TABLE OF CONTENTS CHAPTER 1 COMBINATORIAL PROBABILITY 1

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1 TABLE OF CONTENTS CHAPTER 1 COMBINATORIAL PROBABILITY The Probability Model Finite Discrete Models with Equally Likely Outcomes Tree Diagrams The Multiplication Principle Permutations The Birthday Problem and its Genre Combinations Partitions Sampling and Distribution Sampling Distributions Sampling and Occupancy United More Applications The Binomial and Multinomial Theorems Poker Hands The Powerball Lottery Chapter 1 Sample Examination CHAPTER 2 GENERAL RULES OF PROBABILITY Sets, Sample Spaces, and Events Set Theory Survival Kit The Venn Diagram Basic Rules of Probability DeMorgan s Laws The Venn Box Diagram Conditional Probability Conditional Probability and Tree Diagrams Bayesian Cause and Effect Reasoning Independence Bayes Theorem Credibility Chapter 2 Sample Examination vii

2 viii TABLE OF CONTENTS CHAPTER 3 DISCRETE RANDOM VARIABLES Definition and Examples of Discrete Random Variables Cumulative Probability Distribution Measures of Central Tendency Expected Value (Mean) Median of a Data Set Midrange of a Data Set Mode of a Data Set Quartiles and Percentiles Random Variables and Percentiles Measures of Dispersion Range, Inter-Quartile Range Variance Standard Deviation Standardized Random Variables and Z-Scores Chebychev s Theorem Coefficient of Variation Conditional Expectation and Variance Jointly Distributed Random Variables (Round 1) The Probability Generating Function Chapter 3 Sample Examination CHAPTER 4 SOME DISCRETE DISTRIBUTIONS The Discrete Uniform Distribution Bernoulli Trials and the Binomial Distribution The Geometric Distribution The Negative Binomial Distribution The Hyper-geometric Distribution The Poisson Distribution The Poisson Probability Function Poisson Processes Poisson Process Data Sets Sums of Poisson Random Variables Poisson Approximation to the Binomial Distribution Summary Comparison of Discrete Distributions Chapter 4 Sample Examination

3 TABLE OF CONTENTS ix CHAPTER 5 CALCULUS, PROBABILITY, AND CONTINUOUS DISTRIBUTIONS Cumulative Distribution Functions Density Functions Great Expectations The Variance Formula The Mode of a Continuous Distribution Medians and Percentiles Calculating the Expected Value of X with the CDF Mixed Distributions Applications to Insurance: Deductibles and Caps Deductible Insurance Capped Insurance The CDF Method for Deductibles and Caps The Moment Generating Function The MGF for the Binomial Distribution The MGF for the Geometric and Negative Binomial The MGF for the Poisson Distribution Chapter 5 Sample Examination CHAPTER 6 SOME CONTINUOUS DISTRIBUTIONS Uniform Random Variables The Exponential Distribution Integration Review Exponential and Poisson Relationship Properties of the Exponential Random Variable The Normal Distribution Integration Review The Standard Normal The General Normal Distribution The Lognormal Distribution The Law of Averages and the Central Limit Theorem Random Samples Chebyshev and the Law of Averages Central Limit Theorem (Sum Version) The Continuity Correction Central Limit Theorem (Sample Mean Version) Outline for a Proof of the Central Limit Theorem

4 x TABLE OF CONTENTS 6.5 The Gamma Distribution The Gamma Function Definition and Properties for the Gamma Family Comparing the Gamma and Lognormal Gamma and Poisson Connections The Distribution of Z The Beta Family of Distributions The Beta Function The Beta Family of Distributions More Continuous Distributions The Hazard Rate The Pareto Distribution The Weibull Distribution Chapter 6 Sample Examination CHAPTER 7 MULTIVARIATE DISTRIBUTIONS Joint Distributions for Discrete Random Variables Conditional Distributions The Discrete Case Independence Discrete Case Covariance and Correlation Joint Distributions for Continuous Random Variables Conditional Distributions The Continuous Case Independence and Covariance in the Continuous Case The Multinomial Distribution Bivariate Normal Distributions Moment Generating Function for a Joint Distribution Chapter 7 Sample Examination CHAPTER 8 A PROBABILITY POTPOURRI The Distribution of a Transformed Random Variable The Transformation Formula The Distribution of the Sum of Random Variables The Convolution Integral Simulating Values of the Random Variable X Order Statistics The Moment-Generating Function Method

5 TABLE OF CONTENTS xi 8.3 Covariance Formulas Covariance and the Multinomial Distribution Variance Formula for the Hyper-geometric The Conditioning Formulas and Mixtures Poisson Processes Revisited Non-homogenous Poisson Process The Poisson Thinning Process Chapter 8 Sample Examination CHAPTER 9 SAMPLING DISTRIBUTIONS AND ESTIMATION The Sample Mean as an Estimator Estimating the Population Variance The Sample Variance Sample Variance from a Normal Population Confidence Interval for Population Variance The Student t-distribution The F-Distribution Estimating Proportions Estimating the Difference between Means Large Samples Small Samples Estimating the Sample Size Chapter 9 Sample Examination CHAPTER 10 HYPOTHESIS TESTING Hypothesis Testing Framework Finding the Likelihood of Type I and Type II Errors The Significance Level of a Test The p-value of a Hypothesis Test Hypothesis Testing for Population Means Standard Normal Tests for the Population Mean Student t Tests for the Population Mean Hypothesis Testing for Population Variance Hypothesis Testing for Proportions

6 xii TABLE OF CONTENTS 10.5 Hypothesis Testing for Differences in Population Means Large Samples Small Samples The F Test for Equal Population Variances Chi-Square Tests Contingency Tables Goodness of Fit Tests Chapter 10 Sample Examination CHAPTER 11 THEORY OF ESTIMATION AND HYPOTHESIS TESTING The Bias of an Estimator Building Estimators Method of Moments The Maximum Likelihood Estimator Properties of Estimators Consistent Estimators Efficient Estimators Efficiency and the Cramer-Rao Inequality Sufficient Statistics Hypothesis Testing Theory More General Likelihood Ratio Tests Bayesian Estimation The Bayesian Recipe The Loss Function and Mean Square Error Conjugate Prior Distributions and Credible Intervals Chapter 11 Sample Examination CHAPTER 12 A STATISTICS POTPOURRI Simple Linear Regression: Basic Formulas The Least Squares Model The Analysis of Variance Perspective Simple Linear Regression: Estimation of Parameters Comparing Means Using ANOVA Non-parametric Tests Sign Tests Runs Tests Signed Rank Tests Mann-Whitney-Wilcoxon U Statistic Rank Correlation

7 TABLE OF CONTENTS xiii 12.5 Goodness of Fit: Several Eyeball Tests Chapter 12 Sample Examination APPENDIX I STATISTICAL TABLES 707 ANSWERS TO TEXT EXERCISES 713 INDEX 743