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[null,"1.","Record Nr.",null,"UNINA9910855382403321"]
[null,null,"Autore",null,"Brémaud Pierre"]
[null,null,"Titolo",null,"An Introduction to Applied Probability"]
[null,null,"Pubbl/distr/stampa",null,"Cham : , : Springer International Publishing AG, , 2024"]
[{"$nulls":4},"©2024"]
[null,null,"ISBN",null,"3-031-49306-0"]
[null,null,"Edizione",null,"[1st ed.]"]
[null,null,"Descrizione fisica",null,"1 online resource (498 pages)"]
[null,null,"Collana",null,"Texts in Applied Mathematics Series ; ; v.77"]
[null,null,"Lingua di pubblicazione",null,"Inglese"]
[null,null,"Formato",null,"Materiale a stampa"]
[null,null,null,"Livello bibliografico","Monografia"]
[null,null,"Nota di contenuto",null,"Intro -- Preface -- Part I: The Elementary Calculus -- Part II: The Essential Theory -- Part III: The Important Models -- Contents -- Chapter 1 Basic Notions -- 1.1 Outcomes and Events -- 1.2 Probability of Events -- 1.3 Independence and Conditioning -- 1.4 Counting Models -- 1.5 Exercises -- Chapter 2 Discrete Random Variables -- 2.1 Probability Distribution and Expectation -- Independence and Conditional Independence -- Expectation -- Markov's Inequality -- Jensen's Inequality -- Moment Bounds -- Product Rule for Expectation -- 2.2 Remarkable Discrete Distributions -- Uniform -- Binomial -- Geometric -- Poisson -- Hypergeometric -- Multinomial -- 2.3 Generating Functions -- Moments from the Generating Function -- Random Sums -- Branching Trees -- 2.4 Conditional Expectation I -- 2.5 Exercises -- Chapter 3 Continuous Random Vectors -- 3.1 Random Variables with Real Values -- Expectation -- Mean and Variance -- Remarkable Continuous Random Variables -- Characteristic Functions -- Laplace Transforms -- Random Vectors -- 3.2 Continuous Random Vectors -- Product Formula for Expectations -- Freeze and Integrate -- Characteristic Functions and Laplace Transforms of Random Vectors -- Characteristic Function Test for Independence -- Random Sums and Wald's Identity -- Smooth Change of Variables -- Order Statistics -- Sampling a Distribution -- 3.3 Square-integrable Random Variables -- Inner Product and Schwarz's Inequality -- The Correlation Coefficient -- Covariance Matrices -- Linear Regression -- 3.4 Gaussian Vectors -- Mixed Moments of Gaussian Vectors -- Independence and Non-Correlation -- Probability Density of a Non-degenerate Gaussian Vector -- Empirical Mean and Variance of the Gaussian Distribution -- 3.5 "]
[{"$nulls":4},"Conditional Expectation II -- Properties of the Conditional Expectation -- Bayesian Tests of Hypotheses -- 3.6 Exercises."]
[{"$nulls":4},"Chapter 4 The Lebesgue Integral -- 4.1 Measurable Functions and Measures -- Measurable Functions -- Measure -- μ-negligible sets -- Cumulative Distribution Function -- Caratheodory's Theorem -- 4.2 The Integral -- 4.3 Basic Properties of the Integral -- Beppo Levi, Fatou and Lebesgue -- Differentiation under the Integral Sign -- 4.4 The Big Theorems -- The Image Measure Theorem -- The Radon-Nikod´ym Theorem -- The Fubini-Tonelli Theorem -- The Formula of Integration by Parts -- Lp-spaces and the Riesz-Fischer Theorem -- 4.5 Exercises -- 4.6 Solutions -- Chapter 5 From Integral to Expectation -- 5.1 Translation -- 5.2 The Distribution of a Random Element -- 5.3 Characteristic Functions -- 5.4 Independence -- The Product Formula -- 5.5 Conditional Expectation III -- 5.6 General Theory of Conditional Expectation -- A Special Case -- Properties of the Conditional Expectation -- The L2-theory of Conditional Expectation -- Nonlinear Regression -- 5.7 Exercises -- 5.8 Solutions -- Chapter 6 Convergence Almost Sure -- 6.1 A Sufficient Condition and a Criterion -- The Borel-Cantelli Lemma -- A Sufficient Condition -- A Criterion -- 6.2 The Strong Law of Large Numbers -- Kolmogorov's Strong Law of Large Numbers -- Large Deviations from the Strong Law of Large Numbers -- 6.3 Kolmogorov's Zero-one Law -- 6.4 Related Types of Convergence -- Convergence in Probability -- Convergence in the Quadratic Mean -- 6.5 Uniform Integrability -- 6.6 Exercises -- Chapter 7 Convergence in Distribution -- 7.1 Paul Lévy's Criterion -- Bochner's Theorem -- 7.2 The Central Limit Theorem -- Confidence Intervals -- 7.3 Convergence in Variation -- 7.4 The Rank of Convergence in Distribution -- A Stability Property of the Gaussian Distribution -- Skorokhod's Theorem -- 7.5 Exercises -- Chapter 8 Martingales -- 8.1 The Martingale Property -- Convex Functions of Martingales."]
