Fundamentals of probability with stochastic processes

Cover......Page 1
Half title......Page 2
Title......Page 4
Copyrights......Page 5
Dedication......Page 6
Contents......Page 8
Preface......Page 14
 1.1 Introduction......Page 22
 1.2 Sample Space and Events......Page 24
 1.3 Axioms of Probability......Page 33
 1.4 Basic Theorems......Page 39
 1.5 Continuity of Probability Function......Page 48
 1.6 Probabilities 0 and 1......Page 50
 1.7 Random Selection of Points from Intervals......Page 51
 1.8 What Is Simulation?......Page 56
  Chapter 1 Summary......Page 58
  Review Problems......Page 60
  Self-Test on Chapter 1......Page 63
 2.2 Counting Principle......Page 66
  Tree Diagrams......Page 71
 2.3 Permutations......Page 76
 2.4 Combinations......Page 83
 2.5 Stirling’s Formula......Page 101
  Chapter 2 Summary......Page 102
  Review Problems......Page 103
  Self-Test on Chapter 2......Page 106
 3.1 Conditional Probability......Page 108
  Reduction of Sample Space......Page 112
 3.2 The Multiplication Rule......Page 118
 3.3 Law of Total Probability......Page 122
 3.4 Bayes’ Formula......Page 132
 3.5 Independence......Page 141
  Chapter 3 Summary......Page 158
  Review Problems......Page 160
  Self-Test on Chapter 3......Page 163
 4.1 Random Variables......Page 166
 4.2 Distribution Functions......Page 170
 4.3 Discrete Random Variables......Page 179
 4.4 Expectations of Discrete Random Variables......Page 185
 4.5 Variances and Moments of Discrete Random Variables......Page 199
  Moments......Page 204
 4.6 Standardized Random Variables......Page 208
  Chapter 4 Summary......Page 209
  Review Problems......Page 211
  Self-Test on Chapter 4......Page 213
 5.1 Bernoulli and Binomial Random Variables......Page 216
  Expectations and Variances of Binomial Random Variables......Page 223
  Poisson as an Approximation to Binomial......Page 230
  Poisson Process......Page 234
  Geometric Random Variable......Page 243
  Negative Binomial Random Variable......Page 246
  Hypergeometric Random Variable......Page 248
 Chapter 5 Summary......Page 257
 Review Problems......Page 259
 Self-Test on Chapter 5......Page 261
 6.1 Probability Density Functions......Page 264
 6.2 Density Function of a Function of a Random Variable......Page 274
  Expectations of Continuous Random Variables......Page 280
  Variances of Continuous Random Variables......Page 285
 Chapter 6 Summary......Page 293
 Review Problems......Page 295
 Self-Test on Chapter 6......Page 296
 7.1 Uniform Random Variable......Page 300
 7.2 Normal Random Variable......Page 306
  Correction for Continuity......Page 309
 7.3 Exponential Random Variables......Page 320
 7.4 Gamma Distribution......Page 327
 7.6 Survival Analysis and Hazard Function......Page 338
  Chapter 7 Summary......Page 343
  Review Problems......Page 346
  Self-Test on Chapter 7......Page 347
 7.5 Beta Distribution......Page 332
  Joint Probability Mass Functions......Page 350
  Joint Probability Density Functions......Page 354
 8.2 Independent Random Variables......Page 369
  Independence of Discrete Random Variables......Page 370
  Independence of Continuous Random Variables......Page 372
 8.3 Conditional Distributions......Page 381
  Conditional Distributions: Discrete Case......Page 382
  Conditional Distributions: Continuous Case......Page 387
 8.4 Transformations of Two Random Variables......Page 395
  Chapter 8 Summary......Page 404
  Review Problems......Page 407
  Self-Test on Chapter 8......Page 410
  Joint Probability Mass Functions......Page 414
  Joint Probability Density Functions......Page 422
  Random Sample......Page 427
 9.2 Order Statistics......Page 432
 9.3 Multinomial Distributions......Page 438
  Chapter 9 Summary......Page 443
  Review Problems......Page 446
  Self-Test on Chapter 9......Page 447
 10.1 Expected Values of Sums of Random Variables......Page 450
 10.2 Covariance......Page 460
 10.3 Correlation......Page 471
 10.4 Conditioning on Random Variables......Page 476
 10.5 Bivariate Normal Distribution......Page 491
  Chapter 10 Summary......Page 496
  Review Problems......Page 499
  Self-Test on Chapter 10......Page 501
 11.1 Moment-Generating Functions......Page 504
 11.2 Sums of Independent Random Variables......Page 515
 11.3 Markov and Chebyshev Inequalities......Page 522
  Chebyshev’s Inequality and Sample Mean......Page 526
 11.4 Laws of Large Numbers......Page 532
 11.5 Central Limit Theorem......Page 541
  Chapter 11 Summary......Page 550
  Review Problems......Page 552
  Self-Test on Chapter 11......Page 554
 12.1 Introduction......Page 558
 12.2 More on Poisson Processes......Page 559
  What Is a Queueing System?......Page 569
  PASTA: Poisson Arrivals See Time Average......Page 570
 12.3 Markov Chains......Page 573
  Classifications of States of a Markov Chain......Page 581
  Absorption Probability......Page 589
  Period......Page 591
  Steady-State Probabilities......Page 593
 12.4 Continuous-Time Markov Chains......Page 603
  Steady-State Probabilities......Page 608
  Birth and Death Processes......Page 610
 Chapter 12 Summary......Page 620
 Review Problems......Page 624
 Self-Test on Chapter 12......Page 626
Appendix Tables......Page 628
Answers to Odd-Numbered Exercises......Page 632
Index......Page 644