A First Course in Probability, Global Edition, 10th Edition

Front Cover
Title Page
Copyright Page
Contents
Preface
1 Combinatorial Analysis
	1.1 Introduction
	1.2 The Basic Principle of Counting
	1.3 Permutations
	1.4 Combinations
	1.5 Multinomial Coefficients
	1.6 The Number of Integer Solutions of Equations
		Summary
		Problems
		Theoretical Exercises
		Self-Test Problems and Exercises
2 Axioms of Probability
	2.1 Introduction
	2.2 Sample Space and Events
	2.3 Axioms of Probability
	2.4 Some Simple Propositions
	2.5 Sample Spaces Having Equally Likely Outcomes
	2.6 Probability as a Continuous Set Function
	2.7 Probability as a Measure of Belief
		Summary
		Problems
		Theoretical Exercises
		Self-Test Problems and Exercises
3 Conditional Probability Independence
	3.1 Introduction
	3.2 Conditional Probabilities
	3.3 Bayes’s Formula
	3.4 Independent Events
	3.5 P(·|F) Is a Probability
		Summary
		Problems
		Theoretical Exercises
		Self-Test Problems and Exercises
4 Random Variables
	4.1 Random Variables
	4.2 Discrete Random Variables
	4.3 Expected Value
	4.4 Expectation of a Function of a Random Variable
	4.5 Variance
	4.6 The Bernoulli and Binomial Random Variables
		4.6.1 Properties of Binomial Random Variables
		4.6.2 Computing the Binomial Distribution Function
	4.7 The Poisson Random Variable
		4.7.1 Computing the Poisson Distribution Function
	4.8 Other Discrete Probability Distributions
		4.8.1 The Geometric Random Variable
		4.8.2 The Negative Binomial Random Variable
		4.8.3 The Hypergeometric Random Variable
		4.8.4 The Zeta (or Zipf) Distribution
	4.9 Expected Value of Sums of Random Variables
	4.10 Properties of the Cumulative Distribution Function
		Summary
		Problems
		Theoretical Exercises
		Self-Test Problems and Exercises
5 Continuous Random Variables
	5.1 Introduction
	5.2 Expectation and Variance of Continuous Random Variables
	5.3 The Uniform Random Variable
	5.4 Normal Random Variables
		5.4.1 The Normal Approximation to the Binomial Distribution
	5.5 Exponential Random Variables
		5.5.1 Hazard Rate Functions
	5.6 Other Continuous Distributions
		5.6.1 The Gamma Distribution
		5.6.2 The Weibull Distribution
		5.6.3 The Cauchy Distribution
		5.6.4 The Beta Distribution
		5.6.5 The Pareto Distribution
	5.7 The Distribution of a Function of a Random Variable
		Summary
		Problems
		Theoretical Exercises
		Self-Test Problems and Exercises
6 Jointly Distributed Random Variables
	6.1 Joint Distribution Functions
	6.2 Independent Random Variables
	6.3 Sums of Independent Random Variables
		6.3.1 Identically Distributed Uniform Random Variables
		6.3.2 Gamma Random Variables
		6.3.3 Normal Random Variables
		6.3.4 Poisson and Binomial Random Variables
	6.4 Conditional Distributions: Discrete Case
	6.5 Conditional Distributions: Continuous Case
	6.6 Order Statistics
	6.7 Joint Probability Distribution of Functions of Random Variables
	6.8 Exchangeable Random Variables
		Summary
		Problems
		Theoretical Exercises
		Self-Test Problems and Exercises
7 Properties of Expectation
	7.1 Introduction
	7.2 Expectation of Sums of Random Variables
		7.2.1 Obtaining Bounds from Expectations Via the Probabilistic Method
		7.2.2 The Maximum–Minimums Identity
	7.3 Moments of the Number of Events That Occur
	7.4 Covariance, Variance of Sums, and Correlations
	7.5 Conditional Expectation
		7.5.1 Definitions
		7.5.2 Computing Expectations by Conditioning
		7.5.3 Computing Probabilities by Conditioning
		7.5.4 Conditional Variance
	7.6 Conditional Expectation and Prediction
	7.7 Moment Generating Functions
		7.7.1 Joint Moment Generating Functions
	7.8 Additional Properties of Normal Random Variables
		7.8.1 The Multivariate Normal Distribution
		7.8.2 The Joint Distribution of the Sample Mean and Sample Variance
	7.9 General Definition of Expectation
		Summary
		Problems
		Theoretical Exercises
		Self-Test Problems and Exercises
8 Limit Theorems
	8.1 Introduction
	8.2 Chebyshev’s Inequality and the Weak Law of Large Numbers
	8.3 The Central Limit Theorem
	8.4 The Strong Law of Large Numbers
	8.5 Other Inequalities and a Poisson Limit Result
	8.6 Bounding the Error Probability When Approximating a Sum ofIndependent Bernoulli Random Variables by a Poisson RandomVariable
	8.7 The Lorenz Curve
		Summary
		Problems
		Theoretical Exercises
		Self-Test Problems and Exercises
9 Additional Topics in Probability
	9.1 The Poisson Process
	9.2 Markov Chains
	9.3 Surprise, Uncertainty, and Entropy
	9.4 Coding Theory and Entropy
		Summary
		Problems and Theoretical Exercises
		Self-Test Problems and Exercises
10 Simulation
	10.1 Introduction
	10.2 General Techniques for Simulating Continuous Random Variables
		10.2.1 The Inverse Transformation Method
		10.2.2 The Rejection Method
	10.3 Simulating from Discrete Distributions
	10.4 Variance Reduction Techniques
		10.4.1 Use of Antithetic Variables
			10.4.1 Use of Antithetic Variables
			10.4.2 Variance Reduction by Conditioning
			10.4.3 Control Variates
		Summary
		Problems
		Self-Test Problems and Exercises
Answers to Selected Problems
Solutions to Self-Test Problems and Exercises
Index
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