Probability and random variables : a beginner's guide

This simple and concise introduction to probability theory is written in an informal, tutorial style with concepts and techniques defined and developed as necessary. After an elementary discussion of chance, Stirzaker sets out the central and crucial rules and ideas of probability including independence and conditioning. Counting, combinatorics and the ideas of probability distributions and densities follow. Later chapters present random variables and examine independence, conditioning, covariance and functions of random variables, both discrete and continuous. The final chapter considers generating functions and applies this concept to practical problems including branching processes, random walks and the central limit theorem. Examples, demonstrations, and exercises are used throughout to explore the ways in which probability is motivated by, and applied to, real life problems in science, medicine, gaming and other subjects of interest. Essential proofs of important results are included. Assuming minimal prior technical knowledge on the part of the reader, this book is suitable for students taking introductory courses in probability and will provide a solid foundation for more advanced courses in probability and statistics. It is also a valuable reference to those needing a working knowledge of probability theory and will appeal to anyone interested in this endlessly fascinating and entertaining subject. Contents......Page 5 Synopsis......Page 8 Preface......Page 11 1.2 PROBABILITY......Page 13 1.3 THE SCOPE OF PROBABILITY......Page 15 1.4 BASIC IDEAS: THE CLASSICAL CASE......Page 17 1.5 BASIC IDEAS; THE GENERAL CASE......Page 22 1.6 MODELLING......Page 26 1.7 MATHEMATICAL MODELLING......Page 31 1.8 MODELLING PROBABILITY......Page 33 1.10 APPENDIX I. SOME RANDOMLY SELECTED DEFINITIONS OF PROBABILITY, IN RANDOM ORDER......Page 34 1.11 APPENDIX II. REVIEWOF SETS AND FUNCTIONS......Page 36 1.12 PROBLEMS......Page 39 Part A Probability......Page 41 2.2 NOTATION AND EXPERIMENTS......Page 43 2.3 EVENTS......Page 46 2.4 PROBABILITY; ELEMENTARY CALCULATIONS......Page 49 2.5 THE ADDITION RULES......Page 53 2.6 SIMPLE CONSEQUENCES......Page 56 2.7 CONDITIONAL PROBABILITY; MULTIPLICATION RULE......Page 59 2.8 THE PARTITION RULE AND BAYES' RULE......Page 66 2.9 INDEPENDENCE AND THE PRODUCT RULE......Page 70 2.10 TREES AND GRAPHS......Page 78 2.11 WORKED EXAMPLES......Page 84 2.12 ODDS......Page 90 2.13 POPULAR PARADOXES......Page 94 2.14 REVIEW: NOTATION AND RULES......Page 98 2.15 APPENDIX. DIFFERENCE EQUATIONS......Page 100 2.16 PROBLEMS......Page 101 3.2 FIRST PRINCIPLES......Page 105 3.3 ARRANGING AND CHOOSING......Page 109 3.4 BINOMIAL COEFFICIENTS AND PASCAL'S TRIANGLE......Page 113 3.5 CHOICE AND CHANCE......Page 116 3.6 APPLICATIONS TO LOTTERIES......Page 121 3.7 THE PROBLEM OF THE POINTS......Page 125 3.8 THE GAMBLER'S RUIN PROBLEM......Page 128 3.9 SOME CLASSIC PROBLEMS......Page 130 3.10 STIRLING'S FORMULA......Page 133 3.11 REVIEW......Page 135 3.12 APPENDIX. SERIES AND SUMS......Page 136 3.13 PROBLEMS......Page 138 4.2 INTRODUCTION; SIMPLE EXAMPLES......Page 141 4.3 WAITING; GEOMETRIC DISTRIBUTIONS......Page 148 4.4 THE BINOMIAL DISTRIBUTION AND SOME RELATIVES......Page 151 4.5 SAMPLING......Page 156 4.6 LOCATION AND DISPERSION......Page 159 4.7 APPROXIMATIONS: A FIRST LOOK......Page 166 4.8 SPARSE SAMPLING; THE POISSON DISTRIBUTION......Page 168 4.9 