Basic Probability Theory

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ISBN: 9780486466286 | 0486466280
Cover: Paperback
Copyright: 6/26/2008

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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.

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
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Basic Probability Theory

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