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Introduction to Probability, Second Edition 2nd edition [Kõva köide]

(Harvard University, Cambridge, Massachusetts, USA), (Stanford University, California, USA)
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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 Blitzsteins 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.
Preface xiii
1 Probability and counting
1(44)
1.1 Why study probability?
1(2)
1.2 Sample spaces and Pebble World
3(3)
1.3 Naive definition of probability
6(2)
1.4 How to count
8(12)
1.5 Story proofs
20(1)
1.6 Non-naive definition of probability
21(5)
1.7 Recap
26(3)
1.8 R
29(4)
1.9 Exercises
33(12)
2 Conditional probability
45(58)
2.1 The importance of thinking conditionally
45(1)
2.2 Definition and intuition
46(6)
2.3 Bayes' rule and the law of total probability
52(7)
2.4 Conditional probabilities are probabilities
59(4)
2.5 Independence of events
63(4)
2.6 Coherency of Bayes' rule
67(1)
2.7 Conditioning as a problem-solving tool
68(6)
2.8 Pitfalls and paradoxes
74(5)
2.9 Recap
79(1)
2.10 R
80(3)
2.11 Exercises
83(20)
3 Random variables and their distributions
103(46)
3.1 Random variables
103(3)
3.2 Distributions and probability mass functions
106(6)
3.3 Bernoulli and Binomial
112(3)
3.4 Hypergeometric
115(3)
3.5 Discrete Uniform
118(2)
3.6 Cumulative distribution functions
120(3)
3.7 Functions of random variables
123(6)
3.8 Independence of r.v.s
129(4)
3.9 Connections between Binomial and Hypergeometric
133(3)
3.10 Recap
136(2)
3.11 R
138(2)
3.12 Exercises
140(9)
4 Expectation
149(64)
4.1 Definition of expectation
149(3)
4.2 Linearity of expectation
152(5)
4.3 Geometric and Negative Binomial
157(7)
4.4 Indicator r.v.s and the fundamental bridge
164(6)
4.5 Law of the unconscious statistician (LOTUS)
170(1)
4.6 Variance
171(3)
4.7 Poisson
174(7)
4.8 Connections between Poisson and Binomial
181(3)
4.9 *Using probability and expectation to prove existence
184(5)
4.10 Recap
189(3)
4.11 R
192(2)
4.12 Exercises
194(19)
5 Continuous random variables
213(54)
5.1 Probability density functions
213(7)
5.2 Uniform
220(4)
5.3 Universality of the Uniform
224(7)
5.4 Normal
231(7)
5.5 Exponential
238(6)
5.6 Poisson processes
244(4)
5.7 Symmetry of i.i.d. continuous r.v.s
248(2)
5.8 Recap
250(3)
5.9 R
253(2)
5.10 Exercises
255(12)
6 Moments
267(36)
6.1 Summaries of a distribution
267(5)
6.2 Interpreting moments
272(4)
6.3 Sample moments
276(3)
6.4 Moment generating functions
279(4)
6.5 Generating moments with MGFs
283(3)
6.6 Sums of independent r.v.s via MGFs
286(1)
6.7 *Probability generating functions
287(5)
6.8 Recap
292(1)
6.9 R
293(5)
6.10 Exercises
298(5)
7 Joint distributions
303(64)
7.1 Joint, marginal, and conditional
304(20)
7.2 2D LOTUS
324(2)
7.3 Covariance and correlation
326(6)
7.4 Multinomial
332(11)
7.5 Multivariate Normal . 33?
7.6 Recap
343(3)
7.7 R
346(2)
7.8 Exercises
348(19)
8 Transformations
367(48)
8.1 Change of variables
369(6)
8.2 Convolutions
375(4)
8.3 Beta
379(8)
8.4 Gamma
387(9)
8.5 Beta-Gamma connections
396(2)
8.6 Order statistics
398(4)
8.7 Recap
402(2)
8.8 R
404(3)
8.9 Exercises
407(8)
9 Conditional expectation
415(42)
9.1 Conditional expectation given an event
415(9)
9.2 Conditional expectation given an r.v.
424(2)
9.3 Properties of conditional expectation
426(5)
9.4 *Geometric interpretation of conditional expectation
431(1)
9.5 Conditional variance
432(4)
9.6 Adam and Eve examples
436(3)
9.7 Recap
439(2)
9.8 R
441(2)
9.9 Exercises
443(14)
10 Inequalities and limit theorems
457(40)
10.1 Inequalities
458(9)
10.2 Law of large numbers
467(4)
10.3 Central limit theorem
471(6)
10.4 Chi-Square and Student
477(3)
10.5 Recap
480(3)
10.6 R
483(3)
10.7 Exercises
486(11)
11 Markov chains
497(38)
11.1 Markov property and transit ion matrix
497(5)
11.2 Classification of states
502(4)
11.3 Stationary distribution
506(7)
11.4 Reversibility
513(7)
11.5 Recap
520(1)
11.6 R
521(3)
11.7 Exercises
524(11)
12 Markov chain Monte Carlo
535(24)
12.1 Metropolis-Hastings
536(12)
12.2 Gibbs sampling
548(6)
12.3 Recap
554(1)
12.4 R
555(2)
12.5 Exercises
557(2)
13 Poisson processes
559(22)
13.1 Poisson processes in one dimension
559(2)
13.2 Conditioning, superposition, and thinning
561(12)
13.3 Poisson processes in multiple dimensions
573(2)
13.4 Recap
575(1)
13.5 R
575(2)
13.6 Exercises
577(4)
A Math
581(20)
A.1 Sets
581(4)
A.2 Functions
585(5)
A.3 Matrices
590(2)
A.4 Difference equations
592(1)
A.5 Differential equations
593(1)
A.6 Partial derivatives
594(1)
A.7 Multiple integrals
594(2)
A.8 Sums
596(3)
A.9 Pattern recognition
599(1)
A.10 Common sense and checking answers
599(2)
B R
601(6)
B.1 Vectors
601(1)
B.2 Matrices
602(1)
B.3 Math
602(1)
B.4 Sampling and simulation
603(1)
B.5 Plotting
603(1)
B.6 Programming
603(1)
B.7 Summary statistics
604(1)
B.8 Distributions
604(1)
C Table of distributions
605(2)
References 607(2)
Index 609
Joseph K. Blitzstein, PhD, professor of the practice in statistics, Department of Statistics, Harvard University, Cambridge, Massachusetts, USA

Jessica Hwang is a graduate student in the Stanford statistics department.