Page 11 - Fundamentals of Probability and Statistics for Engineers

viii Contents

3.2.3 Probability Density Function for Continuous Random
Variables 44
3.2.4 Mixed-Type Distribution 46
3.3 Two or More Random Variables 49
3.3.1 Joint Probability Distribution Function 49
3.3.2 Joint Probability Mass Function 51
3.3.3 Joint Probability Density Function 55
3.4 Conditional Distribution and Independence 61
Further Reading and Comments 66
Problems 67

4 EXPECTATIONS AND MOMENTS 75
4.1 Moments of a Single Random Variable 76
4.1.1 Mean, Median, and Mode 76
4.1.2 Central Moments, Variance, and Standard Deviation 79
4.1.3 Conditional Expectation 83
4.2 Chebyshev Inequality 86
4.3 Moments of Two or More Random Variables 87
4.3.1 Covariance and Correlation Coefficient 88
4.3.2 Schwarz Inequality 92
4.3.3 The Case of Three or More Random Variables 92
4.4 Moments of Sums of Random Variables 93
4.5 Characteristic Functions 98
4.5.1 Generation of Moments 99
4.5.2 Inversion Formulae 101
4.5.3 Joint Characteristic Functions 108
Further Reading and Comments 112
Problems 112

5 FUNCTIONS OF RANDOM VARIABLES 119
5.1 Functions of One Random Variable 119
5.1.1 Probability Distribution 120
5.1.2 Moments 134
5.2 Functions of Two or More Random Variables 137
5.2.1 Sums of Random Variables 145
5.3 m Functions of n Random Variables 147
Reference 153
Problems 154

6 SOME IMPORTANT DISCRETE DISTRIBUTIONS 161
6.1 Bernoulli Trials 161
6.1.1 Binomial Distribution 162

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