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Linde, Werner. Probability Theory [[electronic resource].]: De Gruyter, [2017]. — 1 online resource. — (De Gruyter Textbook). — <URL:http://elib.fa.ru/ebsco/1438416.pdf>.

Record create date: 11/25/2016

Subject: Probabilities — Textbooks.; Mathematical statistics — Textbooks.; Measure theory — Textbooks.; MATHEMATICS / Applied; MATHEMATICS / Probability & Statistics / General

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Table of Contents

  • Preface
  • Contents
  • 1 Probabilities
    • 1.1 Probability Spaces
      • 1.1.1 Sample Spaces
      • 1.1.2 3-Fields of Events
      • 1.1.3 Probability Measures
    • 1.2 Basic Properties of Probability Measures
    • 1.3 Discrete Probability Measures
    • 1.4 Special Discrete Probability Measures
      • 1.4.1 Dirac Measure
      • 1.4.2 Uniform Distribution on a Finite Set
      • 1.4.3 Binomial Distribution
      • 1.4.4 Multinomial Distribution
      • 1.4.5 Poisson Distribution
      • 1.4.6 Hypergeometric Distribution
      • 1.4.7 Geometric Distribution
      • 1.4.8 Negative Binomial Distribution
    • 1.5 Continuous Probability Measures
    • 1.6 Special Continuous Distributions
    • 1.6.1 Uniform Distribution on an Interval
      • 1.6.2 Normal Distribution
      • 1.6.3 Gamma Distribution
      • 1.6.4 Exponential Distribution
      • 1.6.5 Erlang Distribution
      • 1.6.6 Chi-Squared Distribution
      • 1.6.7 Beta Distribution
      • 1.6.8 Cauchy Distribution
    • 1.7 Distribution Function
    • 1.8 Multivariate Continuous Distributions
      • 1.8.1 Multivariate Density Functions
      • 1.8.2 Multivariate Uniform Distribution
    • 1.9 *Products of Probability Spaces
      • 1.9.1 Product 3-Fields and Measures
      • 1.9.2 Product Measures: Discrete Case
      • 1.9.3 Product Measures: Continuous Case
    • 1.10 Problems
  • 2 Conditional Probabilities and Independence
    • 2.1 Conditional Probabilities
    • 2.2 Independence of Events
    • 2.3 Problems
  • 3 Random Variables and Their Distribution
    • 3.1 Transformation of Random Values
    • 3.2 Probability Distribution of a Random Variable
    • 3.3 Special Distributed Random Variables
    • 3.4 Random Vectors
    • 3.5 Joint and Marginal Distributions
      • 3.5.1 Marginal Distributions: Discrete Case
      • 3.5.2 Marginal Distributions: Continuous Case
    • 3.6 Independence of Random Variables
      • 3.6.1 Independence of Discrete Random Variables
      • 3.6.2 Independence of Continuous Random Variables
    • 3.7 *Order Statistics
    • 3.8 Problems
  • 4 Operations on Random Variables
    • 4.1 Mappings of Random Variables
    • 4.2 Linear Transformations
    • 4.3 Coin Tossing versus Uniform Distribution
      • 4.3.1 Binary Fractions
      • 4.3.2 Binary Fractions of Random Numbers
      • 4.3.3 Random Numbers Generated by Coin Tossing
    • 4.4 Simulation of Random Variables
    • 4.5 Addition of Random Variables
      • 4.5.1 Sums of Discrete Random Variables
      • 4.5.2 Sums of Continuous Random Variables
    • 4.6 Sums of Certain Random Variables
    • 4.7 Products and Quotients of Random Variables
      • 4.7.1 Student’s t-Distribution
      • 4.7.2 F-Distribution
    • 4.8 Problems
  • 5 Expected Value, Variance, and Covariance
    • 5.1 Expected Value
      • 5.1.1 Expected Value of Discrete Random Variables
      • 5.1.2 Expected Value of Certain Discrete Random Variables
      • 5.1.3 Expected Value of Continuous Random Variables
      • 5.1.4 Expected Value of Certain Continuous Random Variables
      • 5.1.5 Properties of the Expected Value
    • 5.2 Variance
      • 5.2.1 Higher Moments of Random Variables
      • 5.2.2 Variance of Random Variables
      • 5.2.3 Variance of Certain Random Variables
    • 5.3 Covariance and Correlation
      • 5.3.1 Covariance
      • 5.3.2 Correlation Coefficient
    • 5.4 Problems
  • 6 Normally Distributed Random Vectors
    • 6.1 Representation and Density
    • 6.2 Expected Value and Covariance Matrix
    • 6.3 Problems
  • 7 Limit Theorems
    • 7.1 Laws of Large Numbers
      • 7.1.1 Chebyshev’s Inequality
      • 7.1.2 *Infinite Sequences of Independent Random Variables
      • 7.1.3 * Borel–Cantelli Lemma
      • 7.1.4 Weak Law of Large Numbers
      • 7.1.5 Strong Law of Large Numbers
    • 7.2 Central Limit Theorem
    • 7.3 Problems
  • 8 Mathematical Statistics
    • 8.1 Statistical Models
      • 8.1.1 Nonparametric Statistical Models
      • 8.1.2 Parametric Statistical Models
    • 8.2 Statistical Hypothesis Testing
      • 8.2.1 Hypotheses and Tests
      • 8.2.2 Power Function and Significance Tests
    • 8.3 Tests for Binomial Distributed Populations
    • 8.4 Tests for Normally Distributed Populations
      • 8.4.1 Fisher’s Theorem
      • 8.4.2 Quantiles
      • 8.4.3 Z-Tests or Gauss Tests
      • 8.4.4 t-Tests
      • 8.4.5 72-Tests for the Variance
      • 8.4.6 Two-Sample Z-Tests
      • 8.4.7 Two-Sample t-Tests
      • 8.4.8 F-Tests
    • 8.5 Point Estimators
      • 8.5.1 Maximum Likelihood Estimation
      • 8.5.2 Unbiased Estimators
      • 8.5.3 Risk Function
    • 8.6 Confidence Regions and Intervals
    • 8.7 Problems
  • A Appendix
    • A.1 Notations
    • A.2 Elements of Set Theory
    • A.3 Combinatorics
      • A.3.1 Binomial Coefficients
      • A.3.2 Drawing Balls out of an Urn
      • A.3.3 Multinomial Coefficients
    • A.4 Vectors and Matrices
    • A.5 Some Analytic Tools
  • Bibliography
  • Index

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