Introduction to Probability

1 The Concept of Probability 5 1.1 Chance experiments--sample spaces 6 1.2 Operations between events 16 1.3 Probability as relative frequency 35 1.4 Axiomatic definition of probability 48 1.5 Properties of probability 57 1.6 The continuity property of probability 66 1.7 Basic concepts and formulae 75 1.8 Mathematica 76 1.9 Self-assessment exercises 78 1.10 Review problems 82 1.11 Applications 89 2 Finite Sample Spaces--Combinatorial Methods 97 2.1 Finite sample spaces with events of equal probability 98 2.2 Main principles of counting 109 2.3 Permutations 117 2.4 Combinations 128 2.5 The Binomial Theorem 148 2.6 Basic concepts and formulae 159 2.7 Mathematica 161 2.8 Self-assessment exercises 168 2.9 Review problems 173 2.10 Applications 182 3 Conditional Probability--Independent Events 187 3.1 Conditional probability 188 3.2 The multiplicative law of probability 203 3.3 The law of total probability 213 3.4 Bayes' formula 222 3.5 Independent events 229 3.6 Basic concepts and formulae 251 3.7 Mathematica exercises 252 3.8 Self-assessment exercises 256 3.9 Review problems 261 3.10 Applications 271 4 Discrete Random Variables and Distributions 277 4.1 Random variables 278 4.2 Distribution functions 285 4.3 Discrete random variables 302 4.4 Expectation of a discrete random variable 318 4.5 Variance of discrete random variables 340 4.6 Other useful results for the expectation and the variance 353 4.7 Basic concepts and formulae 364 4.8 Mathematica 365 4.9 Self-assessment exercises 372 4.10 Review problems 377 5 The Most Important Discrete Distributions 385 5.1 Bernoulli trials and the binomial distribution 386 5.2 The geometric and the negative binomial distribution 404 5.3 The hypergeometric distribution 428 5.4 The Poisson distribution 443 5.5 The Poisson process 459 5.6 Basic concepts and formulae (or Glossary and new formulae??) 471 5.7 Mathematica exercises 472 5.8 Self-assessment exercises 477 5.9 Review problems 483 5.10 Applications 494 6 Continuous Random Variables 499 6.1 Density functions 500 6.2 Distribution of a function of a random variable 519 6.3 Expectation and variance 531 6.4 Other useful results for the expectation 542 6.5 Mixture distributions 551 6.6 Basic concepts and formulae 563 6.7 Mathematica exercises 564 6.8 Self-assessment exercises 569 6.9 Review problems 575 6.10 Applications 585 7 The Most Important Continuous Distributions 589 7.1 The uniform distribution 590 7.2 The normal distribution 600 7.3 The exponential distribution 634 7.4 Other continuous distributions 646 7.5 Basic concepts and formulae 659 7.6 Mathematica exercises 662 7.7 Self-assessment exercises 666 7.8 Review problems 671