Basic Principles and Applications of Probability TheoryProbability theory arose originally in connection with games of chance and then for a long time it was used primarily to investigate the credibility of testimony of witnesses in the “ethical” sciences. Nevertheless, probability has become a very powerful mathematical tool in understanding those aspects of the world that cannot be described by deterministic laws. Probability has succeeded in ?nding strict determinate relationships where chance seemed to reign and so terming them “laws of chance” combining such contrasting - tions in the nomenclature appears to be quite justi?ed. This introductory chapter discusses such notions as determinism, chaos and randomness, p- dictibility and unpredictibility, some initial approaches to formalizing r- domness and it surveys certain problems that can be solved by probability theory. This will perhaps give one an idea to what extent the theory can - swer questions arising in speci?c random occurrences and the character of the answers provided by the theory. 1. 1 The Nature of Randomness The phrase “by chance” has no single meaning in ordinary language. For instance, it may mean unpremeditated, nonobligatory, unexpected, and so on. Its opposite sense is simpler: “not by chance” signi?es obliged to or bound to (happen). In philosophy, necessity counteracts randomness. Necessity signi?es conforming to law – it can be expressed by an exact law. The basic laws of mechanics, physics and astronomy can be formulated in terms of precise quantitativerelationswhichmustholdwithironcladnecessity. |
Contents
| 5 | |
Probability Space 19 | 18 |
Independence | 53 |
General Theory of Stochastic Processes | 93 |
Limit Theorems | 119 |
I | 127 |
Markov Processes | 139 |
Applied Probability | 193 |
Filtering 257 | 256 |
Historic and Bibliographic Comments | 273 |
Author Index | 275 |
Other editions - View all
Common terms and phrases
algebra assume Borel bounded channel characteristic function compact complete separable metric conditional probability consider continuous function control cost converges countable decision function defined denote density determined diffusion process distribution function dw(s elementary events entropy equation ergodic exists experiment finite finite-dimensional distributions Fn(xo follows formula given independent increments independent random variables integral interval Kolmogorov's large numbers law of large Lemma linear log2 Markov chain Markov process martingale measurable space messages metric space o-algebra observed parameter positive ɛ probability measure probability space probability theory problem Proof random function random walk relative frequency respect right-hand side sample satisfies separable metric space sequence of independent stationary sequence stochastic process stochastically continuous subset sufficient Suppose t₁ transition probability uniformly values Wiener process zero