Probability Via Expectation

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Springer Science & Business Media, May 14, 1992 - Mathematics - 300 pages
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This 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
References
293
Index
295
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About the author (1992)

Peter Whittle is Professor Emeritus at the University of Cambridge. From 1973 to 1986 he was Director of the Statistical Laboratory, Cambridge. He is a Fellow of the Royal Society and this is his 11th book.

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