[{"$nulls":4},"Martingale Transforms and Stopped Martingales -- 8.2 Martingale Inequalities -- Kolmogorov's Inequality -- Doob's Inequality -- Hoeffding's Inequality -- 8.3 The Optional Sampling Theorem -- Wald's Formulas -- 8.4 The Martingale Convergence Theorem -- The Upcrossing Inequality -- Backwards (or Reverse) Martingales -- The Robbins-Sigmund Theorem -- 8.5 Square-integrable Martingales -- Doob's decomposition -- The Martingale Law of Large Numbers -- 8.6 Exercises -- Chapter 9 Markov Chains -- 9.1 The Transition Matrix -- First-step Analysis -- Communication and Period -- Stationary Distributions -- Reversible Chains -- The Strong Markov Property -- The Cycle Independence Property -- 9.2 Recurrence -- The Potential Matrix Criterion -- Invariant Measure -- The Stationary Distribution Criterion of Positive Recurrence -- Birth-and-Death Markov Chain -- Foster's Theorem -- 9.3 Long-run Behavior -- The Markov Chain Ergodic Theorem -- The Markov Chain Convergence Theorem -- 9.4 Absorption -- Before Absorption -- Time to Absorption -- Final Destination -- 9.5 The Markov Property on Graphs -- Gibbs Distributions -- The Hammersley-Clifford Theorem -- 9.6 Monte Carlo Markov Chains -- Simulation of Random Fields -- The Propp-Wilson Algorithm -- 9.7 Exercises -- Chapter 10 Poisson Processes -- 10.1 Poisson Processes on the Line -- The Counting Process of an HPP -- Competing Poisson Processes -- 10.2 Generalities on Point Processes -- Independent Point Processes -- Marked Point Processes -- Point Process Integrals -- The Intensity Measure -- Campbell's Formula -- The Laplace Functional -- 10.3 Spatial Poisson Processes -- Doubly Stochastic Poisson Processes -- The Covariance Formula -- The Exponential Formula -- Marked Spatial Poisson Processes -- 10.4 Operations on Poisson Processes -- Thinning and Coloring -- "]
[{"$nulls":4},"Transportation -- Poisson Shot Noise -- 10.5 Exercises."]
[{"$nulls":4},"Chapter 11 Brownian Motion -- 11.1 Continuous-time Stochastic Processes -- Second-order Stochastic Processes -- Wide-sense Stationarity -- 11.2 Gaussian Processes -- The Wiener Process -- Pathology -- The Brownian Bridge -- Gauss-Markov Processes -- 11.3 The Wiener-Doob Integral -- Gaussian Subspaces -- Construction of the Wiener-Doob Integral -- A Formula of Integration by Parts Theorem 11.3.6 -- 11.4 Two Applications -- Langevin's Equation -- The Cameron-Martin Formula -- 11.5 Fractal Brownian Motion -- 11.6 Exercises -- Chapter 12 Wide-sense Stationary Processes -- 12.1 The Power Spectral Measure -- The General Case -- Special Cases -- 12.2 Filtering of WSS Sochastic Processes -- White Noise -- 12.3 The Cramér-Khinchin Decomposition -- A Plancherel-Parseval Formula -- Linear Operations on WSS Stochastic Processes -- Stochastic Processes -- Linear Transformations of Gaussian Processes -- 12.4 Multivariate WSS Stochastic Processes -- Band-pass Stochastic Processes -- 12.5 Exercises -- Appendix A: A Review of Hilbert Spaces -- Basic Definitions -- Schwarz's Inequality -- Isometric Extension -- Orthogonal Projection -- Bibliography -- Index."]