Linux Command Line Made Easy: A Practical, Step By Step Guide To Linux Commands For Beginners And Intermediates

Developed from celebrated Harvard statistics lectures, **Introduction to Probability** provides essential language and toolsfor understanding statistics, randomness, and uncertainty. The book explores a wide variety of applications and examples, ranging from coincidences and paradoxes to Google PageRank and Markov chain Monte Carlo (MCMC). Additional application areas explored include genetics, medicine, computer science, and information theory. The authors present the material in an accessible style and motivate concepts using real-world examples. Throughout, they use __stories__ to uncover connections between the fundamental distributions in statistics and __conditioning__ to reduce complicated problems to manageable pieces. The book includes many intuitive explanations, diagrams, and practice problems. Each chapter ends with a section showing how to perform relevant simulations and calculations in R, a free statistical software environment. The second edition adds many new examples, exercises, and explanations, to deepen understanding of the ideas, clarify subtle concepts, and respond to feedback from many students and readers. New supplementary online resources have been developed, including animations and interactive visualizations, and the book has been updated to dovetail with these resources. Supplementary material is available on Joseph Blitzstein’s website www. stat110.net. The supplements include: Solutions to selected exercises Additional practice problems Handouts including review material and sample exams Animations and interactive visualizations created in connection with the edX online version of Stat 110. Links to lecture videos available on ITunes U and YouTube There is also a complete instructor's solutions manual available to instructors who require the book for a course. Cover......Page 1 Half Title......Page 2 Title Page......Page 6 Copyright Page......Page 7 Dedication......Page 8 Table of Contents......Page 10 Preface......Page 14 1.1 Why study probability?......Page 18 1.2 Sample spaces and Pebble World......Page 20 1.3 Naive definition of probability......Page 23 1.4 How to count......Page 25 1.5 Story proofs......Page 37 1.6 Non-naive definition of probability......Page 38 1.7 Recap......Page 43 1.8 R......Page 46 1.9 Exercises......Page 50 2.1 The importance of thinking conditionally......Page 62 2.2 Definition and intuition......Page 63 2.3 Bayes' rule and the law of total probability......Page 69 2.4 Conditional probabilities are probabilities......Page 76 2.5 Independence of events......Page 80 2.6 Coherency of Bayes' rule......Page 84 2.7 Conditioning as a problem-solving tool......Page 85 2.8 Pitfalls and paradoxes......Page 91 2.9 Recap......Page 96 2.10 R......Page 97 2.11 Exercises......Page 100 3.1 Random variables......Page 120 3.2 Distributions and probability mass functions......Page 123 3.3 Bernoulli and Binomial......Page 129 3.4 Hypergeometric......Page 132 3.5 Discrete Uniform......Page 135 3.6 Cumulative distribution functions......Page 137 3.7 Functions of random variables......Page 140 3.8 Independence of r.v.s......Page 146 3.9 Connections between Binomial and Hypergeometric......Page 150 3.10 Recap......Page 153 3.11 R......Page 155 3.12 Exercises......Page 157 4.1 Definition of expectation......Page 166 4.2 Linearity of expectation......Page 169 4.3 Geometric and Negative Binomial......Page 174 4.4 Indicator r.v.s and the fundamental bridge......Page 181 4.5 Law of the unconscious statistician (LOTUS)......Page 187 4.6 Variance......Page 188 4.7 Poisson......Page 191 4.8 Connections between Poisson and Binomial......Page 198 4.9 *Using probability and expectation to prove existence......Page 201 4.10 Recap......Page 206 4.11 R......Page 209 4.12 Exercises......Page 211 5.1 Probability density functions......Page 230 5.2 Uniform......Page 237 5.3 Universality of the Uniform......Page 