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Mathematical Statistics with Applications (Wackerly)

 Author(s)  Wackerly, Mendenhall, and Scheaffer
 Title  Mathematical Statistics with Applications
 Edition  Sixth Edition
 Year  2002
 Publisher  Duxbury
 ISBN  0-534-37741-6
 book link

Table of Contents

1. What Is Statistics?
  • Introduction.
  • Characterizing a Set of Measurements: Graphical Methods.
  • Characterizing a Set of Measurements: Numerical Methods.
  • How Inferences Are Made.
  • Theory and Reality.
  • Summary.

2. Probability.
  • Introduction.
  • Probability and Inference.
  • A Review of Set Notation.
  • A Probabilistic Model for an Experiment: The Discrete Case.
  • Calculating the Probability of an Event: The Sample-Point Method.
  • Tools for Counting Sample Points. Conditional Probability and the Independence of Events.
  • Two Laws of Probability.
  • Calculating the Probability of an Event: The Event-Composition Methods.
  • The Law of Total Probability and Bayes''s Rule.
  • Numerical Events and Random Variables.
  • Random Sampling.
  • Summary.

3. Discrete Random Variables and Their Probability Distributions.
  • Basic Definition.
  • The Probability Distribution for Discrete Random Variable.
  • The Expected Value of Random Variable or a Function of Random Variable.
  • The Binomial Probability Distribution.
  • The Geometric Probability Distribution.
  • The Negative Binomial Probability Distribution (Optional).
  • The Hypergeometric Probability Distribution.
  • Moments and Moment-Generating Functions.
  • Probability-Generating Functions (Optional).
  • Tchebysheff''s Theorem. Summary.

4. Continuous Random Variables and Their Probability Distributions.
  • Introduction.
  • The Probability Distribution for Continuous Random Variable.
  • The Expected Value for Continuous Random Variable.
  • The Uniform Probability Distribution.
  • The Normal Probability Distribution.
  • The Gamma Probability Distribution.
  • The Beta Probability Distribution.
  • Some General Comments.
  • Other Expected Values.
  • Tchebysheff''s Theorem.
  • Expectations of Discontinuous Functions and Mixed Probability Distributions (Optional).
  • Summary.

5. Multivariate Probability Distributions.
  • Introduction.
  • Bivariate and Multivariate Probability Distributions.
  •  Independent Random Variables.
  • The Expected Value of a Function of Random Variables.
  • Special Theorems.
  • The Covariance of Two Random Variables.
  • The Expected Value and Variance of Linear Functions of Random Variables.
  • The Multinomial Probability Distribution.
  • The Bivariate Normal Distribution (Optional).
  • Conditional Expectations.
  • Summary.

6. Functions of Random Variables.
  • Introductions.
  • Finding the Probability Distribution of a Function of Random Variables.
  • The Method of Distribution Functions.
  • The Methods of Transformations.
  • Multivariable Transformations Using Jacobians.
  • Order Statistics.
  • Summary.

7. Sampling Distributions and the Central Limit Theorem.
  • Introduction.
  • Sampling Distributions Related to the Normal Distribution.
  • The Central Limit Theorem.
  • A Proof of the Central Limit Theorem (Optional).
  • The Normal Approximation to the Binomial Distributions.
  • Summary.

8. Estimation.
  • Introduction.
  • The Bias and Mean Square Error of Point Estimators.
  • Some Common Unbiased Point Estimators.
  • Evaluating the Goodness of Point Estimator.
  • Confidence Intervals.
  • Large-Sample Confidence Intervals
  • Selecting the Sample Size.
  • Small-Sample Confidence Intervals for u and u1-u2.
  • Confidence Intervals for o2.
  • Summary.

9. Properties of Point Estimators and Methods of Estimation.
  • Introduction.
  • Relative Efficiency.
  • Consistency.
  • Sufficiency.
  • The Rao-Blackwell Theorem and Minimum-Variance Unbiased Estimation.
  • The Method of Moments.
  • The Method of Maximum Likelihood.
  • Some Large-Sample Properties of MLEs (Optional).
  • Summary.

10. Hypothesis Testing.
  • Introduction.
  • Elements of a Statistical Test.
  • Common Large-Sample Tests.
  • Calculating Type II Error Probabilities and Finding the Sample Size for the Z Test.
  • Relationships Between Hypothesis Testing Procedures and Confidence Intervals.
  • Another Way to Report the Results of a Statistical Test: Attained Significance Levels or p-Values.
  • Some Comments on the Theory of Hypothesis Testing.
  • Small-Sample Hypothesis Testing for u and u1-u2.
  • Testing Hypotheses Concerning Variances.
  • Power of Test and the Neyman-Pearson Lemma.
  • Likelihood Ratio Test.
  • Summary.

