viii Contents
4 RANDOM VARIABLES 74
4.1 What is a random variable? 74
4.2 How do random variables come about? 78
4.3 Distribution and expectation 84
4.4 Integer-valued random variables 90
4.5 Random variables with densities 95
4.6 General case 105
Exercises 109
APPENDIX 1: BOREL FIELDS AND GENERAL
RANDOM VARIABLES 115
5 CONDITIONING AND INDEPENDENCE 117
5.1 Examples of conditioning 117
5.2 Basic formulas 122
5.3 Sequential sampling 131
5.4 P´olya’s urn scheme 136
5.5 Independence and relevance 141
5.6 Genetical models 152
Exercises 157
6 MEAN, VARIANCE, AND TRANSFORMS 164
6.1 Basic properties of expectation 164
6.2 The density case 169
6.3 Multiplication theorem; variance and covariance 173
6.4 Multinomial distribution 180
6.5 Generating function and the like 187
Exercises 195
7 POISSON AND NORMAL DISTRIBUTIONS 203
7.1 Models for Poisson distribution 203
7.2 Poisson process 211
7.3 From binomial to normal 222
7.4 Normal distribution 229
7.5 Central limit theorem 233
7.6 Law of large numbers 239
Exercises 246
APPENDIX 2: STIRLING’S FORMULA AND
DE MOIVRE–LAPLACE’S THEOREM 251
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