SYLLABUS for M 378K

Introduction to Mathematical Statistics

Prerequisite: M 362K with grade of C- or better.

Goals and level of course: Goals are to give students some insight into the theory behind the standard statistical procedures and to prepare continuing students for the graduate courses. Within the limits of the prerequisites, students are expected to derive and apply the theoretical results as well as carry out some standard statistical procedures.

Topics covered:

  • Moment-Generating Functions
  • Gamma , Chi-squared, t- and F-distributions
  • Sampling Distributions and the Central Limit Theorem
  • Point Estimation (bias, mean square error, relative efficiency, consistency, sufficiency, Method of Moments, Method of Maximum Likelihood, Rao-Blackwell Theorem and Minimum Variance Unbiased Estimation). Examples should include cases where more than one estimator is possible. Examples involving max and min are probably the easiest to do.
  • Confidence intervals (concepts; small and large sample CIs for means and differences of means; large sample CIs for proportions and differences of proportions; selecting sample sizes)
  • Hypothesis testing (concepts; small and large sample for means and differences of means; large sample for proportions and differences of proportions; small sample for proportions)
  • Errors and power (type I and II errors, power and Neyman Pearson Lemma, calculating sample sizes for desired error level or power)
  • Likelihood ratio tests
  • If time permits: Some selection from: Order statistics, Chi-squared tests, non-parametric tests, least squares regression.

Detailed syllabus based on Wackerly et al (fifth edition): (Chapters 7 - 10 constitute the heart of the course)

 

  • Review of Probability and Introduction to Statistics:
    • Chapter 1, Sections 2.1, 2.2, 2.12, 3.1, 3.11, 3.12, 4.1, 4.2, 4.3, 4.10, 4.12, 5.1, 5.12, 6.1, 6.7
  • Additional Probability Topics: Gamma and Chi-Squared Distributions: Section 4.6
    • Moment Generating Functions: Sections 3.9, 4.9
    • Probability Distributions of Functions of Random Variables: Sections 6.4, 6.5
    • Probability Distributions of Max and Min: First part of Section 6.6
  • Sampling Distributions and the Central Limit Theorem: All of Chapter 7
  • Estimation All of Chapter 8
  • Properties of point Estimators and Methods of Estimation Chapter 9, omitting Section 8
  • Hypothesis Testing Chapter 10 (Section 9 optional)
  • (Additional topics as time permits)