LEADER 05907nam 2200553 450 001 9910568249603321 005 20230717151949.0 010 $a3-030-97963-6 035 $a(MiAaPQ)EBC6975909 035 $a(Au-PeEL)EBL6975909 035 $a(CKB)21957562900041 035 $a(OCoLC)1314629464 035 $a(PPN)269154973 035 $a(EXLCZ)9921957562900041 100 $a20221126d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aApplied probability $efrom random experiments to random sequences and statistics /$fVale?rie Girardin and Nikolaos Limnios 210 1$aCham, Switzerland :$cSpringer,$d[2022] 210 4$d©2022 215 $a1 online resource (265 pages) 311 08$aPrint version: Girardin, Valérie Applied Probability Cham : Springer International Publishing AG,c2022 9783030979621 327 $aIntro -- Preface -- Contents -- Notation -- 1 Events and Probability Spaces -- 1.1 Sample Space -- 1.2 Measure Spaces -- 1.2.1 ?-Algebras -- Properties of ?-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. 327 $aProperties 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. 327 $a5.2.1 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. 606 $aDistribution (Probability theory) 606 $aStatistics 606 $aStochastic processes 606 $aProbabilitats$2thub 606 $aEstadística matemàtica$2thub 608 $aLlibres electrònics$2thub 615 0$aDistribution (Probability theory) 615 0$aStatistics. 615 0$aStochastic processes. 615 7$aProbabilitats 615 7$aEstadística matemàtica 676 $a519.2 700 $aGirardin$b Vale?rie$0768207 702 $aLimnios$b N. 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910568249603321 996 $aApplied Probability$92070204 997 $aUNINA