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E-raamat: Probability: With Applications and R

(Carleton College), (Amherst College)
  • Formaat: EPUB+DRM
  • Ilmumisaeg: 11-Jun-2021
  • Kirjastus: John Wiley & Sons Inc
  • Keel: eng
  • ISBN-13: 9781119692416
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  • Formaat: EPUB+DRM
  • Ilmumisaeg: 11-Jun-2021
  • Kirjastus: John Wiley & Sons Inc
  • Keel: eng
  • ISBN-13: 9781119692416
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"This book is ideal for courses on Probability typically taught in Mathematics and/or Statistics departments but could also be used in Engineering or Data Science departments. This book could also serve as a supplemental or review text for courses on Stochastic Processes or Markov Chains or Brownian Motion, since those require a strong foundation in probability. The text is also preparatory for the Probability Actuarial Exam -- students who successfully complete a course with this text and do well are well-positioned to pass the P exam. Some major features of the new edition include an addition of supplemental materials for coding and simulation, improved exposition and examples for some topics, and addressing issues with errata. These features increase the value of the text especially in an era where developing computing skills has become a staple of statistical practice, and desirable for many other fields as well"--

Discover the latest edition of a practical introduction to the theory of probability, complete with R code samples

In the newly revised Second Edition of Probability: With Applications and R, distinguished researchers Drs. Robert Dobrow and Amy Wagaman deliver a thorough introduction to the foundations of probability theory. The book includes a host of chapter exercises, examples in R with included code, and well-explained solutions. With new and improved discussions on reproducibility for random numbers and how to set seeds in R, and organizational changes, the new edition will be of use to anyone taking their first probability course within a mathematics, statistics, engineering, or data science program.

New exercises and supplemental materials support more engagement with R, and include new code samples to accompany examples in a variety of chapters and sections that didn’t include them in the first edition.

The new edition also includes for the first time: 

  • A thorough discussion of reproducibility in the context of generating random numbers
  • Revised sections and exercises on conditioning, and a renewed description of specifying PMFs and PDFs
  • Substantial organizational changes to improve the flow of the material
  • Additional descriptions and supplemental examples to the bivariate sections to assist students with a limited understanding of calculus

Perfect for upper-level undergraduate students in a first course on probability theory, Probability: With Applications and R is also ideal for researchers seeking to learn probability from the ground up or those self-studying probability for the purpose of taking advanced coursework or preparing for actuarial exams.