241 5.4 Normal......Page 248 5.5 Exponential......Page 255 5.6 Poisson processes......Page 261 5.7 Symmetry of i.i.d. continuous r.v.s......Page 265 5.8 Recap......Page 267 5.9 R......Page 270 5.10 Exercises......Page 272 6.1 Summaries of a distribution......Page 284 6.2 Interpreting moments......Page 289 6.3 Sample moments......Page 293 6.4 Moment generating functions......Page 296 6.5 Generating moments with MGFs......Page 300 6.6 Sums of independent r.v.s via MGFs......Page 303 6.7 *Probability generating functions......Page 304 6.8 Recap......Page 309 6.9 R......Page 310 6.10 Exercises......Page 315 7: Joint distributions......Page 320 7.1 Joint, marginal, and conditional......Page 321 7.2 2D LOTUS......Page 341 7.3 Covariance and correlation......Page 343 7.4 Multinomial......Page 349 7.5 Multivariate Normal......Page 354 7.6 Recap......Page 360 7.7 R......Page 363 7.8 Exercises......Page 365 8: Transformations......Page 384 8.1 Change of variables......Page 386 8.2 Convolutions......Page 392 8.3 Beta......Page 396 8.4 Gamma......Page 404 8.5 Beta-Gamma connections......Page 413 8.6 Order statistics......Page 415 8.7 Recap......Page 419 8.8 R......Page 421 8.9 Exercises......Page 424 9.1 Conditional expectation given an event......Page 432 9.2 Conditional expectation given an r.v.......Page 441 9.3 Properties of conditional expectation......Page 443 9.4 *Geometric interpretation of conditional expectation......Page 448 9.5 Conditional variance......Page 449 9.6 Adam and Eve examples......Page 453 9.7 Recap......Page 456 9.8 R......Page 458 9.9 Exercises......Page 460 10: Inequalities and limit theorems......Page 474 10.1 Inequalities......Page 475 10.2 Law of large numbers......Page 484 10.3 Central limit theorem......Page 488 10.4 Chi-Square and Student-t......Page 494 10.5 Recap......Page 497 10.6 R......Page 500 10.7 Exercises......Page 503 11.1 Markov property and transition matrix......Page 514 11.2 Classification of states......Page 519 11.3 Stationary distribution......Page 523 11.4 Reversibility......Page 530 11.5 Recap......Page 537 11.6 R......Page 538 11.7 Exercises......Page 541 12: Markov chain Monte Carlo......Page 552 12.1 Metropolis-Hastings......Page 553 12.2 Gibbs sampling......Page 565 12.3 Recap......Page 571 12.4 R......Page 572 12.5 Exercises......Page 574 13.1 Poisson processes in one dimension......Page 576 13.2 Conditioning, superposition, and thinning......Page 578 13.3 Poisson processes in multiple dimensions......Page 590 13.5 R......Page 592 13.6 Exercises......Page 594 A.1 Sets......Page 598 A.2 Functions......Page 602 A.3 Matrices......Page 607 A.4 Difference equations......Page 609 A.5 Differential equations......Page 610 A.7 Multiple integrals......Page 611 A.8 Sums......Page 613 A.10 Common sense and checking answers......Page 616 B.1 Vectors......Page 618 B.3 Math......Page 619 B.6 Programming......Page 620 B.8 Distributions......Page 621 C: Table of distributions......Page 622 References......Page 624 Index......Page 626 "Undergraduate probability book that assumes one-semester of calculus. One key is the emphasis on "stories" for the probability distributions (which I mean in both an intuitive and technical sense): there are a dozen or so key distributions (Normal, Binomial, Poisson, etc.) that are incredibly widely-used in statistics, but a lot of books just write down formulas for them without explaining clearly why these particular distributions are so important, or how they are all connected. Each of these distributions has a "story" (a natural application where it arises), and thinking about stories makes the distributions easier to remember, understand, and work with"-- Provided by publisher Assumes one-semester of calculus. "Stories" make distributions (Normal, Binomial, Poisson that are widely-used in statistics) easier to remember, understand. Many books write down formulas without explaining clearly why these particular distributions are important or how they are all connected.