Probability Concepts and Theory for Engineers

Title\n......Page 5
Contents......Page 9
Preface......Page 13
Introduction......Page 15
Part I: The Basic Model......Page 19
Part I: Introduction......Page 20
SECTION 1: Dealing with ‘Real-World’ Problems......Page 21
SECTION 2: The Probabilistic Experiment......Page 24
SECTION 3: Outcome......Page 29
SECTION 4: Events......Page 32
SECTION 5: The Connection to the Mathematical World......Page 35
SECTION 6: Elements and Sets......Page 38
SECTION 7: Classes of Sets......Page 41
SECTION 8: Elementary Set Operations......Page 44
SECTION 9: Additional Set Operations......Page 48
SECTION 10: Functions......Page 51
SECTION 11: The Size of a Set......Page 54
SECTION 12: Multiple and Infinite Set Operations......Page 58
SECTION 13: More About Additive Classes......Page 62
SECTION 14: Additive Set Functions......Page 67
SECTION 15: More about Probabilistic Experiments......Page 71
SECTION 16: The Probability Function......Page 76
SECTION 17: Probability Space......Page 80
SECTION 18: Simple Probability Arithmetic......Page 83
Part I: Summary......Page 89
Part II: The Approach to Elementary Probability Problems......Page 91
Part II: Introduction......Page 92
SECTION 19: About Probability Problems......Page 93
SECTION 20: Equally Likely Possible Outcomes......Page 99
SECTION 21: Conditional Probability......Page 104
SECTION 22: Conditional Probability Distributions......Page 109
SECTION 23: Independent Events......Page 117
SECTION 24: Classes of Independent Events......Page 122
SECTION 25: Possible Outcomes Represented as Ordered k-Tuples......Page 127
SECTION 26: Product Experiments and Product Spaces......Page 132
SECTION 27: Product Probability Spaces......Page 138
SECTION 28: Dependence Between the Components in an Ordered k-Tuple......Page 143
SECTION 29: Multiple Observations Without Regard to Order......Page 146
SECTION 30: Unordered Sampling with Replacement......Page 150
SECTION 31: More Complicated Discrete Probability Problems......Page 153
SECTION 32: Uncertainty and Randomness......Page 158
SECTION 33: Fuzziness......Page 164
Part II: Summary......Page 170
Part III: Introduction to Random Variables......Page 171
Part III: Introduction......Page 172
SECTION 34: Numerical-Valued Outcomes......Page 173
SECTION 35: The Binomial Distribution......Page 179
SECTION 36: The Real Numbers......Page 183
SECTION 37: General Definition of a Random Variable......Page 187
SECTION 38: The Cumulative Distribution Function......Page 191
SECTION 39: The Probability Density Function......Page 198
SECTION 40: The Gaussian Distribution......Page 204
SECTION 41: Two Discrete Random Variables......Page 209
SECTION 42: Two Arbitrary Random Variables......Page 215
SECTION 43: Two-Dimensional Distribution Functions......Page 220
SECTION 44: Two-Dimensional Density Functions......Page 226
SECTION 45: Two Statistically Independent Random Variables......Page 234
SECTION 46: Two Statistically Independent Random Variables—Absolutely Continuous Case......Page 239
Part III: Summary......Page 244
Part IV: Transformations and Multiple Random Variables......Page 245
Part IV: Introduction......Page 246
a) Transformation of a discrete random variable......Page 247
b) Transformation of an arbitrary random variable......Page 249
c) Transformation of an absolutely continuous random variable......Page 253
SECTION 48: Transformation of a Two-Dimensional Random Variable......Page 256
SECTION 49: The Sum of Two Discrete Random Variables......Page 261
SECTION 50: The Sum of Two Arbitrary Random Variables......Page 265
SECTION 51: n-Dimensional Random Variables......Page 271
SECTION 52: Absolutely Continuous n-Dimensional R.V.’s......Page 277
SECTION 53: Coordinate Transformations......Page 281
SECTION 54: Rotations and the Bivariate Gaussian Distribution......Page 286
SECTION 55: Several Statistically Independent Random Variables......Page 292
SECTION 56: Singular Distributions in One Dimension......Page 297
SECTION 57: Conditional Induced Distribution, Given an Event......Page 302
SECTION 58: Resolving a Distribution into Components of Pure Type......Page 308
SECTION 59: Conditional Distribution Given the Value of a Random Variable......Page 311
SECTION 60: Random Occurrences in Time......Page 316
Part IV: Summary......Page 322
Part V: Parameters for Describing Random Variables and Induced Distributions......Page 323
Part V: Introduction......Page 324
SECTION 61: Some Properties of a Random Variable......Page 325
SECTION 62: Higher Moments......Page 332
b) General formulation......Page 338
c) Sum of random variables......Page 340
d) Powers of a random variable......Page 341
e) Product of random variables......Page 343
SECTION 64: The Variance of a Function of a Random Variable......Page 346
SECTION 65: Bounds on the Induced Distribution......Page 350
a) A simple random sample......Page 354
b) Unbiased estimators......Page 356
c) Variance of the sample average......Page 357
d) Estimating the population variance......Page 359
e) Sampling with replacement......Page 360
SECTION 67: Conditional Expectation with Respect to an Event......Page 363
SECTION 68: Covariance and Correlation Coefficient......Page 368
SECTION 69: The Correlation Coefficient as Parameter in a Joint Distribution......Page 374
SECTION 70: More General Kinds of Dependence Between Random Variables......Page 380
SECTION 71: The Covariance Matrix......Page 385
SECTION 72: Random Variables as the Elements of a Vector Space......Page 392
b) Optimum constant estimates......Page 397
c) Mean-square estimation using random variables......Page 399
d) Linear mean-square estimation......Page 400
SECTION 74: The Stieltjes Integral......Page 404
Part V: Summary......Page 411
Part VI: Further Topics in Random Variables......Page 413
Part VI: Introduction......Page 414
SECTION 75: Complex Random Variables......Page 415
SECTION 76: The Characteristic Function......Page 420
SECTION 77: Characteristic Function of a Transformed Random Variable......Page 426
SECTION 78: Characteristic Function of a Multidimensional Random Variable......Page 430
SECTION 79: The Generating Function......Page 435
SECTION 80: Several Jointly Gaussian Random Variables......Page 440
SECTION 81: Spherically Symmetric Vector Random Variables......Page 446
a) Discrete random variables......Page 453
b) Absolutely continuous random variables......Page 456
SECTION 83: Copulas......Page 461
a) Preliminaries......Page 472
b) Simple gambling schemes......Page 473
c) Operations on sequences......Page 476
a) Convergence of Sequences......Page 479
b) Laws of Large Numbers......Page 482
c) Connection with Statistical Regularity......Page 486
SECTION 86: Convergence of Probability Distributions and the Central Limit Theorem......Page 488
Part VI: Summary......Page 495
Answers to Queries......Page 497
Table of the Gaussian Integral......Page 500
Part I: Problems......Page 501
Part II: Problems......Page 518
Part III: Problems......Page 539
Part IV: Problems......Page 555
Part V: Problems......Page 574
Part VI: Problems......Page 592
Notation and Abbreviations......Page 605
References......Page 613
Subject Index......Page 615