Exercises in Probability

I Elementary Probabilities.- I Basic Probabilities. Discrete Spaces.- Basic Definitions and Formulas Exercises.- Exercises.- 1. Sets. Events: 1–8.- 2. Combinatorics: 9–16.- 3. Properties of Binomial Coefficients: 17–25.- 4. Properties of Probability: 26–34.- 5. Classical Probabilities. Equally Likely Cases: 35–53.- 6. Independent Events. Conditional Probability: 54–79.- 2 Distributions. Random Variables.- Elements of Theory.- Exercises.- 1. Discrete Distributions: 80–89.- 2. Continuous Distributions: 90–100.- 3 Expectation. Variance. Moments.- Elements of Theory.- Exercises.- 1. Theoretical Exercises: 101–113.- 2. Mean and Variance: 114–125.- 4 General Problems: 126–170.- II Advanced Topics.- 5 Multivariate Distributions.- Elements of Theory.- Exercises: 171–190.- 6 Generating Functions. Characteristic Functions.- Elements of Theory.- Exercises: 191–215.- 7 Distribution of Functions of Random Variables.- Elements of Theory.- Exercises: 216–250.- 8 Limit Theorems. Laws of Large Numbers. Central Limit Theorems.- Elements of Theory.- Exercises: 251–269.- 9 Special Topics: Inequalities, Geometrical Probabilities, Difference Equations.- Elements of Theory.- A. Inequalities.- B. Geometrical Probabilities.- C. Difference Equations.- Exercises.- A. Inequalities: 270–282.- B. Geometrical Probabilities: 283–289.- C. Difference Equations: 290–300.- 10 General Exercises: 301–329.- Supplements.- Supplement I Miscellaneous Exercises: I-1–I-56.- Supplement II Complements and Problems.- 1. Multivariate Distributions: 1.1–1.27.- 2. Generating Functions: 2.1–2.22.- 3. Transformation of Random Variables: 3.1–3.15.- 4. Convergence of Random Variables: 4.1–4.19.- 5. Miscellaneous Complements and Problems: 5.1–5.29.- III Solutions.- Solutions: 1–329.