An Introduction to Probability and Statistical Inference

by Roussas

• ISBN13: 9780125990202
• ISBN10: 0125990200
• eBook ISBN(s): 9780080495750
• Format: Hardcover
• Copyright: 2002-12-19
• Publisher: Elsevier Science
• More Book Details
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Summary

Probability models, statistical methods, and the information to be gained from them is vital for work in business, engineering, sciences (including social and behavioral), and other fields. Data must be properly collected, analyzed and interpreted in order for the results to be used with confidence. Roussas introduces readers with no prior knowledge in probability or statistics, to a thinking process to guide them toward the best solution to a posed question or situation. An Introduction to Probability and Statistical Inference provides a plethora of examples for each topic discussed, giving the reader more experience in applying statistical methods to different situations.

Table of Contents

Preface	xi
Some Motivating Examples and Some Fundamental Concepts	1	(22)
Some Motivating Examples	1	(7)
Some Fundamental Concepts	8	(11)
Random Variables	19	(4)
The Concept of Probability and Basic Results	23	(45)
Definition of Probability and Some Basic Results	24	(9)
Distribution of a Random Variable	33	(8)
Conditional Probability and Related Results	41	(10)
Independent Events and Related Results	51	(8)
Basic Concepts and Results in Counting	59	(9)
Numerical Characteristics of a Random Variable, Some Special Random Variables	68	(41)
Expectation, Variance, and Moment Generating Function of a Random Variable	68	(9)
Some Probability Inequalities	77	(2)
Some Special Random Variables	79	(23)
Median and Mode of a Random Variable	102	(7)
Joint and Conditional P.D.F.'S, Conditional Expectation and Variance, Moment Generating, Function, Covariance, and Correlation Coefficient	109	(41)
Joint d.f. and Joint p.d.f. of Two Random Variables	110	(7)
Marginal and Conditional p.d.f.'s, Conditional Expectation and Variance	117	(9)
Expectation of a Function of Two r.v.'s, Joint and Marginal m.g.f.'s, Covariance, and Correlation Coefficient	126	(11)
Some Generalizations to k Random Variables	137	(2)
The Multinomial, the Bivariate Normal, and the Multivariate Normal Distributions	139	(11)
Independence of Random Variables and Some Applications	150	(18)
Independence of Random Variables and Criteria of Independence	150	(9)
The Reproductive Property of Certain Distributions	159	(9)
Transformation of Random Variables	168	(34)
Transforming a Single Random Variable	168	(5)
Transforming Two or More Random Variables	173	(12)
Linear Transformations	185	(7)
The Probability Integral Transform	192	(1)
Order Statistics	193	(9)
Some Modes of Convergence of Random Variables, Applications	202	(25)
Convergence in Distribution or in Probability and Their Relationship	202	(6)
Some Applications of Convergence in Distribution: The Weak Law of Large Numbers and the Central Limit Theorem	208	(14)
Further Limit Theorems	222	(5)
An Overview of Statistical Inference	227	(13)
The Basics of Point Estimation	228	(2)
The Basics of Interval Estimation	230	(1)
The Basics of Testing Hypotheses	231	(4)
The Basics of Regression Analysis	235	(1)
The Basics of Analysis of Variance	236	(2)
The Basics of Nonparametric Inference	238	(2)
Point Estimation	240	(41)
Maximum Likelihood Estimation: Motivation and Examples	240	(13)
Some Properties of Maximum Likelihood Estimates	253	(8)
Uniformly Minimum Variance Unbiased Estimates	261	(9)
Decision-Theoretic Approach to Estimation	270	(7)
Other Methods of Estimation	277	(4)
Confidence Intervals and Confidence Regions	281	(18)
Confidence Intervals	282	(7)
Confidence Intervals in the Presence of Nuisance Parameters	289	(3)
A Confidence Region for (μ, σ2) in the N(μ, σ2) Distribution	292	(2)
Confidence Intervals with Approximate Confidence Coefficient	294	(5)
Testing Hypotheses	299	(44)
General Concepts, Formulation of Some Testing Hypotheses	300	(2)
Neyman--Pearson Fundamental Lemma, Exponential Type Families, Uniformly Most Powerful Tests for Some Composite Hypotheses	302	(13)
Some Applications of Theorems 2 and 3	315	(9)
Likelihood Ratio Tests	324	(19)
More About Testing Hypotheses	343	(20)
Likelihood Ratio Tests in the Multinomial Case and Contingency Tables	343	(6)
A Goodness-of-Fit Test	349	(4)
Decision-Theoretic Approach to Testing Hypotheses	353	(7)
Relationship Between Testing Hypotheses and Confidence Regions	360	(3)
A Simple Linear Regression Model	363	(34)
Setting-up the Model --- The Principle of Least Squares	364	(2)
The Least Squares Estimates of β1 and β2, and Some of Their Properties	366	(8)
Normally Distributed Errors: MLE's of β1, β2, and σ2, Some Distributional Results	374	(9)
Confidence Intervals and Hypotheses Testing Problems	383	(6)
Some Prediction Problems	389	(4)
Proof of Theorem 5	393	(2)
Concluding Remarks	395	(2)
Two Models of Analysis of Variance	397	(31)
One-Way Layout with the Same Number of Observations per Cell	398	(9)
A Multicomparison Method	407	(5)
Two-Way Layout with One Observation per Cell	412	(16)
Some Topics in Nonparametric Inference	428	(22)
Some Confidence Intervals with Given Approximate Confidence Coefficient	429	(2)
Confidence Intervals for Quantiles of a Distribution Function	431	(2)
The Two-Sample Sign Test	433	(2)
The Rank Sum and the Wilcoxon--Mann--Whitney Two-Sample Tests	435	(7)
Nonparametric Curve Estimation	442	(8)
Appendix	450	(30)
Some Notation and Abbreviations	480	(3)
Answers to Even-Numbered Exercises	483	(32)
Index	515