Unlike traditional introductory math/stat textbooks, Probability and Statistics: The Science of Uncertainty brings a modern flavor based on incorporating the computer to the course and an integrated approach to inference. From the start the book integrates simulations into its theoretical coverage, and emphasizes the use of computer-powered computation throughout.* Math and science majors with just one year of Calculus can use this text and experience a refreshing blend of applications and theory that goes beyond merely mastering the technicalities. They'll get a thorough grounding in probability theory, and go beyond that to the theory of statistical inference and its applications. An integrated approach to inference is presented that includes the frequency approach as well as Bayesian methodology. Bayesian inference is developed as a logical extension of likelihood methods. A separate chapter is devoted to the important topic of model checking and this is applied in the context of the standard applied statistical techniques. Examples of data analyses using real-world data are presented throughout the text. A final chapter introduces a number of the most important stochastic process models using elementary methods.

Contents

1. Probability Models
    1.1 Probability: A Measure of Uncertainty
       1.1.1 Why Do We Need Probability Theory?
    1.2 Probability Models
    1.3 Basic Results for Probability Models
    1.4 Uniform Probability on Finite Spaces
       1.4.1 Combinatorial Principles
    1.5 Conditional Probability and Independence
       1.5.1 Conditional Probability
       1.5.2 Independence of Events
    1.6 Continuity of P
    1.7 Further Proofs (Advanced)
    
  2. Random Variables and Distributions
    2.1 Random Variables
    2.2 Distribution of Random Variables
    2.3 Discrete Distributions
       2.3.1 Important Discrete Distributions
    2.4 Continuous Distributions
       2.4.1 Important Absolutely Continuous Distributions
    2.5 Cumulative Distribution Functions (cdfs)
       2.5.1 Properties of Distribution Functions
       2.5.2 Cdf's of Discrete Distributions
       2.5.3 Cdf's of Absolutely Continuous Distributions
       2.5.4 Mixture Distributions
       2.5.5 Distributions Neither Discrete Nor Continuous (Advanced)
    2.6 One-dimensional Change of Variable
       2.6.1 The Discrete Case
       2.6.2 The Continuous Case
    2.7 Joint Distributions
       2.7.1 Joint Cumulative Distribution Functions
       2.7.2 Marginal Distributions
       2.7.3 Joint Probability Functions
       2.7.4 Joint Density Functions
    2.8 Conditioning and Independence
       2.8.1 Conditioning on Discrete Random Variables
       2.8.2 Conditioning on Continuous Random Variables
       2.8.3 Independence of Random Variables
       2.8.4 Sampling From a Population
    2.9 Multi-dimensional Change of Variable
       2.9.1 The Discrete Case
       2.9.2 The Continuous Case (Advance)
       2.9.3 Convolution
    2.10 Simulating Probability Distributions
       2.10.1 Simulating Discrete Distributions
       2.10.2 Simulating Continuous Distributions
    2.11 Further Proofs (Advanced)
    
  3. Expectation
    3.1 The Discrete Case
    3.2 The Absolutely Continuous Case
    3.3 Variance, Covariance and Correlation
    3.4 Generating Functions
       3.4.1 Characteristic Functions (Advanced)
    3.5 Conditional Expectation
       3.5.1 Discrete Case
       3.5.2 Absolutely Continuous Case
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