M 462 PROBABILITY AND STATISTICS II (4-0-4)(F). A review of the concept of probability space and random variable; expectation and moment-generating functions leading to the central limit theorem: multiple factor analysis of variance; multiple linear regression; nonparametric tests. Offered fall of odd-numbered years, subject to sufficient demand. PREREQ: M 301, M 361, and either M 272 or M 275.
This course assumes that the student has facility with multi-dimensional mathematics including multiple integration and elementary linear algebra. It also assumes that the student has understands introductory ideas of probability theory, inferential statistics. interval estimation and hypothesis testing. This understanding must include an understanding of the elementary proofs involving expectation.
The instructor has complete control over the teaching of this course, including the examinations, homework, and grading system.
Probability and statistics are two of the most widely used of the mathematical sciences. On completion of this course the student should have an understanding of the techniques and philosophy of the field. The student also should gain an appreciation of both the power and the limitations of statistics.
Upon completion of the course students should be able:
1. to demonstrate an understanding of the techniques and philosophy of the field.
2 to explain the power and limitations of statistics
3. to correctly utilize some of the standard mathematical tools of the field e.g. maximum likelyhood, sufficiency, Neyman-Pearson lemma
Students will be evaluated on their ability top do problems, both as homework and as formal examinations. The problems will of two types:
1 to reinforce the material in the text, and
2. to extend the results and ideas to new situations
The following table is a rough schedule for 60 class meeting of 50 minutes each. The actual topics and the amount of time spent on each topic will vary slightly from semester to semester and instructor to instructor. It is understood that the instructor is able to guide the students to an adequate list of supplementary references.
The note column, below, serves to reference the connection of this course with a junior statistics course such as our M 361. ``rev'' = brief review with selected problems and independent reading assigned. ``new'' = essentially new material. ``ext'' = an extension with some overlap of previously developed material but with a more theoretical perspective than in a course like M 361.
The Mend column refers to proposed material from the Mendenhall-et-al
text, below. The Hogg column refers to proposed material from the
Hogg-Craig text
| No. of | ||||
| Topic | note | hours | Menden | Hogg |
| Basic Definition and Properties of Probability | rev | 2 | 2.1-2.5 | 1.2-1.7 |
| Conditional Probability and Independence | rev | 1 | 2.7 | 2 |
| Expectation and Moment Generating Functions | new | 6 | 3.9,4.9,4.10
| 1.9,1.10,4.7 |
| toward to Central Limit Theorem | 6.5,7.3,7.4 | 5.3,5.4 | ||
| Multivariate distribution w. Transformations | new | 6 | 5, 6 | 4 |
| Distributions of Continuous Random Variables | ext | 6 | ||
| Basic Sampling Theory | rev | 2 | 7.1, 7.2 | 4.1 |
| Efficiency, Consistency, Sufficiency | new | 3 | 9 | 10 |
| Estimation including confidence intervals | ext | 6 | 8, 9 | 6, 11 |
| Hypothesis Testing including Neyman-Pearson Thm. | ext | 6 | 10 | 7 |
| Multiple Regression Analysis | new | 4 | 11 | |
| Analysis of Variance including distributions | ext | 6 | 13,14 | 8,12 |
| of certain quadratic forms | ||||
| Nonparametric inference | new | 4 | 15 | 9 |
| Further material | new | 2 | ||
| Exams | 3 |
Each of the following texts can serve as primary sources of material appropriate for this course:
Mathematical Statistics, Freund, J. & Walpole, R., Prentice Hall, 1987.
Introduction to Mathematical Statistics, Hogg, R. & Craig, A., . Macmillan Pub. Co., 1978.
Probability and Statistical Inference, Hogg, R. & Tanis E.., . Macmillan Pub. Co., 1983.
Mathematical Statistics with Applications, Mendenhall, Scheaffer & Wackerly, Duxbury, 1986
Class meetings involve a combination of lecture, questions and discussion.
Homework is an important part of the course. The instructor chooses the
exact grading
scheme, but a typical distribution would be:
| 3 Exams | 300 | |
| Final Exam | 250 | |
| Homework | 100 | |
| Total | 650
|