# MAT 321 Probability and Statistics

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### Course Description

A calculus-based survey of topics in probability and statistics emphasizing applications.

MAT 321 meets for three hours of lecture per week.

### Prerequisites

MAT 193 and MAT 271 with grade C or better.

### Objectives

After completing MAT 321 the student should be able to

• understand the relationship between a question that arises in the natural, computer, economic, and social sciences and the nature of the numerical data that are needed in order to provide an answer to the question
• formulate the question in a mathematical context, set up the required mathematical procedure and carry out the required mathematical analysis and calculations, with appropriate use of a calculator, to answer the questions.
• understand why some statistical procedures are better than others in certain contexts.

### Expected outcomes

Students should be able to demonstrate through written assignments, tests, and/or oral presentations, that they have achieved the objectives of MAT 321.

### Method of Evaluating Outcomes

Evaluations are based on homework, class participation, short tests scheduled examinations covering students' understanding of topics that are covered in MAT 321. Most instructors give "closed-book" in-class examinations with no books, notes, "crib sheets", etc. allowed. However they encourage students to use the most advanced calculators available in both homework assignments and examinations. Statistical tables and a few of the more complex formulas are provided for use on exams at the instructors' discretion

### Text

An Introduction to Mathematical Statistics and its Applications, by R.J. Larsen and M.L. Marx. Prentice-Hall, 2nd edition, 1986.

#### Suggested Course Outline

The following outline would be appropriate for a 15 week course with 2 lectures per week.

• Lecture 1. Arithmetic of events, distributive laws, deMorgan's laws
• Lecture 2. Probability
• Lecture 3. Conditional probability, independent events, Bayes' theorem
• Lecture 4. Sequences and subsets, permutations and combinations, Pascal's triangle
• Lecture 5. Discrete distributions: Bernoulli, binomial, geometric, hypergeometric
• Lecture 6. Poisson distribution, Poisson's limit theorem
• Lecture 7. Continuous random variables; exponential, gamma, uniform distributions
• Lecture 8. FIRST EXAM
• Lecture 9. Joint distribution functions, independent random variables
• Lecture 10. The normal (Gaussian) distribution
• Lecture 11. Transformations of random variables, random numbers
• Lecture 12. Sums of independent random variables, convolutions
• Lecture 13. Conditional densities
• Lecture 14. Expected values, moments, covariance, variance
• Lecture 15. Chebyshev's inequality, the "law of averages"
• Lecture 16. Moment-generating functions, the central limit theorem
• Lecture 17. Maximum-likelihood estimation of parameters
• Lecture 18. SECOND EXAM
• Lecture 19. Unbiased estimators, efficiency, Cramer-Rao lower bound
• Lecture 20. Consistent estimators, confidence intervals for means, required sample size
• Lecture 21. Confidence intervals for proportions, required sample size
• Lecture 22. The statistical decision-making process, Type I and Type II errors
• Lecture 23. The classical sampling distributions
• Lecture 24. Large- and small-sample hypothesis tests of means, one- and two-sample tests, paired-sample t test
• Lecture 25. The multinomial distribution, chi-square test of independence
• Lecture 26. THIRD EXAM
• Lecture 27. Covariance, correlation, linearity
• Lecture 28. Linear regression, the least-squares principle
• Lecture 29. Testing for linearity, confidence intervals for regression predictions
• Lecture 30. REVIEW
• FINAL EXAM

Students' grades are based on homework, class participation, short tests, and scheduled examinations covering students' understanding of the topics covered in MAT 321. The instructor determines the relative weights of these factors.

### Attendance Requirements

Attendance policy is set by the instructor.

### Policy on Due Dates and Make-Up Work

Due dates and policy regarding make-up work are set by the instructor.

### Schedule of Examinations

The instructor sets all test dates except the date of the final exam. The final exam is given at the date and time announced in the Schedule of Classes.