Probability : with applications and R

Cover
Title Page
Copyright
Contents
Preface
Acknowledgments
About the Companion Website
Introduction
Chapter 1 First Principles
	1.1 Random Experiment, Sample Space, Event
	1.2 What Is a Probability?
	1.3 Probability Function
	1.4 Properties of Probabilities
	1.5 Equally likely outcomes
	1.6 Counting I
		1.6.1 Permutations
	1.7 Counting II
		1.7.1 Combinations and Binomial Coefficients
	1.8 Problem‐Solving Strategies: Complements and Inclusion–Exclusion
	1.9 A First Look at Simulation
	1.10 Summary
	Exercises
Chapter 2 Conditional Probability and Independence
	2.1 Conditional Probability
	2.2 New Information Changes the Sample Space
	2.3 Finding P(A and B)
		2.3.1 Birthday Problem
	2.4 Conditioning and the Law of Total Probability
	2.5 Bayes Formula and Inverting a Conditional Probability
	2.6 Independence and Dependence
	2.7 Product Spaces*
	2.8 Summary
	Exercises
Chapter 3 INTRODUCTION TO DISCRETE RANDOM VARIABLES
	Learning Outcomes
	3.1 Random Variables
	3.2 Independent Random Variables
	3.3 Bernoulli Sequences
	3.4 Binomial Distribution
	3.5 Poisson Distribution
		3.5.1 Poisson Approximation of Binomial Distribution
		3.5.2 Poisson as Limit of Binomial Probabilities*
	3.6 Summary
	Exercises
Chapter 4 Expectation and More with Discrete Random Variables
	4.1 Expectation
	4.2 Functions of Random Variables
	4.3 Joint distributions
	4.4 Independent Random Variables
		4.4.1 Sums of Independent Random Variables
	4.5 Linearity of expectation
	4.6 Variance and Standard Deviation
	4.7 Covariance and Correlation
	4.8 Conditional Distribution
		4.8.1 Introduction to Conditional Expectation
	4.9 Properties of Covariance and Correlation*
	4.10 Expectation of a Function of a Random Variable*
	4.11 Summary
	Exercises
Chapter 5 More Discrete Distributions and Their Relationships
	5.1 Geometric Distribution
		5.1.1 Memorylessness
		5.1.2 Coupon Collecting and Tiger Counting
	5.2 Moment‐Generating Functions
	5.3 Negative Binomial—Up from the Geometric
	5.4 Hypergeometric—Sampling Without Replacement
	5.5 From Binomial to Multinomial
	5.6 Benford's Law*
	5.7 Summary
	Exercises
Chapter 6 Continuous Probability
	6.1 Probability Density Function
	6.2 Cumulative Distribution Function
	6.3 Expectation and Variance
	6.4 Uniform Distribution
	6.5 Exponential Distribution
		6.5.1 Memorylessness
	6.6 Joint Distributions
	6.7 Independence
		6.7.1 Accept–Reject Method
	6.8 Covariance, Correlation
	6.9 Summary
	Exercises
Chapter 7 Continuous Distributions
	7.1 Normal Distribution
		7.1.1 Standard Normal Distribution
		7.1.2 Normal Approximation of Binomial Distribution
		7.1.3 Quantiles
		7.1.4 Sums of Independent Normals
	7.2 Gamma Distribution
		7.2.1 Probability as a Technique of Integration
	7.3 Poisson Process
	7.4 Beta Distribution
	7.5 Pareto Distribution*
	7.6 Summary
	Exercises
Chapter 8 Densities of Functions of Random Variables
	8.1 Densities via CDFs
		8.1.1 Simulating a Continuous Random Variable
		8.1.2 Method of Transformations
	8.2 Maximums, Minimums, and Order Statistics
	8.3 Convolution
	8.4 Geometric Probability
	8.5 Transformations of Two Random Variables*
	8.6 Summary
	Exercises
Chapter 9 Conditional Distribution, Expectation, and Variance
	9.1 Conditional Distributions
	9.2 DISCRETE AND CONTINUOUS: MIXING IT UP
	9.3 CONDITIONAL EXPECTATION
		9.3.1 From Function to Random Variable
		9.3.2 Random Sum of Random Variables
	9.4 COMPUTING PROBABILITIES BY CONDITIONING
	9.5 CONDITIONAL VARIANCE
	9.6 BIVARIATE NORMAL DISTRIBUTION*
	9.7 SUMMARY
	Exercises
Chapter 10 Limits
	10.1 WEAK LAW OF LARGE NUMBERS
		10.1.1 Markov and Chebyshev Inequalities
	10.2 STRONG LAW OF LARGE NUMBERS
	10.3 METHOD OF MOMENTS*
	10.4 MONTE CARLO INTEGRATION
	10.5 CENTRAL LIMIT THEOREM
		10.5.1 Central Limit Theorem and Monte Carlo
	10.6 A PROOF OF THE CENTRAL LIMIT THEOREM
	10.7 SUMMARY
	Exercises
Chapter 11 Beyond Random Walks And Markov Chains
	11.1 RANDOM WALKS ON GRAPHS
		11.1.1 Long‐Term Behavior
	11.2 RANDOM WALKS ON WEIGHTED GRAPHS AND MARKOV CHAINS
		11.2.1 Stationary Distribution
	11.3 FROM MARKOV CHAIN TO MARKOV CHAIN MONTE CARLO
	11.4 SUMMARY
	Exercises
Chapter A Probability Distributions in R
Chapter B Summary of Probability Distributions
Chapter C Mathematical Reminders
Chapter D Working with Joint Distributions
SOLUTIONS TO EXERCISES
References
Index
EULA