## Probability Via ExpectationThis book is a complete revision of the earlier work Probability which ap peared in 1970. While revised so radically and incorporating so much new material as to amount to a new text, it preserves both the aim and the approach of the original. That aim was stated as the provision of a 'first text in probability, de manding a reasonable but not extensive knowledge of mathematics, and taking the reader to what one might describe as a good intermediate level'. In doing so it attempted to break away from stereotyped applications, and consider applications of a more novel and significant character. The particular novelty of the approach was that expectation was taken as the prime concept, and the concept of expectation axiomatized rather than that of a probability measure. In the preface to the original text of 1970 (reproduced below, together with that to the Russian edition of 1982) I listed what I saw as the advantages of the approach in as unlaboured a fashion as I could. I also took the view that the text rather than the author should persuade, and left the text to speak for itself. It has, indeed, stimulated a steady interest, to the point that Springer-Verlag has now commissioned this complete reworking. |

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### Contents

Uncertainty Intuition and Expectation | 1 |

2 The Empirical Basis | 2 |

3 Averages over a Finite Population | 5 |

Expectation | 8 |

5 More on Sample Spaces and Variables | 10 |

Observables | 11 |

Expectation | 13 |

2 Axioms for the Expectation Operator | 14 |

6 Random Walks Random Stopping and Ruin | 162 |

7 Auguries of Martingales | 168 |

8 Recurrence and Equilibrium | 169 |

9 Recurrence and Dimension | 173 |

Markov Processes in Continuous Time | 175 |

2 The Case of a Discrete State Space | 176 |

3 The Poisson Process | 179 |

4 Birth and Death Processes | 180 |

Probability | 16 |

4 Some Examples of an Expectation | 17 |

5 Moments | 21 |

Optimization Problems | 22 |

Sample Surveys | 24 |

Least Square Estimation of Random Variables | 28 |

9 Some Implications of the Axioms | 32 |

Probability | 38 |

2 Probability Measure | 42 |

3 Expectation as a Probability Integral | 44 |

4 Some History | 45 |

5 Subjective Probability | 47 |

Some Basic Models | 49 |

2 The Multinomial Binomial Poisson and Geometric Distributions | 52 |

3 Independence | 56 |

4 Probability Generating Functions | 59 |

5 The St Petersburg Paradox | 63 |

6 Matching and Other Combinatorial Problems | 66 |

7 Conditioning | 68 |

the Exponential and Gamma Distributions | 73 |

Conditioning | 77 |

2 Conditional Probability | 81 |

3 A Conditional Expectation as a Random Variable | 85 |

4 Conditioning on a aField | 89 |

6 Statistical Decision Theory | 92 |

7 Information Transmission | 94 |

8 Acceptance Sampling | 97 |

Applications of the Independence Concept | 100 |

Regeneration Points | 105 |

the Gibbs Distribution | 109 |

4 Branching Processes | 113 |

The Two Basic Limit Theorems | 119 |

2 Properties of the Characteristic Function | 122 |

3 The Law of Large Numbers | 126 |

4 Normal Convergence the Central Limit Theorem | 127 |

5 The Normal Distribution | 129 |

Continuous Random Variables and Their Transformations | 136 |

2 Functions of Random Variables | 139 |

3 Conditional Densities | 142 |

Markov Processes in Discrete Time | 145 |

the Kolmogorov Equations | 151 |

Ruin Survival and Runs | 156 |

Detailed Balance | 159 |

5 Some Examples We Should Like to Defer | 161 |

5 Processes on Nondiscrete State Spaces | 183 |

6 The Filing Problem | 186 |

7 Some ContinuousTime Martingales | 187 |

8 Stationarity and Reversibility | 189 |

9 The Ehrenfest Model | 192 |

10 Processes of Independent Increments | 194 |

Diffusion Processes | 198 |

12 First Passage and Recurrence for Brownian Motion | 202 |

SecondOrder Theory | 206 |

2 Linear Least Square Approximation | 208 |

Innovation | 210 |

4 The GaussMarkov Theorem | 213 |

5 The Convergence of Linear Least Square Estimates | 215 |

6 Direct and Mutual Mean Square Convergence | 217 |

Martingale Convergence | 218 |

Consistency and Extension the FiniteDimensional Case | 221 |

2 Convex Sets | 222 |

3 The Consistency Condition for Expectation Values | 227 |

4 The Extension of Expectation Values | 228 |

5 Examples of Extension | 230 |

Chernoff Bounds | 233 |

Stochastic Convergence | 235 |

2 Types of Convergence | 237 |

3 Some Consequences | 239 |

4 Convergence in rth Mean | 240 |

Martingales | 243 |

the Law of Large Numbers | 247 |

Applications | 250 |

4 The Optional Stopping Theorem | 253 |

5 Examples of Stopped Martingales | 255 |

Extension Examples of the InfiniteDimensional Case | 258 |

2 Fields and aFields of Events | 259 |

3 Extension on a Linear Lattice | 260 |

4 Integrable Functions of a Scalar Random Variable | 263 |

Weak Convergence | 265 |

Some Interesting Processes | 270 |

More on the Shannon Measure | 273 |

Sequential Interrogation and Questionnaires | 275 |

4 Dynamic Optimization | 277 |

the Static Case | 283 |

the Dynamic Case | 289 |

293 | |

295 | |

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### Common terms and phrases

analogue appeal assertion assumption average axioms binomial bound Chapter coefficient concept conditional expectation consider constant continuous convergence in distribution convex convex set corresponding cov(X deduce defined definition denote determined distribution function dynamic elements equal equilibrium distribution equivalent evaluation event example Exercises and Comments expression fact finite follows geometric distribution given holds identity implies increasing independent indicator function inequality infinite integral large numbers limit linear LLS estimate Markov process Markov property martingale matrix maximal mean square measure minimizing multinomial multinomial distribution nonnegative normal distribution Note number of molecules observation occurs optimal orthogonal Petersburg paradox Poisson process population positive possible prescribed probabilistic probability density problem Proof random variable random vector random walk realization recurrent regarded relation respectively sample space scalar r.v. Section sequence Show solution specified statistical stochastic Suppose Theorem theory tion transformation transition valid variance X(co