Elementary Probability Theory with Stochastic Processes

Description

Chapter 1 Set 1.1 Sample sets 1.2 Operations with sets 1.3 Various relations 1.4 Indicator Exercises Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 De Moivre-Laplace's Theorem Chapter 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 Table 1 Values of the standard normal distribution function Index