11. Linear Models and Estimation by Least Squares.
  • Introduction.
  • Linear Statistical Models.
  • The Method of Least Squares.
  • Properties of the Least Squares Estimators for the Simple Linear Regression Model.
  • Inference Concerning the Parameters BI.
  • Inferences Concerning Linear Functions of the Model Parameters: Simple Linear Regression.
  • Predicting a Particular Value of Y Using Simple Linear Regression. Correlation.
  • Some Practical Examples.
  • Fitting the Linear Model by Using Matrices.
  • Properties of the Least Squares Estimators for the Multiple Linear Regression Model.
  • Inferences Concerning Linear Functions of the Model Parameters: Multiple Linear Regression.
  • Prediction a Particular Value of Y Using Multiple Regression.
  • A Test for H0: Bg+1 + Bg+2 = . = Bk = 0.
  • Summary and Concluding Remarks.

12. Considerations in Designing Experiments.
  • The Elements Affecting the Information in a Sample.
  • Designing Experiment to Increase Accuracy.
  • The Matched Pairs Experiment.
  • Some Elementary Experimental Designs.
  • Summary.

13. The Analysis of Variance.
  • Introduction.
  • The Analysis of Variance Procedure.
  • Comparison of More than Two Means: Analysis of Variance for a One-way Layout.
  • An Analysis of Variance Table for a One-Way Layout.
  • A Statistical Model of the One-Way Layout.
  • Proof of Additivity of the Sums of Squares and E (MST) for a One-Way Layout (Optional).
  • Estimation in the One-Way Layout.
  • A Statistical Model for the Randomized Block Design.
  • The Analysis of Variance for a Randomized Block Design.
  • Estimation in the Randomized Block Design.
  • Selecting the Sample Size.
  • Simultaneous Confidence Intervals for More than One Parameter.
  • Analysis of Variance Using Linear Models. Summary.

14. Analysis of Categorical Data.
  • A Description of the Experiment.
  • The Chi-Square Test.
  • A Test of Hypothesis Concerning Specified Cell Probabilities: A Goodness-of-Fit Test.
  • Contingency Tables.
  • r x c Tables with Fixed Row or Column Totals.
  • Other Applications.
  • Summary and Concluding Remarks.

15. Nonparametric Statistics.
  • Introduction.
  • A General Two-Sampling Shift Model.
  • A Sign Test for a Matched Pairs Experiment.
  • The Wilcoxon Signed-Rank Test for a Matched Pairs Experiment.
  • The Use of Ranks for Comparing Two Population Distributions: Independent Random Samples.
  • The Mann-Whitney U Test: Independent Random Samples.
  • The Kruskal-Wallis Test for One-Way Layout.
  • The Friedman Test for Randomized Block Designs.
  • The Runs Test: A Test for Randomness.
  • Rank Correlation Coefficient.
  • Some General Comments on Nonparametric Statistical Test.

16. Introduction to Bayesian Methods for Inference.
  • Introduction.
  • Bayesian Priors, Posteriors and Estimators.
  • Bayesian Credible Intervals.
  • Bayesian Tests of Hypotheses.
  • Summary and Additional Comments.

Appendix 1. Matrices and Other Useful Mathematical Results.
  • Matrices and Matrix Algebra.
  • Addition of Matrices.
  • Multiplication of a Matrix by a Real Number.
  • Matrix Multiplication.
  • Identity Elements.
  • The Inverse of a Matrix.
  • The Transpose of a Matrix.
  • A Matrix Expression for a System of Simultaneous Linear Equations.
  • Inverting a Matrix.
  • Solving a System of Simultaneous Linear Equations.
  • Other Useful Mathematical Results.

Appendix 2. Common Probability Distributions, Means, Variances, and Moment-Generating Functions.
  • Discrete Distributions.
  • Continuous Distributions.

Appendix 3. Tables.
  • Binomial Probabilities.
  • Table of e-x. Poisson Probabilities.
  • Normal Curve Areas.
  • Percentage Points of the t Distributions.
  • Percentage Points of the F Distributions.
  • Distribution of Function U.
  • Critical Values of T in the Wilcoxon Matched-Pairs, Signed-Ranks Test.
  • Distribution of the Total Number of Runs R in Sample Size (n1,n2); P(R < a).
  • Critical Values of Pearman's Rank Correlation Coefficient.
  • Random Numbers.
Answer to Exercises.

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