Models for probability and statistical inference: theory and applications

Abstract

1. Probability Models. 1.1 Discrete Probability Models. 1.2 Conditional Probability and Independence. 1.3 Random Variables. 1.4 Expectation. 1.5 The Variance. 1.6 Covariance and Correlation. 2. Special Discrete Distributions. 2.1 The Binomial Distribution. 2.2 The Hypergeometric Distribution. 2.3 The Geometric and Negative Binomial Distributions. 2.4 The Poisson Distribution. 3. Continuous Random Variables. 4.1 Continuous RV's and Their Distributions. 4.2 Expected Values and Variances. 4.3 Transformations of Random Variables. 4.4Joint Densities. 4 Special Continuous Distributions. 4.1 The Normal Distribution. 4.2 The Gamma Distribution. 5. Conditional Distributions. 5.1 The Discrete Case. 5.2 Conditional Expectations for the Discrete Case. 5.3 Conditional Densities and Expectations for Continuous RV's. 6. Limit Laws. 6.1 Moment Generating Functions. 6.2 Convergence in Probability and in Distribution. 6.3 The Central Limit Theorem. 6.4 The Delta-Method. 7. Estimation. 7.1 Point Estimation. 7.2 The Method of Moments. 7.3 Maximum Likelihood. 7.4 Consistency. 7.5 The OMEGA-Method. 7.6 Confidence Intervals. 7.7 Fisher Information, The Cramer-Rao Bound, and Asymptotic Normality of MLE's. 7.8 Sufficiency. 8. Testing Hypotheses. 8.1 Introduction. 8.2 The Neyman-Pearson Lemma. 8.3 The Likelihood Ratio Test. 8.4 The p-Value and the Relationship Between Tests of Hypotheses and Confidence Intervals. 9. The Multivariate Normal, Chi-square, t, and F-Distributions. 9.1 The Multivariate Normal Distribution. 9.2 The Central and Noncentral Chi-Square Distributions. 9.3 Student's t-Distribution. 9.4 The F-Distribution. 10.3 Nonparametric Statistics. 10.1 The Wilcoxon Test and Estimator. 10.2 One Sample Methods. 10.3 The Kolmogorov-Smirnov Tests. 11. Linear Models. 11.1 The Principle of Least Squares. 11.2 Linear Models. 11.3 F-Tests for H0. 11.4 Two-Way Analysis of Variance. 12. Frequency Data. 12.1 Logistic Regression. 12.2 Two-Way Frequency Tables. 12.3 Chi-Square Goodness of Fit Tests. 13. Miscellaneous Topics. 13.1 Survival Analysis. 13.2 Bootstrapping. 13.3 Bayesian Statistics. 13.4 Sampling.