[Schaum's Outline 01] • Probability and Statistics · 4th Edition by Spiegel, Murray R. -- Read -- Imperial Library of Trantor

Index

Cover Page Schaum's Outline of Probability and Statistics, 4th Edition Copyright Page Preface to the Third Edition Preface to the Second Edition Preface to the First Edition Contents Part I Probability

Chapter 1 Basic Probability

Random Experiments Sample Spaces Events The Concept of Probability The Axioms of Probability Some Important Theorems on Probability Assignment of Probabilities Conditional Probability Theorems on Conditional Probability Independent Events Bayes’ Theorem or Rule Combinatorial Analysis Fundamental Principle of Counting Tree Diagrams Permutations Combinations Binomial Coefficients Stirling’s Approximation to n!

Chapter 2 Random Variables and Probability Distributions

Random Variables Discrete Probability Distributions Distribution Functions for Random Variables Distribution Functions for Discrete Random Variables Continuous Random Variables Graphical Interpretations Joint Distributions Independent Random Variables Change of Variables Probability Distributions of Functions of Random Variables Convolutions Conditional Distributions Applications to Geometric Probability

Chapter 3 Mathematical Expectation

Definition of Mathematical Expectation Functions of Random Variables Some Theorems on Expectation The Variance and Standard Deviation Some Theorems on Variance Standardized Random Variables Moments Moment Generating Functions Some Theorems on Moment Generating Functions Characteristic Functions Variance for Joint Distributions. Covariance Correlation Coefficient Conditional Expectation, Variance, and Moments Chebyshev’s Inequality Law of Large Numbers Other Measures of Central Tendency Percentiles Other Measures of Dispersion Skewness and Kurtosis

Chapter 4 Special Probability Distributions

The Binomial Distribution Some Properties of the Binomial Distribution The Law of Large Numbers for Bernoulli Trials The Normal Distribution Some Properties of the Normal Distribution Relation Between Binomial and Normal Distributions The Poisson Distribution Some Properties of the Poisson Distribution Relation Between the Binomial and Poisson Distributions Relation Between the Poisson and Normal Distributions The Central Limit Theorem The Multinomial Distribution The Hypergeometric Distribution The Uniform Distribution The Cauchy Distribution The Gamma Distribution The Beta Distribution The Chi-Square Distribution Student’s t Distribution The F Distribution Relationships Among Chi-Square, t, and F Distributions The Bivariate Normal Distribution Miscellaneous Distributions

Part II Statistics

Chapter 5 Sampling Theory

Population and Sample. Statistical Inference Sampling With and Without Replacement Random Samples. Random Numbers Population Parameters Sample Statistics Sampling Distributions The Sample Mean Sampling Distribution of Means Sampling Distribution of Proportions Sampling Distribution of Differences and Sums The Sample Variance Sampling Distribution of Variances Case Where Population Variance Is Unknown Sampling Distribution of Ratios of Variances Other Statistics Frequency Distributions Relative Frequency Distributions Computation of Mean, Variance, and Moments for Grouped Data

Chapter 6 Estimation Theory

Unbiased Estimates and Efficient Estimates Point Estimates and Interval Estimates. Reliability Confidence Interval Estimates of Population Parameters Confidence Intervals for Means Confidence Intervals for Proportions Confidence Intervals for Differences and Sums Confidence Intervals for the Variance of a Normal Distribution Confidence Intervals for Variance Ratios Maximum Likelihood Estimates

Chapter 7 Tests of Hypotheses and Significance

Statistical Decisions Statistical Hypotheses. Null Hypotheses Tests of Hypotheses and Significance Type I and Type II Errors Level of Significance Tests Involving the Normal Distribution One-Tailed and Two-Tailed Tests P Value Special Tests of Significance for Large Samples Special Tests of Significance for Small Samples Relationship Between Estimation Theory and Hypothesis Testing Operating Characteristic Curves. Power of a Test Quality Control Charts Fitting Theoretical Distributions to Sample Frequency Distributions The Chi-Square Test for Goodness of Fit Contingency Tables Yates’ Correction for Continuity Coefficient of Contingency

Chapter 8 Curve Fitting, Regression, and Correlation

Curve Fitting Regression The Method of Least Squares The Least-Squares Line The Least-Squares Line in Terms of Sample Variances and Covariance The Least-Squares Parabola Multiple Regression Standard Error of Estimate The Linear Correlation Coefficient Generalized Correlation Coefficient Rank Correlation Probability Interpretation of Regression Probability Interpretation of Correlation Sampling Theory of Regression Sampling Theory of Correlation Correlation and Dependence

Chapter 9 Analysis of Variance

The Purpose of Analysis of Variance One-Way Classification or One-Factor Experiments Total Variation. Variation Within Treatments. Variation Between Treatments Shortcut Methods for Obtaining Variations Linear Mathematical Model for Analysis of Variance Expected Values of the Variations Distributions of the Variations The F Test for the Null Hypothesis of Equal Means Analysis of Variance Tables Modifications for Unequal Numbers of Observations Two-Way Classification or Two-Factor Experiments Notation for Two-Factor Experiments Variations for Two-Factor Experiments Analysis of Variance for Two-Factor Experiments Two-Factor Experiments with Replication Experimental Design

Chapter 10 Nonparametric Tests

Introduction The Sign Test The Mann–Whitney U Test The Kruskal–Wallis H Test The H Test Corrected for Ties The Runs Test for Randomness Further Applications of the Runs Test Spearman’s Rank Correlation Chapter 11 Bayesian Methods Subjective Probability Prior and Posterior Distributions Sampling From a Binomial Population Sampling From a Poisson Population Sampling From a Normal Population with Known Variance Improper Prior Distributions Conjugate Prior Distributions Bayesian Point Estimation Bayesian Interval Estimation Bayesian Hypothesis Tests Bayes Factors Bayesian Predictive Distributions

Appendix A Mathematical Topics

Special Sums Euler’s Formulas The Gamma Function The Beta Function Special Integrals