Preface xi
Acknowledgments xv
About the Companion Website xvii
Introduction xix
1 First Principles
1(44)
1.1 Random Experiment, Sample Space, Event
1(2)
1.2 What Is a Probability?
3(1)
1.3 Probability Function
4(3)
1.4 Properties of Probabilities
7(4)
1.5 Equally Likely Outcomes
11(1)
1.6 Counting I
12(4)
1.6.1 Permutations
13(3)
1.7 Counting II
16(10)
1.7.1 Combinations and Binomial Coefficients
17(9)
1.8 Problem-Solving Strategies: Complements and Inclusion-Exclusion
26(3)
1.9 A First Look at Simulation
29(5)
1.10 Summary
34(11)
Exercises
36(9)
2 Conditional Probability and Independence
45(48)
2.1 Conditional Probability
45(5)
2.2 New Information Changes the Sample Space
50(1)
2.3 Finding P (A and B)
51(9)
2.3.1 Birthday Problem
56(4)
2.4 Conditioning and the Law of Total Probability
60(7)
2.5 Bayes Formula and Inverting a Conditional Probability
67(5)
2.6 Independence and Dependence
72(8)
2.7 Product Spaces
80(2)
2.8 Summary
82(11)
Exercises
83(10)
3 Introduction to Discrete Random Variables
93(32)
3.1 Random Variables
93(4)
3.2 Independent Random Variables
97(2)
3.3 Bernoulli Sequences
99(2)
3.4 Binomial Distribution
101(7)
3.5 Poisson Distribution
108(8)
3.5.1 Poisson Approximation of Binomial Distribution
113(2)
3.5.2 Poisson as Limit of Binomial Probabilities
115(1)
3.6 Summary
116(9)
Exercises
118(7)
4 Expectation and More with Discrete Random Variables
125(60)
4.1 Expectation
127(3)
4.2 Functions of Random Variables
130(4)
4.3 Joint Distributions
134(5)
4.4 Independent Random Variables
139(5)
4.4.1 Sums of Independent Random Variables
142(2)
4.5 Linearity of Expectation
144(5)
4.6 Variance and Standard Deviation
149(9)
4.7 Covariance and Correlation
158(7)
4.8 Conditional Distribution
165(6)
4.8.1 Introduction to Conditional Expectation
168(3)
4.9 Properties of Covariance and Correlation
171(2)
4.10 Expectation of a Function of a Random Variable
173(1)
4.11 Summary
174(11)
Exercises
176(9)
5 More Discrete Distributions and Their Relationships
185(42)
5.1 Geometric Distribution
185(8)
5.1.1 Memorylessness
188(1)
5.1.2 Coupon Collecting and Tiger Counting
189(4)
5.2 Moment-Generating Functions
193(3)
5.3 Negative Binomial--Up from the Geometric
196(6)
5.4 Hypergeometric--Sampling Without Replacement
202(5)
5.5 From Binomial to Multinomial
207(6)
5.6 Benford's Law
213(3)
5.7 Summary
216(11)
Exercises
218(9)
6 Continuous Probability
227(46)
6.1 Probability Density Function
229(4)
6.2 Cumulative Distribution Function
233(4)
6.3 Expectation and Variance
237(2)
6.4 Uniform Distribution
239(3)
6.5 Exponential Distribution
242(5)
6.5.1 Memorylessness
243(4)
6.6 Joint Distributions
247(9)
6.7 Independence
256(6)
6.7.1 Accept-Reject Method
258(4)
6.8 Covariance, Correlation
262(2)
6.9 Summary
264(9)
Exercises
266(7)
7 Continuous Distributions
273(46)
7.1 Normal Distribution
273(15)
7.1.1 Standard Normal Distribution
276(2)
7.1.2 Normal Approximation of Binomial Distribution
278(4)
7.1.3 Quantiles
282(3)
7.1.4 Sums of Independent Normals
285(3)
7.2 Gamma Distribution
288(6)
7.2.1 Probability as a Technique of Integration
292(2)
7.3 Poisson Process
294(8)
7.4 Beta Distribution
302(3)
7.5 Pareto Distribution
305(3)
7.6 Summary
308(11)
Exercises
311(8)
8 Densities of Functions of Random Variables
319(38)
8.1 Densities via CDFs
320(10)
8.1.1 Simulating a Continuous Random Variable
326(3)
8.1.2 Method of Transformations
329(1)
8.2 Maximums, Minimums, and Order Statistics
330(5)
8.3 Convolution
335(3)
8.4 Geometric Probability
338(6)
8.5 Transformations of Two Random Variables
344(4)
8.6 Summary
348(9)
Exercises
349(8)
9 Conditional Distribution, Expectation, and Variance
357(50)
Introduction
357(1)
9.1 Conditional Distributions
358(6)
9.2 Discrete and Continuous: Mixing it Up
364(5)
9.3 Conditional Expectation
369(9)
9.3.1 From Function to Random Variable
371(7)
9.3.2 Random Sum of Random Variables
378(1)
9.4 Computing Probabilities by Conditioning
378(4)
9.5 Conditional Variance
382(5)
9.6 Bivariate Normal Distribution
387(9)
9.7 Summary
396(11)
Exercises
398(9)
10 Limits
407(40)
10.1 Weak Law of Large Numbers
409(6)
10.1.1 Markov and Chebyshev Inequalities
411(4)
10.2 Strong Law of Large Numbers
415(6)
10.3 Method of Moments
421(3)
10.4 Monte Carlo Integration
424(4)
10.5 Central Limit Theorem
428(9)
10.5.1 Central Limit Theorem and Monte Carlo
436(1)
10.6 A Proof of the Central Limit Theorem
437(2)
10.7 Summary
439(8)
Exercises
440(7)
11 Beyond Random Walks and Markov Chains
447(34)
11.1 Random Walks on Graphs
447(8)
11.1.1 Long-Term Behavior
451(4)
11.2 Random Walks on Weighted Graphs and Markov Chains
455(7)
11.2.1 Stationary Distribution
458(4)
11.3 From Markov Chain to Markov Chain Monte Carlo
462(12)
11.4 Summary
474(7)
Exercises
476(5)
Appendix A Probability Distributions in R 481(2)
Appendix B Summary of Probability Distributions 483(4)
Appendix C Mathematical Reminders 487(2)
Appendix D Working with Joint Distributions 489(8)
Solutions to Exercises 497(14)
References 511(4)
Index 515
Amy S. Wagaman, PhD, is Associate Professor of Statistics at Amherst College. She received her doctorate in Statistics at the University of Michigan in 2008. Her research interests include nonparametric statistics, statistics education, dimension reduction and estimation, and covariance estimation and regularization.

Robert P. Dobrow, PhD, is Emeritus Professor of Mathematics at Carleton College. He has over 15 years of experience teaching probability and has authored numerous papers in probability theory, Markov chains, and statistics.