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Basic Concepts | |
Introduction | p. 1 |
Algebra of Events (Boolean Algebra) | p. 3 |
Probability | p. 10 |
Combinatorial Problems | p. 15 |
Independence | p. 25 |
Conditional Probability | p. 33 |
Some Fallacies in Combinatorial Problems | p. 39 |
Appendix: Stirling's Formula | p. 43 |
Random Variables | |
Introduction | p. 46 |
Definition of a Random Variable | p. 48 |
Classification of Random Variables | p. 51 |
Functions of a Random Variable | p. 58 |
Properties of Distribution Functions | p. 66 |
Joint Density Functions | p. 70 |
Relationship Between Joint and Individual Densities; Independence of Random Variables | p. 76 |
Functions of More Than One Random Variable | p. 85 |
Some Discrete Examples | p. 95 |
Expectation | |
Introduction | p. 100 |
Terminology and Examples | p. 107 |
Properties of Expectation | p. 114 |
Correlation | p. 119 |
The Method of Indicators | p. 122 |
Some Properties of the Normal Distribution | p. 124 |
Chebyshev's Inequality and the Weak Law of Large Numbers | p. 126 |
Conditional Probability and Expectation | |
Introduction | p. 130 |
Examples | p. 133 |
Conditional Density Functions | p. 135 |
Conditional Expectation | p. 140 |
Appendix: The General Concept of Conditional Expectation | p. 152 |
Characteristic Functions | |
Introduction | p. 154 |
Examples | p. 158 |
Properties of Characteristic Functions | p. 166 |
The Central Limit Theorem | p. 169 |
Infinite Sequences of Random Variables | |
Introduction | p. 178 |
The Gambler's Ruin Problem | p. 182 |
Combinatorial Approach to the Random Walk; the Reflection Principle | p. 186 |
Generating Functions | p. 191 |
The Poisson Random Process | p. 196 |
The Strong Law of Large Numbers | p. 203 |
Markov Chains | |
Introduction | p. 211 |
Stopping Times and the Strong Markov Property | p. 217 |
Classification of States | p. 220 |
Limiting Probabilities | p. 230 |
Stationary and Steady-State Distributions | p. 236 |
Introduction to Statistics | |
Statistical Decisions | p. 241 |
Hypothesis Testing | p. 243 |
Estimation | p. 258 |
Sufficient Statistics | p. 264 |
Unbiased Estimates Based on a Complete Sufficient Statistic | p. 268 |
Sampling from a Normal Population | p. 274 |
The Multidimensional Gaussian Distribution | p. 279 |
Tables | p. 286 |
A Brief Bibliography | p. 289 |
Solutions to Problems | p. 290 |
Index | p. 333 |
Table of Contents provided by Ingram. All Rights Reserved. |
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This introduction to more advanced courses in probability and real analysis emphasizes the probabilistic way of thinking, rather than measure-theoretic concepts. Geared toward advanced undergraduates and graduate students, its sole prerequisite is calculus.
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