FUNDAMENTALS OF PROBABILITY AND STATISTICS PDF

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FUNDAMENTALS OF PROBABILITY AND STATISTICS FOR ENGINEERS

TABLE OF CONTENT

1 INTRODUCTION 1

1.1 Organization of Text 2
1.2 Probability Tables and Computer Software 3
1.3 Prerequisites 3

PART A: PROBABILITY AND RANDOM VARIABLES 5

2 BASIC PROBABILITY CONCEPTS 7


2.1 Elements of Set Theory 8
2.1.1 Set Operations 9
2.2 Sample Space and Probability Measure 12
2.2.1 Axioms of Probability 13
2.2.2 Assignment of Probability 16
2.3 Statistical Independence 17
2.4 Conditional Probability 20

Reference 28

Further Reading 28

Problems 28

3 RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS 37


3.1 Random Variables 37
3.2 Probability Distributions 39
3.2.1 Probability Distribution Function 39
3.2.2 Probability Mass Function for Discrete Random
Variables 41

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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6.1.2 Geometric Distribution 167
6.1.3 Negative Binomial Distribution 169
6.2 Multinomial Distribution 172
6.3 Poisson Distribution 173
6.3.1 Spatial Distributions 181
6.3.2 The Poisson Approximation to the Binomial Distribution 182
6.4 Summary 183
Further Reading 184
Problems 185

7 SOME IMPORTANT CONTINUOUS DISTRIBUTIONS 191


7.1 Uniform Distribution 191
7.1.1 Bivariate Uniform Distribution 193
7.2 Gaussian or Normal Distribution 196
7.2.1 The Central Limit Theorem 199
7.2.2 Probability Tabulations 201
7.2.3 Multivariate Normal Distribution 205
7.2.4 Sums of Normal Random Variables 207
7.3 Lognormal Distribution 209
7.3.1 Probability Tabulations 211
7.4 Gamma and Related Distributions 212
7.4.1 Exponential Distribution 215
7.4.2 Chi-Squared Distribution 219
7.5 Beta and Related Distributions 221
7.5.1 Probability Tabulations 223
7.5.2 Generalized Beta Distribution 225
7.6 Extreme-Value Distributions 226
7.6.1 Type-I Asymptotic Distributions of Extreme Values 228
7.6.2 Type-II Asymptotic Distributions of Extreme Values 233
7.6.3 Type-III Asymptotic Distributions of Extreme Values 234
7.7 Summary 238
References 238
Further Reading and Comments 238
Problems 239

PART B: STATISTICAL INFERENCE, PARAMETER


ESTIMATION, AND MODEL VERIFICATION 245
8 OBSERVED DATA AND GRAPHICAL REPRESENTATION 247
8.1 Histogram and Frequency Diagrams 248
References 252
Problems 253
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9 PARAMETER ESTIMATION 259


9.1 Samples and Statistics 259
9.1.1 Sample Mean 261
9.1.2 Sample Variance 262
9.1.3 Sample Moments 263
9.1.4 Order Statistics 264
9.2 Quality Criteria for Estimates 264
9.2.1 Unbiasedness 265
9.2.2 Minimum Variance 266
9.2.3 Consistency 274
9.2.4 Sufficiency 275
9.3 Methods of Estimation 277
9.3.1 Point Estimation 277
9.3.2 Interval Estimation 294
References 306
Further Reading and Comments 306
Problems 307

10 MODEL VERIFICATION 315


10.1 Preliminaries 315
10.1.1 Type-I and Type-II Errors 316
10.2 Chi-Squared Goodness-of-Fit Test 316
10.2.1 The Case of Known Parameters 317
10.2.2 The Case of Estimated Parameters 322
10.3 Kolmogorov–Smirnov Test 327
References 330
Further Reading and Comments 330
Problems 330
11 LINEAR MODELS AND LINEAR REGRESSION 335
11.1 Simple Linear Regression 335
11.1.1 Least Squares Method of Estimation 336
11.1.2 Properties of Least-Square Estimators 342
11.1.3 Unbiased Estimator for 2 345
11.1.4 Confidence Intervals for Regression Coefficients 347
11.1.5 Significance Tests 351
11.2 Multiple Linear Regression 354
11.2.1 Least Squares Method of Estimation 354
11.3 Other Regression Models 357
Reference 359
Further Reading 359
Problems 359
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APPENDIX A: TABLES 365


A.1 Binomial Mass Function 365
A.2 Poisson Mass Function 367
A.3 Standardized Normal Distribution Function 369
A.4 Student’s t Distribution with n Degrees of Freedom 370
A.5 Chi-Squared Distribution with n Degrees of Freedom 371
A.6 D2 Distribution with Sample Size n 372
References 373

APPENDIX B: COMPUTER SOFTWARE 375

APPENDIX C: ANSWERS TO SELECTED PROBLEMS 379


Chapter 2 379
Chapter 3 380
Chapter 4 381
Chapter 5 382
Chapter 6 384
Chapter 7 385
Chapter 8 385
Chapter 9 385
Chapter 10 386
Chapter 11 386

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