First Course in Probability, A

<div class="c-un-numbered-headers-and-contents-list__container"> <h3>1. Combinatorial Analysis</h3> <ul> <li>1.1 Introduction</li> <li>1.2 The Basic Principle of Counting</li> <li>1.3 Permutations</li> <li>1.4 Combinations</li> <li>1.5 Multinomial Coefficients</li> <li>1.6 The Number of Integer Solutions of Equations</li> <li>Summary</li> <li>Problems</li> <li>Theoretical Exercises</li> <li>Self-Test Problems and Exercises</li> </ul> <h3>2. Axioms of Probability</h3> <ul> <li>2.1 Introduction</li> <li>2.2 Sample Space and Events</li> <li>2.3 Axioms of Probability</li> <li>2.4 Some Simple Propositions</li> <li>2.5 Sample Spaces Having Equally Likely Outcomes</li> <li>2.6 Probability as a Continuous Set Function</li> <li>2.7 Probability as a Measure of Belief</li> <li>Summary</li> <li>Problems</li> <li>Theoretical Exercises</li> <li>Self-Test Problems and Exercises</li> </ul> <h3>3. Conditional Probability and Inference</h3> <ul> <li>3.1 Introduction</li> <li>3.2 Conditional Probabilities</li> <li>3.3 Bayes's Formula</li> <li>3.4 Independent Events</li> <li>3.5 P(·|F) Is a Probability</li> <li>Summary</li> <li>Problems</li> <li>Theoretical Exercises</li> <li>Self-Test Problems and Exercises</li> </ul> <h3>4. Random Variables</h3> <ul> <li>4.1 Random Variables</li> <li>4.2 Discrete Random Variables</li> <li>4.3 Expected Value</li> <li>4.4 Expectation of a Function of a Random Variable</li> <li>4.5 Variance</li> <li>4.6 The Bernoulli and Binomial Random Variables</li> <li>4.7 The Poisson Random Variable</li> <li>4.8 Other Discrete Probability Distributions</li> <li>4.9 Expected Value of Sums of Random Variables</li> <li>4.10 Properties of the Cumulative Distribution Function</li> <li>Summary</li> <li>Problems</li> <li>Theoretical Exercises</li> <li>Self-Test Problems and Exercises</li> </ul> <h3>5. Continuous Random Variables</h3> <ul> <li>5.1 Introduction</li> <li>5.2 Expectation and Variance of Continuous Random Variables</li> <li>5.3 The Uniform Random Variable</li> <li>5.4 Normal Random Variables</li> <li>5.5 Exponential Random Variables</li> <li>5.6 Other Continuous Distributions</li> <li>5.7 The Distribution of a Function of a Random Variable</li> <li>Summary</li> <li>Problems</li> <li>Theoretical Exercises</li> <li>Self-Test Problems and Exercises</li> </ul> <h3>6. Jointly Distributed Random Variables</h3> <ul> <li>6.1 Joint Distribution Functions</li> <li>6.2 Independent Random Variables</li> <li>6.3 Sums of Independent Random Variables</li> <li>6.4 Conditional Distributions: Discrete Case</li> <li>6.5 Conditional Distributions: Continuous Case</li> <li>6.6 Order Statistics</li> <li>6.7 Joint Probability Distribution of Functions of Random Variables</li> <li>6.8 Exchangeable Random Variables</li> <li>Summary</li> <li>Problems</li> <li>Theoretical Exercises</li> <li>Self-Test Problems and Exercises</li> </ul> <h3>7. Properties of Expectation</h3> <ul> <li>7.1 Introduction</li> <li>7.2 Expectation of Sums of Random Variables</li> <li>7.3 Moments of the Number of Events that Occur</li> <li>7.4 Covariance, Variance of Sums, and Correlations</li> <li>7.5 Conditional Expectation</li> <li>7.6 Conditional Expectation and Prediction</li> <li>7.7 Moment Generating Functions</li> <li>7.8 Additional Properties of Normal Random Variables</li> <li>7.9 General Definition of Expectation</li> <li>Summary</li> <li>Problems</li> <li>Theoretical Exercises</li> <li>Self-Test Problems and Exercises</li> </ul> <h3>8. Limit Theorems</h3> <ul> <li>8.1 Introduction</li> <li>8.2 Chebyshev's Inequality and the Weak Law of Large Numbers</li> <li>8.3 The Central Limit Theorem</li> <li>8.4 The Strong Law of Large Numbers</li> <li>8.5 Other Inequalities and a Poisson Limit Result</li> <li>8.6 Bounding the Error Probability When Approximating a Sum of Independent Bernoulli Random Variables by a Poisson Random Variable</li> <li>8.7 The Lorenz Curve</li> <li>Summary</li> <li>Problems</li> <li>Theoretical Exercises</li> <li>Self-Test Problems and Exercises</li> </ul> <h3>9. Additional Topics in Probability</h3> <ul> <li>9.1 The Poisson Process</li> <li>9.2 Markov Chains</li> <li>9.3 Surprise, Uncertainty, and Entropy</li> <li>9.4 Coding Theory and Entropy</li> <li>Summary</li> <li>Problems and Theoretical Exercises</li> <li>Self-Test Problems and Exercises</li> </ul> <h3>10. Simulation</h3> <ul> <li>10.1 Introduction</li> <li>10.2 General Techniques for Simulating Continuous Random Variables</li> <li>10.3 Simulating from Discrete Distributions</li> <li>10.4 Variance Reduction Techniques</li> <li>Summary</li> <li>Problems</li> <li>Self-Test Problems and Exercises</li> </ul> <h4 class="h5">Answers to Selected Problems</h4> <h4 class="h5">Solutions to Self-Test Problems and Exercises</h4> <h4 class="h5">Index</h4> <h4 class="h5">Common Discrete Distributions</h4> <h4 class="h5">Common Continuous Distributions</h4> </div>