Basic Probability Theory

Cover ......Page 1
Title ......Page 2
Copyright Page ......Page 3
Preface ......Page 4
Contents ......Page 6
1.1 Introduction ......Page 12
1.2 Algebra of Events (Boolean Algebra) ......Page 14
1.3 Probability ......Page 21
1.4 Combinatorial Problems ......Page 26
1.5 Independence ......Page 36
1.6 Conditional Probability ......Page 44
1.7 Some Fallacies in Combinatorial Problems ......Page 50
1.8 Appendix: Stirling\'s Formula ......Page 54
2.1 Introduction ......Page 57
2.2 Definition of a Random Variable ......Page 59
2.3 Classification of Random Variables ......Page 62
2.4 Functions of a Random Variable ......Page 69
2.5 Properties of Distribution Functions ......Page 77
2.6 Joint Density Functions ......Page 81
2.7 Relationship Between Joint and Individual Densities; Independence of Random Variables ......Page 87
2.8 Functions of More Than One Random Variable ......Page 96
2.9 Some Discrete Examples ......Page 106
3.1 Introduction ......Page 111
3.2 Terminology and Examples ......Page 118
3.3 Properties of Expectation ......Page 125
3.4 Correlation ......Page 130
3.5 The Method of Indicators ......Page 133
3.6 Some Properties of the Normal Distribution ......Page 135
3.7 Chebyshev\'s Inequality at?-d the Weak Law of Large Numbers ......Page 137
4.1 Introduction ......Page 141
4.2 Examples ......Page 144
4.3 Conditional Density Functions ......Page 146
4.4 Conditional Expectation ......Page 151
4.5 Appendix : The General Concept of Conditional Expectation ......Page 163
5.1 Introduction ......Page 165
5.2 Examples ......Page 169
5.3 Properties of Characteristic Functions ......Page 177
5.4 The Central Limit Theorem ......Page 180
6.1 Introduction ......Page 189
6.2 The Gambler\'s Ruin Problem ......Page 193
6.3 Combinatorial Approach to the Random Walk; the Reflection Principle ......Page 197
6.4 Generating Functions ......Page 202
6.5 The Poisson Random Process ......Page 207
6.6 The Strong Law of Large Numbers ......Page 214
7.1 Introduction ......Page 222
7.2 Stopping Times and the Strong Markov Property ......Page 228
7.3 Classification of States ......Page 231
7.4 Limiting Probabilities ......Page 241
7.5 Stationary and Steady-State Distributions ......Page 247
8.1 Statistical Decisions ......Page 252
8.2 Hypothesis Testing ......Page 254
8.3 Estimation ......Page 269
8.4 Sufficient Statistics ......Page 275
8.5 Unbiased Estimates Based on a Complete Sufficient Statistic ......Page 279
8.6 Sampling from a Normal Population ......Page 285
8.7 The Multidimensional Gaussian Distribution ......Page 290
Tables ......Page 297
A Brief Bibliography ......Page 300
Solutions to Problems ......Page 301
Index ......Page 344