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Random Variables and Probability Distributions
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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
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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
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Special Probability Distributions
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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
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Contents STATISTICS Sampling Theory 151 153
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Part II CHAPTER 5
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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
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Estimation Theory
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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
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
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
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
Contents 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
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
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