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TEST BANK FOR FIRST COURSE IN PROBABILITY 8TH EDITION BY SHELDON ROSS ISBN;9780136033134
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TEST BANK FOR FIRST COURSE IN PROBABILITY 8TH EDITION BY SHELDON ROSS ISBN;9780136033134
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,A FIRST COURSE IN PROBABILITY,Contents
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
, viii Contents
4.7 The Poisson Random Variable . . . . . . . . . . . . . . . . . . . . . . . 143
4.7.1 Computing the Poisson Distribution Function . . . . . . . . . . 154
4.8 Other Discrete Probability Distributions . . . . . . . . . . . . . . . . . 155
4.8.1 The Geometric Random Variable . . . . . . . . . . . . . . . . . 155
4.8.2 The Negative Binomial Random Variable . . . . . . . . . . . . 157
4.8.3 The Hypergeometric Random Variable . . . . . . . . . . . . . 160
4.8.4 The Zeta (or Zipf) Distribution . . . . . . . . . . . . . . . . . . 163
4.9 Expected Value of Sums of Random Variables . . . . . . . . . . . . . 164
4.10 Properties of the Cumulative Distribution Function . . . . . . . . . . . 168
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
Theoretical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
Self-Test Problems and Exercises . . . . . . . . . . . . . . . . . . . . . 183
5 Continuous Random Variables 186
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.2 Expectation and Variance of Continuous Random Variables . . . . . 190
5.3 The Uniform Random Variable . . . . . . . . . . . . . . . . . . . . . . 194
5.4 Normal Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . 198
5.4.1 The Normal Approximation to the Binomial Distribution . . . 204
5.5 Exponential Random Variables . . . . . . . . . . . . . . . . . . . . . . 208
5.5.1 Hazard Rate Functions . . . . . . . . . . . . . . . . . . . . . . . 212
5.6 Other Continuous Distributions . . . . . . . . . . . . . . . . . . . . . . 215
5.6.1 The Gamma Distribution . . . . . . . . . . . . . . . . . . . . . 215
5.6.2 The Weibull Distribution . . . . . . . . . . . . . . . . . . . . . 216
5.6.3 The Cauchy Distribution . . . . . . . . . . . . . . . . . . . . . . 217
5.6.4 The Beta Distribution . . . . . . . . . . . . . . . . . . . . . . . 218
5.7 The Distribution of a Function of a Random Variable . . . . . . . . . 219
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
Theoretical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Self-Test Problems and Exercises . . . . . . . . . . . . . . . . . . . . . 229
6 Jointly Distributed Random Variables 232
6.1 Joint Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . 232
6.2 Independent Random Variables . . . . . . . . . . . . . . . . . . . . . . 240
6.3 Sums of Independent Random Variables . . . . . . . . . . . . . . . . . 252
6.3.1 Identically Distributed Uniform Random Variables . . . . . . 252
6.3.2 Gamma Random Variables . . . . . . . . . . . . . . . . . . . . 254
6.3.3 Normal Random Variables . . . . . . . . . . . . . . . . . . . . 256
6.3.4 Poisson and Binomial Random Variables . . . . . . . . . . . . 259
6.3.5 Geometric Random Variables . . . . . . . . . . . . . . . . . . . 260
6.4 Conditional Distributions: Discrete Case . . . . . . . . . . . . . . . . . 263
6.5 Conditional Distributions: Continuous Case . . . . . . . . . . . . . . . 266
6.6 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
6.7 Joint Probability Distribution of Functions of Random Variables . . . 274
6.8 Exchangeable Random Variables . . . . . . . . . . . . . . . . . . . . . 282
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
Theoretical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
Self-Test Problems and Exercises . . . . . . . . . . . . . . . . . . . . . 293
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