Table of Contents

• I. FUNDAMENTALS OF PROBABILITY.
• 1. Probability Basics.
• Biography: Girolamo Cardano.
• From Percentages to Probabilities.
• Set Theory.
• 2. Mathematical Probability.
• Biography: Andrei Kolmogorov.
• Sample Space and Events.
• Axioms of Probability.
• Specifying Probabilities.
• Basic Properties of Probability.
• 3. Combinatorial Probability.
• Biography: James Bernoulli.
• The Basic Counting Rule.
• Permutations and Combinations.
• Applications of Counting Rules to Probability.
• 4. Conditional Probability and Independence.
• Biography: Thomas Bayes.
• Conditional Probability.
• The General Multiplication Rule.
• Independent Events.
• Bayes' Rule.
• III. DISCRETE RANDOM VARIABLES.
• 5. Discrete Random Variables and Their Distributions.
• Biography: Siméon-Dennis Poisson.
• From Variables to Random Variables.
• Probability Mass Functions.
• Binomial Random Variables.
• Hypergeometric Random Variables.
• Poisson Random Variables.
• Geometric Random Variables.
• Other Important Discrete Random Variables.
• Functions of a Discrete Random Variable.
• 6. Jointly Discrete Random Variables.
• Biography: Blaise Pascal.
• Joint and Marginal Probability Mass Functions: Bivariate Case.
• Joint and Marginal Probability Mass Functions: Multivariate Case.
• Conditional Probability Mass Functions.
• Independent Random Variables.
• Functions of Two or More Discrete Random Variables.
• Sums of Discrete Random Variables.
• 7. Expected Value of Discrete Random Variables.
• Biography: Christiaan Huygens.
• From Averages to Expected Values.
• Basic Properties of Expected Value.
• Variance of Discrete Random Variables.
• Variance, Covariance, and Correlation.
• Conditional Expectation.
• 8. Continuous Random Variables and Their Distributions.
• Biography: Carl Friedrich Gauss.
• Introducing Continuous Random Variables.
• Cumulative Distribution Functions.
• Probability Density Functions.
• Uniform and Exponential Random Variables.
• Normal Random Variables.
• Other Important Continuous Random Variables.
• Functions of a Continuous Random Variable.
• 9. Jointly Continuous Random Variables.
• Biography: Pierre de Fermat.
• Joint Cumulative Distribution Functions.
• Introducing Joint Probability Density Functions.
• Basic Properties of Joint Probability Density Functions.
• Marginal and Conditional Probability Density Functions.
• Independent Continuous Random Variables.
• Functions of Two or More Continuous Random Variables.
• Sums and Quotients of Continuous Random Variables.
• Multidimensional Transformation Theorem.
• 10. Expected Value of Continuous Random Variables.
• Biography: Pafnuty Chebyshev.
• Expected Value of a Continuous Random Variable.
• Basic Properties of Expected Value.
• Variance, Covariance, and Correlation.
• Conditional Expectation.
• The Bivariate Normal Distribution.
• IV. LIMIT THEOREMS AND ADVANCED TOPICS.
• 11. Generating Functions and Limit Theorems.
• Biography: William Feller.
• Moment Generating Functions.
• Joint Moment Generating Functions.
• Laws of Large Numbers.
• The Central Limit Theorem.
• 12. Additional Topics.
• Biography: Sir Ronald Fisher.
• The Poisson Process.
• Basic Queueing Theory.
• The Multivariate Normal Distribution.
• Sampling Distributions. Appendices.