Contents Preface 1 Combinatorial Analysis 12 12 13 14 Combinations 6 efficients 591215 Problems ercise 6180 2 iomsofProbabilty tion 2. Sample Space and Events ome Simple Pr 93 2.6 Probability as a Continuous Set Function 44 2.7 Probability as a Measure of Belief. 8190 Theoretical exercises 4 Self-Test Problems and Exercises 56 3 Conditional Probability and Independence Probabilities 33 Baves's Formula 65 3.4 3.5 。， 990 2 4 Random Variables 42 dom Variables 123 43 Expected Value Epcation ofa Function of Random Variable 125 The 34 4.6.1 Properties of Binomial Random Variables. 1 4.6.2 Computing the Binomial Distribution Function. 142Contents Preface xi 1 Combinatorial Analysis 1 1.1 Introduction . . . . 1 1.2 The Basic Principle of Counting . . . . 1 1.3 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Combinations . . . 5 1.5 Multinomial Coefficients . . . . 9 1.6 The Number of Integer Solutions of Equations . . . . . . . . . . . . . 12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Theoretical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Self-Test Problems and Exercises . . . . . . . . . . . . . . . . . . . . . 20 2 Axioms of Probability 22 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Sample Space and Events . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Axioms of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4 Some Simple Propositions . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.5 Sample Spaces Having Equally Likely Outcomes . . . . . . . . . . . . 33 2.6 Probability as a Continuous Set Function . . . . . . . . . . . . . . . . . 44 2.7 Probability as a Measure of Belief . . . . . . . . . . . . . . . . . . . . . 48 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Theoretical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Self-Test Problems and Exercises . . . . . . . . . . . . . . . . . . . . . 56 3 Conditional Probability and Independence 58 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2 Conditional Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3 Bayes’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.5 P(·|F) Is a Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Theoretical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Self-Test Problems and Exercises . . . . . . . . . . . . . . . . . . . . . 114 4 Random Variables 117 4.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.2 Discrete Random Variables . . . . . . . . . . . . . . . . . . . . . . . . 123 4.3 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.4 Expectation of a Function of a Random Variable . . . . . . . . . . . . 128 4.5 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.6 The Bernoulli and Binomial Random Variables . . . . . . . . . . . . . 134 4.6.1 Properties of Binomial Random Variables . . . . . . . . . . . . 139 4.6.2 Computing the Binomial Distribution Function . . . . . . . . . 142 vii