Applied Probability : From Random Experiments to Random Sequences and Statistics / / by Valérie Girardin, Nikolaos Limnios

• Autore: Girardin Valérie
• Pubblicazione: Cham : , : Springer International Publishing : , : Imprint : Springer, , 2022
• Edizione: 1st ed. 2022.
• Descrizione fisica: 1 online resource (265 pages)
• Disciplina: 519.2
• Soggetto topico: Probabilities; Statistics; Probability Theory; Statistical Theory and Methods; Applied Probability; Applied Statistics
• Persona (resp. second.): LimniosN
• Nota di contenuto: Intro -- Preface -- Contents -- Notation -- 1 Events and Probability Spaces -- 1.1 Sample Space -- 1.2 Measure Spaces -- 1.2.1 σ-Algebras -- Properties of σ-Algebras -- 1.2.2 Measures -- Properties of Measures -- Dirac Measure -- Counting Measure -- Lebesgue Measure -- 1.3 Probability Spaces -- 1.3.1 General Case -- 1.3.2 Conditional Probabilities -- 1.3.3 Discrete Case: Combinatorial Analysis and Entropy -- Properties of Shannon Entropy -- 1.4 Independence of Finite Collections -- 1.5 Exercises -- 2 Random Variables -- 2.1 Random Variables -- 2.1.1 Measurable Functions -- Properties of Measurable Functions -- 2.1.2 Distributions and Distribution Functions -- Properties of Distribution Functions -- Properties of Quantiles -- 2.2 Expectation -- 2.2.1 Lebesgue Integral -- Properties of Lebesgue Integrals -- 2.2.2 Expectation -- 2.3 Discrete Random Variables -- 2.3.1 General Properties -- 2.3.2 Classical Discrete Distributions -- Dirac Distribution -- Uniform Distribution -- Bernoulli Distribution -- Binomial Distribution -- Hyper-Geometric Distribution -- Geometric and Negative Binomial Distributions -- Poisson Distribution -- 2.4 Continuous Random Variables -- 2.4.1 Absolute Continuity of Measures -- 2.4.2 Densities -- Properties of Densities of Random Variables -- 2.4.3 Classical Distributions with Densities -- Uniform Distribution -- Gaussian Distribution -- Gamma, Exponential, Chi-Squared, Erlang Distributions -- Log-Normal Distribution -- Weibull Distribution -- Inverse-Gaussian Distribution -- Beta Distribution -- Fisher Distribution -- Student and Cauchy Distributions -- 2.4.4 Determination of Distributions -- 2.5 Analytical Tools -- 2.5.1 Generating Functions -- Properties of Generating Functions -- 2.5.2 Fourier Transform and Characteristic Functions -- Properties of Characteristic Functions -- 2.5.3 Laplace Transform. ; Properties of Laplace Transforms -- 2.5.4 Moment Generating Functions and Cramér Transform -- Properties of Cramér Transform -- 2.6 Reliability and Survival Analysis -- 2.7 Exercises and Complements -- 3 Random Vectors -- 3.1 Relations Between Random Variables -- 3.1.1 Covariance -- Properties of Covariance and Correlation Coefficients -- 3.1.2 Independence of Random Variables -- 3.1.3 Stochastic Order Relation -- 3.1.4 Entropy -- Properties of Entropy -- 3.2 Characteristics of Random Vectors -- 3.2.1 Product of Probability Spaces -- 3.2.2 Distribution of Random Vectors -- Properties of Multi-dimensional Distribution Functions -- Properties of Densities of Random Vectors -- Properties of Covariance Matrices -- 3.2.3 Independence of Random Vectors -- Properties of Covariance Matrices of Two Vectors -- 3.3 Functions of Random Vectors -- 3.3.1 Order Statistics -- 3.3.2 Sums of Independent Variables or Vectors -- Properties of Convolution -- 3.3.3 Determination of Distributions -- 3.4 Gaussian Vectors -- 3.5 Exercises and Complements -- 4 Random Sequences -- 4.1 Enumerable Sequences -- 4.1.1 Sequences of Events -- Properties of Superior and Inferior Limits of Events -- 4.1.2 Independence of Sequences -- 4.2 Stochastic Convergence -- 4.2.1 Different Types of Convergence -- 4.2.2 Convergence Criteria -- 4.2.3 Links Between Convergences -- 4.2.4 Convergence of Sequences of Random Vectors -- 4.3 Limit Theorems -- 4.3.1 Asymptotics of Discrete Distributions -- 4.3.2 Laws of Large Numbers -- 4.3.3 Central Limit Theorem -- 4.4 Stochastic Simulation Methods -- 4.4.1 Generating Random Variables -- 4.4.2 Monte Carlo Simulation Method -- 4.5 Exercises and Complements -- 5 Introduction to Statistics -- 5.1 Non-parametric Statistics -- 5.1.1 Empirical Distribution Function -- 5.1.2 Confidence Intervals -- 5.1.3 Non-parametric Testing -- 5.2 Parametric Statistics. ; 5.2.1 Point Estimation -- 5.2.2 Maximum Likelihood Method -- 5.2.3 Precision of the Estimators -- 5.2.4 Parametric Confidence Intervals -- 5.2.5 Testing in a Parametric Model -- 5.3 The Linear Model -- 5.3.1 Linear and Quadratic Approximations -- 5.3.2 The Simple Linear Model -- 5.3.3 ANOVA -- For Two Samples -- One Way Model -- Two Way Model -- 5.4 Exercises and Complements -- Further Reading -- Measure and Probability -- Probability Theory and Statistics -- Applications -- Index.
• Sommario/riassunto: This textbook presents the basics of probability and statistical estimation, with a view to applications. The didactic presentation follows a path of increasing complexity with a constant concern for pedagogy, from the most classical formulas of probability theory to the asymptotics of independent random sequences and an introduction to inferential statistics. The necessary basics on measure theory are included to ensure the book is self-contained. Illustrations are provided from many applied fields, including information theory and reliability theory. Numerous examples and exercises in each chapter, all with solutions, add to the main content of the book. Written in an accessible yet rigorous style, the book is addressed to advanced undergraduate students in mathematics and graduate students in applied mathematics and statistics. It will also appeal to students and researchers in other disciplines, including computer science, engineering, biology, physicsand economics, who are interested in a pragmatic introduction to the probability modeling of random phenomena.
• Titolo autorizzato: Applied Probability
• ISBN: 3-030-97963-6
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910568249603321
• Lo trovi qui: Univ. Federico II