Fundamentals of Probability......Page 1
Letter to the Regents of the University of California......Page 2
Table Of Content......Page 5
1.1 - Introduction......Page 9
1.2 - Sample Space and Events......Page 12
1.3 - Axioms of Probability......Page 19
1.4 - Basic Theorems......Page 26
1.5 - Continuity of Probability Function......Page 35
1.6 - Probabilities 0 and 1......Page 37
1.7 - Random Selection of Points from Intervals......Page 39
Review Problems......Page 43
2.2 - Counting Principle......Page 46
Number of Subsets of a Set......Page 50
Tree Diagrams......Page 51
2.3 - Permutations......Page 55
2.4 - Combinations......Page 62
2.5 - Stirling's Formula......Page 78
Review Problems......Page 80
3.1 Conditional Probability......Page 83
Reduction of Sample Space......Page 87
3.2 - Law of Multiplication......Page 93
3.3 - Law of Total Probability......Page 97
3.4 - Bayes' Formula......Page 107
3.5 - Independence......Page 114
Review Problems......Page 134
4.1 - Random Variables......Page 137
4.2 - Distribution Functions......Page 142
4.3 - Discrete Random Variables......Page 151
4.4 - Expectations of Discrete Random Variables......Page 158
4.5 - Variances and Moments of Discrete Random Variables......Page 169
Moments......Page 175
4.6 - Standardized Random Variables......Page 178
Review Problems......Page 179
5.1 - Bernoulli and Binomial Random Variables......Page 182
Expectations and Variances of Binomial Random Variables......Page 188
Poisson as an Approximation to Binomial......Page 196
Poisson Process......Page 202
Geometric Random Variables......Page 211
Negative Binomial Random Variable......Page 213
Hypergeometric Random Variable......Page 215
Review Problems......Page 223
6.1 - Probability Density Functions......Page 226
6.2 - Density Function of a Function of a Random Variable......Page 235
Expectations of Continuous Random Variables......Page 241
Variances of Continuous Random Variables......Page 247
Review Problems......Page 253
7.1 - Uniform Random Variable......Page 255
7.2 - Normal Random Variable......Page 262
Correction for Continuity......Page 265
7.3 - Exponential Random Variable......Page 277
7.4 - Gamma Distribution......Page 284
7.5 - Beta Distribution......Page 289
Review Problems......Page 293
Joint Probability Functions......Page 295
Joint Probability Density Functions......Page 298
8.2 - Independent Random Variables......Page 313
8.3 - Conditional Distributions......Page 324
8.4 - Multivariate Distributions......Page 337
8.5 - Order Statistics......Page 353
8.6 - Multinomial Distributions......Page 360
8.7 - Transformations of Two Random Variables......Page 364
Review Problems......Page 370
9.1 - Expected Values of Sums of Random Variables......Page 375
Pattern Appearance......Page 384
9.2 - Covariance......Page 391
9.3 - Correlation......Page 400
9.4 - Conditioning on Random Variables......Page 405
9.5 - Bivariate Normal Distribution......Page 416
Review Problems......Page 422
10.1 - Moment-Generating Functions......Page 424
10.2 - Sums of Independent Random Variables......Page 435
10.3 - Markov and Chebyshev Inequalities......Page 445
10.4 - Laws of Large Numbers......Page 451
10.5 - Central Limit Theorem......Page 460
Review Questions......Page 466
11.1 - Introduction......Page 469
11.2 - Simulations of Combinatorial Problems......Page 474
11.3 - Simulations of Conditional Probabilities......Page 477
11.4 - Simulation of Random Variables......Page 481
11.5 - Monte Carlo Method......Page 489
Appendix......Page 495
Answers to Odd-Numbered Exercises......Page 499
Index......Page 509