence, expectation, convergence in law and almost-sure convergence. Short expositions of more advanced topics such as Markov Chains, Stochastic Processes, Bayesian Decision Theory and Information Theory.

An Introduction to Probabilistic Modeling Summary

An Introduction to Probabilistic Modeling by Pierre Bremaud

Introduction to the basic concepts of probability theory: independence, expectation, convergence in law and almost-sure convergence. Short expositions of more advanced topics such as Markov Chains, Stochastic Processes, Bayesian Decision Theory and Information Theory.

Table of Contents

1 Basic Concepts and Elementary Models.- 1. The Vocabulary of Probability Theory.- 2. Events and Probability.- 2.1. Probability Space.- 2.2. Two Elementary Probabilistic Models.- 3. Random Variables and Their Distributions.- 3.1. Random Variables.- 3.2. Cumulative Distribution Function.- 4. Conditional Probability and Independence.- 4.1. Independence of Events.- 4.2. Independence of Random Variables.- 5. Solving Elementary Problems.- 5.1. More Formulas.- 5.2. A Small Bestiary of Exercises.- 6. Counting and Probability.- 7. Concrete Probability Spaces.- Illustration 1. A Simple Model in Genetics: Mendel's Law and Hardy-Weinberg's Theorem.- Illustration 2. The Art of Counting: The Ballot Problem and the Reflection Principle.- Illustration 3. Bertrand's Paradox.- 2 Discrete Probability.- 1. Discrete Random Elements.- 1.1. Discrete Probability Distributions.- 1.2. Expectation.- 1.3. Independence.- 2. Variance and Chebyshev's Inequality.- 2.1. Mean and Variance.- 2.2. Chebyshev's Inequality.- 3. Generating Functions.- 3.1. Definition and Basic Properties.- 3.2. Independence and Product of Generating Functions.- Illustration 4. An Introduction to Population Theory: Galton-Watson's Branching Process.- Illustration 5. Shannon's Source Coding Theorem: An Introduction to Information Theory.- 3 Probability Densities.- I. Expectation of Random Variables with a Density.- 1.1. Univariate Probability Densities.- 1.2. Mean and Variance.- 1.3. Chebyshev's Inequality.- 1.4. Characteristic Function of a Random Variable.- 2. Expectation of Functionals of Random Vectors.- 2.1. Multivariate Probability Densities.- 2.2. Covariance, Cross-Covariance, and Correlation.- 2.3. Characteristic Function of a Random Vector.- 3. Independence.- 3.1. Independent Random Variables.- 3.2. Independent Random Vectors.- 4. Random Variables That Are Not Discrete and Do Not Have a pd.- 4.1. The Abstract Definition of Expectation.- 4.2. Lebesgue's Theorems and Applications.- Illustration 6. Buffon's Needle: A Problem in Random Geometry.- 4 Gauss and Poisson.- 1. Smooth Change of Variables.- 1.1. The Method of the Dummy Function.- 1.2. Examples.- 2. Gaussian Vectors.- 2.1. Characteristic Function of Gaussian Vectors.- 2.2. Probability Density of a Nondegenerate Gaussian Vector.- 2.3. Moments of a Centered Gaussian Vector.- 2.4. Random Variables Related to Gaussian Vectors.- 3. Poisson Processes.- 3.1. Homogeneous Poisson Processes Over the Positive Half Line.- 3.2. Nonhomogeneous Poisson Processes Over the Positive Half Line.- 3.3. Homogeneous Poisson Processes on the Plane.- 4. Gaussian Stochastic Processes.- 4.1. Stochastic Processes and Their Law.- 4.2. Gaussian Stochastic Processes.- Illustration 7. An Introduction to Bayesian Decision Theory: Tests of Gaussian Hypotheses.- 5 Convergences.- 1. Almost-Sure Convergence.- 1.1. The Borel-Cantelli Lemma.- 1.2. A Criterion for Almost-Sure Convergence.- 1.3. The Strong Law of Large Numbers.- 2. Convergence in Law.- 2.1. Criterion of the Characteristic Function.- 2.2. The Central Limit Theorem.- 3. The Hierarchy of Convergences.- 3.1. Almost-Sure Convergence Versus Convergence in Probability.- 3.2. Convergence in the Quadratic Mean.- 3.3. Convergence in Law in the Hierarchy of Convergences.- 3.4. The Hierarchical Tableau.- Illustration 8. A Statistical Procedure: The Chi-Square Test.- Illustration 9. Introduction to Signal Theory: Filtering.- Additional Exercises.- Solutions to Additional Exercises.

Additional information

NLS9781461269960
9781461269960
1461269962
An Introduction to Probabilistic Modeling by Pierre Bremaud
Pierre Bremaud
Undergraduate Texts in Mathematics
New
Paperback
Springer-Verlag New York Inc.
2012-10-17
208
N/A
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