CONTINUOUS APPROXIMATIONS......Page 170 4.10 BINOMIAL DISTRIBUTIONS AND THE NORMAL APPROXIMATION......Page 175 4.11 DENSITY......Page 181 4.12 DISTRIBUTIONS IN THE PLANE......Page 184 4.13 REVIEW......Page 186 4.14 APPENDIX. CALCULUS......Page 188 4.15 APPENDIX. SKETCH PROOF OF THE NORMAL LIMIT THEOREM......Page 190 4.16 PROBLEMS......Page 192 Part B Random Variables......Page 199 5.2 INTRODUCTION TO RANDOM VARIABLES......Page 201 5.3 DISCRETE RANDOM VARIABLES......Page 206 5.4 CONTINUOUS RANDOM VARIABLES; DENSITY......Page 210 5.5 FUNCTIONS OFACONTINUOUS RANDOM VARIABLE......Page 216 5.6 EXPECTATION......Page 219 5.7 FUNCTIONS AND MOMENTS......Page 224 5.8 CONDITIONAL DISTRIBUTIONS......Page 230 5.9 CONDITIONAL DENSITY......Page 237 5.10 REVIEW......Page 241 5.11 APPENDIX. DOUBLE INTEGRALS......Page 244 5.12 PROBLEMS......Page 245 6.2 JOINT DISTRIBUTIONS......Page 250 6.3 JOINT DENSITY......Page 257 6.4 INDEPENDENCE......Page 262 6.5 FUNCTIONS......Page 266 6.6 SUMS OF RANDOM VARIABLES......Page 272 6.7 EXPECTATION; THE METHOD OF INDICATORS......Page 279 6.8 INDEPENDENCE AND COVARIANCE......Page 285 6.9 CONDITIONING AND DEPENDENCE, DISCRETE CASE......Page 292 6.10 CONDITIONING AND DEPENDENCE, CONTINUOUS CASE......Page 298 6.11 APPLICATIONS OF CONDITIONAL EXPECTATION......Page 303 6.12 BIVARIATE NORMAL DENSITY......Page 306 6.13 CHANGE-OF-VARIABLES TECHNIQUE; ORDER STATISTICS......Page 310 6.14 REVIEW......Page 313 6.15 PROBLEMS......Page 314 7.2 INTRODUCTION......Page 321 7.3 EXAMPLES OF GENERATING FUNCTIONS......Page 324 7.4 APPLICATIONS OF GENERATING FUNCTIONS......Page 327 7.5 RANDOM SUMS AND BRANCHING PROCESSES......Page 331 7.6 CENTRAL LIMIT THEOREM......Page 335 7.7 RANDOM WALKS AND OTHER DIVERSIONS......Page 336 7.9 APPENDIX. TABLES OF GENERATING FUNCTIONS......Page 341 7.10 PROBLEMS......Page 342 Hints and solutions for selected exercises and problems......Page 348 Index......Page 377 "This is a simple and concise introduction to probability theory. Self-contained and readily accessible, it is written in an informal tutorial style with concepts and techniques defined and developed as necessary. Examples, demonstrations and exercises are used throughout to explore the ways in which probability is motivated by, and applied to, real-life problems in science, medicine, gaming and other subjects of interest. Essential proofs of important results are included." "Since it assumes minimal prior technical knowledge on the part of the reader, this book is suitable for students taking introductory courses in probability and will provide a solid foundation for more advanced courses in probability and statistics. It would also be a valuable reference for those needing a working knowledge of probability theory and will appeal to anyone interested in this endlessly fascinating and entertaining subject."--Jacket I'm using this book for a graduate course in probability for political science, and I have been very disappointed with it. The main problem is that it skips steps involved in problems and then tells you that "obviously" or "clearly" or "intuitively" the solution to the problem follows. For example, my professor has had to distribute his own detailed explanations of sample problems in the book, which have been much clearer (and longer) than what the book offers. If math is not intuitive for you or your quantitative skills are rusty, this book does not give you the kind of step-by-step guidance you'll need.