A new feature of this edition consists of photogra phs of eight masters in the contemporary development of probability theory. All of them appear in the body of the book, though the few references there merely serve to give a glimpse of their manifold contributions. It is hoped that these vivid pictures will inspire in the reader a feeling that our science is a live endeavor created and pursued by real personalities. I have had the privilege of meeting and knowing most of them after studying their works and now take pleasure in introducing them to a younger generation. In collecting the photographs I had the kind assistance of Drs Marie-Helene Schwartz, Joanne Elliot, Milo Keynes and Yu. A. Rozanov, to whom warm thanks are due. A German edition of the book has just been published. I am most grateful to Dr. Herbert Vogt for his careful translation which resulted also in a consid­ erable number of improvements on the text of this edition. Other readers who were kind enough to send their comments include Marvin Greenberg, Louise Hay, Nora Holmquist, H. -E. Lahmann, and Fred Wolock. Springer-Verlag is to be complimented once again for its willingness to make its books "immer besser. " K. L. C. September 19, 1978 Preface to the Second Edition A determined effort was made to correct the errors in the first edition. This task was assisted by: Chao Hung-po, J. L. Doob, R. M. Exner, W. H.

Inhalt

1: Set.- 1.1 Sample sets.- 1.2 Operations with sets.- 1.3 Various relations.- 1.4 Indicator.- Exercises.- 2: Probability.- 2.1 Examples of probability.- 2.2 Definition and illustrations.- 2.3 Deductions from the axioms.- 2.4 Independent events.- 2.5 Arithmetical density.- Exercises.- 3: Counting.- 3.1 Fundamental rule.- 3.2 Diverse ways of sampling.- 3.3 Allocation models; binomial coefficients.- 3.4 How to solve it.- Exercises.- 4: Random Variables.- 4.1 What is a random variable?.- 4.2 How do random variables come about?.- 4.3 Distribution and expectation.- 4.4 Integer-valued random variables.- 4.5 Random variables with densities.- 4.6 General case.- Exercises.- Appendix 1: Borel Fields and General Random Variables.- 5: Conditioning and Independence.- 5.1 Examples of conditioning.- 5.2 Basic formulas.- 5.3 Sequential sampling.- 5.4 Pólya's urn scheme.- 5.5 Independence and relevance.- 5.6 Genetical models.- Exercises.- 6: Mean, Variance and Transforms.- 6.1 Basic properties of expectation.- 6.2 The density case.- 6.3 Multiplication theorem; variance and covariance.- 6.4 Multinomial distribution.- 6.5 Generating function and the like.- Exercises.- 7: Poisson and Normal Distributions.- 7.1 Models for Poisson distribution.- 7.2 Poisson process.- 7.3 From binomial to normal.- 7.4 Normal distribution.- 7.5 Central limit theorem.- 7.6 Law of large numbers.- Exercises.- Appendix 2: Stirling's Formula and DeMoivre-Laplace's Theorem.- 8: From Random Walks to Markov Chains.- 8.1 Problems of the wanderer or gambler.- 8.2 Limiting schemes.- 8.3 Transition probabilities.- 8.4 Basic structure of Markov chains.- 8.5 Further developments.- 8.6 Steady state.- 8.7 Winding up (or down?).- Exercises.- Appendix 3: Martingale.- General References.- Answers to Problems.