![[A First Course in Probability.png]] ## 1 Combinatorial Analysis - 1.1 Introduction - 1.2 The [[Basic Principle of Counting]] - 1.3 [[Permutation|Permutations]] - 1.4 [[Combination|Combinations]] - 1.5 [[Multinomial Coefficients]] - 1.6 The Number of Integer Solutions of Equations ## 2 Axioms of Probability - 2.1 Introduction - 2.2 [[Sample Space]] and [[Event|Events]] - 2.3 [[Axioms of Probability]] - 2.4 Some Simple Propositions - 2.5 Sample Spaces Having Equally Likely Outcomes - 2.6 Probability as a Continuous Set Function - 2.7 Probability as a Measure of Belief ## 3 Conditional Probability and Independence - 3.1 Introduction - 3.2 [[Conditional Probability|Conditional Probabilities]] - 3.3 [[Bayes' Theorem|Bayes’s Formula]] - 3.4 [[Independent Events]] - 3.5 P(·|F) Is a Probability ## 4 Random Variables - 4.1 [[Random Variable|Random Variables]] - 4.2 Discrete Random Variables - 4.3 [[Expected Value]] - 4.4 Expectation of a Function of a Random Variable - 4.5 [[Variance]] - 4.6 The [[Bernoulli]] and [[Binomial]] Random Variables - 4.6.1 Properties of Binomial Random Variables - 4.6.2 Computing the Binomial Distribution Function - 4.7 The [[Poisson]] Random Variable - 4.7.1 Computing the Poisson Distribution Function - 4.8 Other Discrete Probability Distributions - 4.8.1 The [[Geometric Distribution|Geometric Random Variable]] - 4.8.2 [[Negative Binomial Distribution|Negative Binomial Random Variable]] - 4.8.3 [[Hypergeometric Distribution|Hypergeometric Random Variable]] - 4.8.4 [[Zipf's Law|Zeta (Zipf) Distribution]] - 4.9 Expected Value of Sums of Random Variables - 4.10 Properties of the Cumulative Distribution Function ## 5 Continuous Random Variables - 5.1 Introduction - 5.2 Expectation and Variance of Continuous Random Variables - 5.3 [[Uniform Distribution|Uniform Random Variable]] - 5.4 [[Normal Distribution|Normal Random Variables]] - 5.4.1 Normal Approximation to the Binomial Distribution - 5.5 [[Exponential Distribution|Exponential Random Variables]] - 5.5.1 Hazard Rate Functions - 5.6 Other Continuous Distributions - 5.6.1 [[Gamma Distribution]] - 5.6.2 [[Weibull Distribution]] - 5.6.3 [[Cauchy Distribution]] - 5.6.4 [[Beta Distribution]] - 5.6.5 [[Pareto Distribution]] - 5.7 Distribution of a Function of a Random Variable ## 6 Jointly Distributed Random Variables - 6.1 [[Joint Distribution Function|Joint Distribution Functions]] - 6.2 Independent Random Variables - 6.3 Sums of Independent Random Variables - 6.3.1 Identically Distributed Uniform Random Variables - 6.3.2 Gamma Random Variables - 6.3.3 Normal Random Variables - 6.3.4 Poisson and Binomial Random Variables - 6.4 Conditional Distributions: Discrete Case - 6.5 Conditional Distributions: Continuous Case - 6.6 [[Order Statistics]] - 6.7 Joint Distribution of Functions of Random Variables - 6.8 Exchangeable Random Variables ## 7 Properties of Expectation - 7.1 Introduction - 7.2 Expectation of Sums of Random Variables - 7.2.1 Probabilistic Method (Bounds via Expectation) - 7.2.2 Maximum–Minimum Identity - 7.3 Moments of the Number of Events - 7.4 [[Covariance]] and [[Correlation]] - 7.5 Conditional Expectation - 7.5.1 Definitions - 7.5.2 Computing Expectations by Conditioning - 7.5.3 Computing Probabilities by Conditioning - 7.5.4 Conditional Variance - 7.6 Conditional Expectation and Prediction - 7.7 Moment Generating Functions - 7.7.1 Joint Moment Generating Functions - 7.8 Additional Properties of Normal Random Variables - 7.8.1 Multivariate Normal Distribution - 7.8.2 Sample Mean and Variance Distribution - 7.9 General Definition of Expectation ## 8 Limit Theorems - 8.1 Introduction - 8.2 [[Chebyshev's Inequality]] and Weak Law of Large Numbers - 8.3 [[Central Limit Theorem]] - 8.4 Strong Law of Large Numbers - 8.5 Other Inequalities and Poisson Limit Result - 8.6 Error Bounds for Poisson Approximation - 8.7 Lorenz Curve ## 9 Additional Topics in Probability - 9.1 [[Poisson Process]] - 9.2 [[Markov Chain|Markov Chains]] - 9.3 [[Entropy (information theory)|Entropy]] - 9.4 Coding Theory and Entropy ## 10 Simulation - 10.1 Introduction - 10.2 Simulation of Continuous Random Variables - 10.2.1 Inverse Transformation Method - 10.2.2 Rejection Method - 10.3 Simulating Discrete Distributions - 10.4 Variance Reduction Techniques - 10.4.1 Antithetic Variables - 10.4.2 Conditioning - 10.4.3 Control Variates ## Appendices - Answers to Selected Problems - Solutions to Self-Test Problems and Exercises - Index - Common Discrete Distributions - Common Continuous Distributions ## Links - [worldcat.org](https://search.worldcat.org/title/1023819820)