Probability with Statistical Applications

Preface to the Third Edition
Contents
1 Probability Space
	1 Equally Likely Outcomes
	2 The Axioms of Probability
	Problems
2 Conditional Probabilities
	1 Definition
	2 Bayes\' Method
	3 Symmetry
	Problems
	4 Independence
	Problems
	5 The Birthday Problem
	Problems
3 Discrete Random Variables
	1 Discrete Distributions
		1.1 Bernoulli Random Variables
		1.2 Geometric Random Variables
	Problems
	2 Expectation
		2.1 The Expectation of a Sum
	Problems
	3 Variance
		3.1 Variance and Independence
	Problems
	4 Coupon Collector\'s Problem
	Problems
4 Binomial Random Variables
	1 Binomial Probability Distribution
	Problems
	2 Mean and Variance
		2.1 Derivation of the Binomial Distribution
	3 Normal Approximation
		3.1 The Normal Table
		3.2 Normal Approximation
	Problems
	4 The Negative Binomial
	Problems
5 Poisson Random Variables
	1 Poisson Probability Distribution
	2 Poisson Scatter Theorem
	3 Poisson Approximation to the Binomial
	4 Approximation to a Sum of Binomials
	Problems
	5 Mean and Variance
6 Simulations of Discrete Random Variables
	1 Random Numbers
	2 Bernoulli Random Variables
	3 Binomial Random Variables
		3.1 Computational Formula for the Binomial Distribution
	4 Poisson Random Variables
	Problems
7 Combinatorics
	1 Counting Principle
	Problems
	2 Properties of the Binomial Coefficients
	Problems
	3 Hypergeometric Random Variables
	Problems
	4 Mean and Variance of a Hypergeometric
	5 Conditioning on the Number of Successes
8 Continuous Random Variables
	1 Probability Densities
	2 Uniform Random Variables
	3 Exponential Random Variables
		3.1 Memoryless Property
	Problems
	4 Expected Value
		4.1 Symmetric Probability Density
		4.2 Function of a Random Variable
	5 The Median
	Problems
	6 Variance
	Problems
	7 Normal Random Variables
		7.1 The Standard Normal
		7.2 Normal Random Variables
		7.3 Applications of Normal Random Variables
		7.4 Expectation and Variance of a Standard Normal
	Problems
	More Problems for Chap.8
9 The Sample Average and Variance
	1 The Sample Average
	2 The Central Limit Theorem
	3 The Sample Variance
	Problems
	4 Monte Carlo Integration
	Problem
10 Estimating and Testing Proportions
	1 Testing a Proportion
	2 Confidence Interval for a Proportion
	Problems
	3 Testing Two Proportions
	4 Confidence Interval for Two Proportions
	Problems
11 Estimating and Testing Means
	1 Testing a Mean
	2 Confidence Interval for a Mean
	3 Testing Two Means
	4 Two Means Confidence Interval
	Problems
12 Small Samples
	1 Student Tests
	2 Two Means Student Tests
	3 Student Tests for Matched Pairs
	4 The Sign Test
	Problems
13 Chi-Squared Tests
	1 Testing Independence
	2 Goodness of Fit Test
	Problems
14 Design of Experiments
	1 Double Blind Design
	2 Data Dredging
	Problems
15 The Cumulative Distribution Function
	1 Definition and Examples
	2 Transformations of Random Variables
	Problems
	3 Sample Maximum and Minimum
	Problems
	4 Simulations
	Problems
16 Continuous Joint Distributions
	1 Joint and Marginal Densities
	2 Independence
	3 Transformations of Random Vectors
	Problems
	4 Gamma and Beta Random Variables
		4.1 The Function Gamma
		4.2 Gamma Random Variables
		4.3 The Ratio of Two Gamma Random Variables
		4.4 Beta Random Variables
	Problems
17 Covariance and Independence
	1 Covariance
	2 Independence
	3 Correlation
	4 Variance of a Sum
	Problems
	5 Proof That the Expectation Is Linear
	6 Proof That the Correlation Is Bounded
18 Conditional Distribution and Expectation
	1 The Discrete Case
	Problems
	2 Continuous Case
	Problems
	3 Conditional Expectation
		3.1 Conditional Expectation and Prediction
	Problems
19 The Bivariate Normal Distribution
	1 The Correlation
	2 An Application
		2.1 Best Predictor
	Problems
	3 The Joint Probability Density
	4 The Conditional Probability Density
	Problems
20 Sums of Bernoulli Random Variables
	1 The Expected Number of Birthdays
	2 The Matching Problem
		2.1 Expected Number of Matches
		2.2 Variance of a Sum
		2.3 Variance of the Number of Matches
	3 The Moments of the Hypergeometric
	4 The Number of Records
	Problems
21 Coupling Random Variables
	1 Coupling Two Bernoulli Random Variables
	2 Coupling Two Poisson Random Variables
	3 The Coupling Inequality
	4 Poisson Approximation of a Sum
		4.1 Poisson Approximation of a Binomial
	5 Proof of the Poisson Approximation
	Problems
22 The Moment Generating Function
	1 Definition and Examples
		1.1 Sum of i.i.d. Bernoulli Random Variables
		1.2 Sum of Independent Poisson Random Variables
	Problems
	2 The m.g.f. of a Normal
	3 Moment Computations
	Problems
	4 Convergence in Distribution
		4.1 Binomial Convergence to a Poisson
		4.2 Proof of the Central Limit Theorem
	Problems
23 Chi-Squared, Student, and F Distributions
	1 The m.g.f. of a Gamma Random Variable
		1.1 Sum of i.i.d. Exponential Random Variables
	Problems
	2 The Chi-Squared Distribution
	3 The Student Distribution
	4 The F Distribution
	Problems
24 Sampling from a Normal Distribution
	1 The Sample Average and Variance
	Problems
	2 The Sample Average Is Normal
	3 The Sample Variance Distribution
	4 The Standardized Average
	Problem
25 Finding Estimators
	1 The Method of Moments
	Problems
	2 The Maximum Likelihood Method
	Problems
26 Comparing Estimators
	1 The Mean Squared Error
	2 Biased and Unbiased Estimators
	3 Two Estimators for a Normal Variance
	4 Two Estimators for a Uniform Distribution
	5 Proof of the M.S.E. Formula
	Problems
27 Best Unbiased Estimators
	1 Exponential Families of Distributions
	2 Minimum Variance Unbiased Estimators
	Problems
	3 Sufficient Statistics
	4 A Factorization Theorem
	5 Conditional Expectation and Sufficiency
	Problems
28 Bayes\' Estimators
	1 The Prior and Posterior Distributions
	2 Bayes\' Estimators
	Problems
29 Multiple Linear Regression
	1 The Least Squares Estimate
	2 Statistical Tests
		2.1 Sums of Squares
		2.2 The R Statistic
		2.3 Significance of the Model
		2.4 Estimating the Variance
		2.5 Testing Individual Regression Coefficients
	Problems
	3 Proofs
		3.1 The Normal Equations
		3.2 Partitioning the Sum of Squares
		3.3 Expectation and Variance of a Random Vector
		3.4 Normal Random Vectors
	Problems
List of Common Discrete Distributions
List of Common Continuous Distributions
Further Reading
	Probability
	Statistics
Standard Normal Table
Student Table
Chi-Squared Table
Index