Probability and random variables. A beginner's guide

Front Cover......Page 1
Back Cover......Page 2
Contents......Page 7
Synopsis......Page 10
Preface......Page 13
1.2 PROBABILITY......Page 15
1.3 THE SCOPE OF PROBABILITY......Page 17
1.4 BASIC IDEAS: THE CLASSICAL CASE......Page 19
1.5 BASIC IDEAS; THE GENERAL CASE......Page 24
1.6 MODELLING......Page 28
1.7 MATHEMATICAL MODELLING......Page 33
1.8 MODELLING PROBABILITY......Page 35
1.10 APPENDIX I. SOME RANDOMLY SELECTED DEFINITIONS OF PROBABILITY, IN RANDOM ORDER......Page 36
1.11 APPENDIX II. REVIEWOF SETS AND FUNCTIONS......Page 38
1.12 PROBLEMS......Page 41
Part A Probability......Page 43
2.2 NOTATION AND EXPERIMENTS......Page 45
2.3 EVENTS......Page 48
2.4 PROBABILITY; ELEMENTARY CALCULATIONS......Page 51
2.5 THE ADDITION RULES......Page 55
2.6 SIMPLE CONSEQUENCES......Page 58
2.7 CONDITIONAL PROBABILITY; MULTIPLICATION RULE......Page 61
2.8 THE PARTITION RULE AND BAYES\' RULE......Page 68
2.9 INDEPENDENCE AND THE PRODUCT RULE......Page 72
2.10 TREES AND GRAPHS......Page 80
2.11 WORKED EXAMPLES......Page 86
2.12 ODDS......Page 92
2.13 POPULAR PARADOXES......Page 96
2.14 REVIEW: NOTATION AND RULES......Page 100
2.15 APPENDIX. DIFFERENCE EQUATIONS......Page 102
2.16 PROBLEMS......Page 103
3.2 FIRST PRINCIPLES......Page 107
3.3 ARRANGING AND CHOOSING......Page 111
3.4 BINOMIAL COEFFICIENTS AND PASCAL\'S TRIANGLE......Page 115
3.5 CHOICE AND CHANCE......Page 118
3.6 APPLICATIONS TO LOTTERIES......Page 123
3.7 THE PROBLEM OF THE POINTS......Page 127
3.8 THE GAMBLER\'S RUIN PROBLEM......Page 130
3.9 SOME CLASSIC PROBLEMS......Page 132
3.10 STIRLING\'S FORMULA......Page 135
3.11 REVIEW......Page 137
3.12 APPENDIX. SERIES AND SUMS......Page 138
3.13 PROBLEMS......Page 140
4.2 INTRODUCTION; SIMPLE EXAMPLES......Page 143
4.3 WAITING; GEOMETRIC DISTRIBUTIONS......Page 150
4.4 THE BINOMIAL DISTRIBUTION AND SOME RELATIVES......Page 153
4.5 SAMPLING......Page 158
4.6 LOCATION AND DISPERSION......Page 161
4.7 APPROXIMATIONS: A FIRST LOOK......Page 168
4.8 SPARSE SAMPLING; THE POISSON DISTRIBUTION......Page 170
4.9 CONTINUOUS APPROXIMATIONS......Page 172
4.10 BINOMIAL DISTRIBUTIONS AND THE NORMAL APPROXIMATION......Page 177
4.11 DENSITY......Page 183
4.12 DISTRIBUTIONS IN THE PLANE......Page 186
4.13 REVIEW......Page 188
4.14 APPENDIX. CALCULUS......Page 190
4.15 APPENDIX. SKETCH PROOF OF THE NORMAL LIMIT THEOREM......Page 192
4.16 PROBLEMS......Page 194
Part B Random Variables......Page 201
5.2 INTRODUCTION TO RANDOM VARIABLES......Page 203
5.3 DISCRETE RANDOM VARIABLES......Page 208
5.4 CONTINUOUS RANDOM VARIABLES; DENSITY......Page 212
5.5 FUNCTIONS OFACONTINUOUS RANDOM VARIABLE......Page 218
5.6 EXPECTATION......Page 221
5.7 FUNCTIONS AND MOMENTS......Page 226
5.8 CONDITIONAL DISTRIBUTIONS......Page 232
5.9 CONDITIONAL DENSITY......Page 239
5.10 REVIEW......Page 243
5.11 APPENDIX. DOUBLE INTEGRALS......Page 246
5.12 PROBLEMS......Page 247
6.2 JOINT DISTRIBUTIONS......Page 252
6.3 JOINT DENSITY......Page 259
6.4 INDEPENDENCE......Page 264
6.5 FUNCTIONS......Page 268
6.6 SUMS OF RANDOM VARIABLES......Page 274
6.7 EXPECTATION; THE METHOD OF INDICATORS......Page 281
6.8 INDEPENDENCE AND COVARIANCE......Page 287
6.9 CONDITIONING AND DEPENDENCE, DISCRETE CASE......Page 294
6.10 CONDITIONING AND DEPENDENCE, CONTINUOUS CASE......Page 300
6.11 APPLICATIONS OF CONDITIONAL EXPECTATION......Page 305
6.12 BIVARIATE NORMAL DENSITY......Page 308
6.13 CHANGE-OF-VARIABLES TECHNIQUE; ORDER STATISTICS......Page 312
6.14 REVIEW......Page 315
6.15 PROBLEMS......Page 316
7.2 INTRODUCTION......Page 323
7.3 EXAMPLES OF GENERATING FUNCTIONS......Page 326
7.4 APPLICATIONS OF GENERATING FUNCTIONS......Page 329
7.5 RANDOM SUMS AND BRANCHING PROCESSES......Page 333
7.6 CENTRAL LIMIT THEOREM......Page 337
7.7 RANDOM WALKS AND OTHER DIVERSIONS......Page 338
7.9 APPENDIX. TABLES OF GENERATING FUNCTIONS......Page 343
7.10 PROBLEMS......Page 344
Hints and solutions for selected exercises and problems......Page 350
Index......Page 379