Exercises and Solutions in Biostatistical Theory

Basic Probability Theory Counting Formulas (N-tuples, permutations, combinations, Pascal's identity, Vandermonde's identity) Probability Formulas (union, intersection, complement, mutually exclusive events, conditional probability, independence, partitions, Bayes' theorem) Univariate Distribution Theory Discrete and Continuous Random Variables Cumulative Distribution Functions Median and Mode Expectation Theory Some Important Expectations (mean, variance, moments, moment generating function, probability generating function) Inequalities Involving Expectations Some Important Probability Distributions for Discrete Random Variables Some Important Distributions (i.e., Density Functions) for Continuous Random Variables Multivariate Distribution Theory Discrete and Continuous Multivariate Distributions Multivariate Cumulative Distribution Functions Expectation Theory (covariance, correlation, moment generating function) Marginal Distributions Conditional Distributions and Expectations Mutual Independence among a Set of Random Variables Random Sample Some Important Multivariate Discrete and Continuous Probability Distributions Special Topics of Interest (mean and variance of a linear function, convergence in distribution and the Central Limit Theorem, order statistics, transformations) Estimation Theory Point Estimation of Population Parameters (method of moments, unweighted and weighted least squares, maximum likelihood) Data Reduction and Joint Sufficiency (Factorization Theorem) Methods for Evaluating the Properties of a Point Estimator (mean-squared error, Cramer--Rao lower bound, efficiency, completeness, Rao--Blackwell theorem) Interval Estimation of Population Parameters (normal distribution-based exact intervals, Slutsky's theorem, consistency, maximum-likelihood-based approximate intervals) Hypothesis Testing Theory Basic Principles (simple and composite hypotheses, null and alternative hypotheses, Type I and Type II errors, power, P-value) Most Powerful (MP) and Uniformly Most Powerful (UMP) Tests (Neyman--Pearson Lemma) Large-Sample ML-Based Methods for Testing a Simple Null Hypothesis versus a Composite Alternative Hypothesis (likelihood ratio, Wald, and score tests) Large-Sample ML-Based Methods for Testing a Composite Null Hypothesis versus a Composite Alternative Hypothesis (likelihood ratio, Wald, and score tests) Appendix: Useful Mathematical Results References Index Exercises and Solutions appear at the end of each chapter.