## MTH451-0001 Introduction to Probability and Statistics

### University of Rhode Island         Spring 2020

Instructor: Mark Comerford

Office: Lippitt Hall 102F

Tel: 874 5984

Email: mcomerford@math.uri.edu

Class Schedule: TuTh 12:30-1:45 pm, Morrill Hall 323

Office hours: Mon, Wed 2-3pm or by appointment

Textbook:   I. Miller, M. Miller, John E. Freund's Mathematical Statistics with Applications, Eighth Edition, Pearson Prentice Hall, ISBN 978-0-321-80709-0

Prerequisite:   MTH 243 (Multivariable Calculus) or equivalent.

Description:   MTH 451 is an introduction to the mathematical theory of probability using calculus. Probability theory has a tremendous variety of applications in all the sciences, including the social sciences, business and economics, and provides the mathematical foundation for statistics. It uses a wide variety of mathematical techniques and concepts, especially elementary set theory, combinatorics, and calculus. A main goal of this course is that you will be able to read more advanced material on probability and its applications and go on to courses in mathematical statistics and stochastic processes.

The class is designed for an audience with quite diverse interests, for example:

If you are an engineering, science, economics or business major, probability will be a basic part of your mathematical toolkit;

If you are a secondary math education major, you will most likely need to take the Praxis content exam, which contains material on discrete mathematics and probability for which this course is great preparation;

If you are interested in taking the actuarial exams, this course is absolutely fundamental. We will discuss problems similar to problems on the actuarial exams during the course. For information about careers in actuarial science see careers in actuarial science; actuarial exams.

Finally, probability theory is a fundamental discipline in mathematics itself and well as the foundation for all of statistics. It can be entertaining, enlightening and sometimes surprising.

Syllabus and Homework Problems: Clicking on the section in the table below will bring up the scanned notes for that section.

 Reading Problems Chapter 1 - Combinatorics 7th Ed.  1.1, 1.2, 1.5, 1.6, 1.8, 1.14, 1.25, 1.26, 1.27 8th Ed.  1.1, 1.2, 1.5, 1.6, 1.8, 1.14, 1.24, 1.26, 1,27 2.1 Introduction 2.2 Sample Spaces 2.3 Events 7th Ed.  2.2, 2.3, 2.4 8th Ed.  2.2, 2.4, 2.3 2.4 The Probability of an Event 2.5 Some Rules of Probability 7th Ed.  2.5, 2.8, 2.9, 2.11, 2.12, 2.14, 2.15 8th Ed.  2.5, 2.8, 2.9, 2.11, 2.12, 2.14, 2.15 2.6 Conditional Probability 2.7 Indpendence 7th Ed.  2.17, 2.18, 2.19, 2.22, 2.102 8th Ed.  2.17, 2.18, 2.19, 2.22, 2.102 2.8 Bayes' Theorem 7th Ed.  2.106, 2.109, 2.111 8th Ed.  2.106, 2.110, 2.111 3.1 Discrete Random Variables 3.2 Probability Distributions for Discrete Random Variables 7th Ed.  3.1, 3.2, 3.3, 3.8, 3.11, 3.13, 3.15 8th Ed.  3.1, 3.2, 3.3, 3.8, 3.12, 3.13, 3.15 3.3 Continuous Random Variables 3.4 Probability Distributions for Continuous Random Variables 7th Ed.  3.18, 3.19, 3.22, 3.23, 3.32, 3.33, 3.41 8th Ed.  3.19, 3.18, 3.22, 3.23, 3.32, 3.33, 3.41 3.5 I Multivariate Distributions 7th Ed.  3.42, 3.43, 3.44, 3.45, 3.46, 3.47 8th Ed.  3.42, 3.43, 3.44, 3.45, 3.46, 3.47 3.5 II Multivariate Densities 7th Ed.  3.49, 3.51, 3.52, 3.54, 3.65 8th Ed.  3.49, 3.51, 3.52, 3.54, 3.65 3.6 Marginal Distributions 3.7 Conditional Distributions 7th Ed.  3.69, 3.71, 3.73, 3.76 8th Ed.  3.69, 3.71, 3.73, 3.76 4.1, 4.2 Mathematical Expectation 7th Ed.  4.1, 4.7, 4.9 8th Ed.  4.1, 4.7, 4.9 4.3 Moments 4.4 Markov's and Chebyshev's Inequalities 7th Ed.  4.17, 4.19, 4.23 8th Ed.  4.17, 4.19, 4.23 4.5 Moment Generating Functions 7th Ed.  4.33, 4.34, 4.35, 4.37 8th Ed.  4.34, 4.33, 4.35, 4.37 4.6 Product Moments 7th Ed.  4.41, 4.45, 4.46 8th Ed.  4.42, 4.46, 4.47 4.7 Moments of Linear Combinations of Random Variables 7th Ed.  4.48, 4.49, 4.50, 4.53 8th Ed.  4.49, 4.50, 4.51, 4.54 4.8 Conditional Expectations 7th Ed.  4.55, 4.56, 4.57 8th Ed.  4.56, 4.57, 4.58 5 Special Probability Distributions 7th Ed.  5.1, 5.2, 5.5, 5.20, 5.21, 5.23, 5.33 8th Ed.  5.1, 5.2, 5.5, 5.20, 5.21, 5.23, 5.33 6.1-6.4 Special Probability Densities 7th Ed.  6.1, 6.2, 6.3, 6.15, 6.16 8th Ed.  6.1, 6.2, 6.3, 6.15, 6.16 6.5 The Normal Distribution 7th Ed.  6.31, 6.37 8th Ed.  6.31, 6.37 7 The Weak Law of Large Numbers and the Central Limit Theorem

Important dates:
Exam 1:     Tuesday March 3, in class.
Exam 2:     Tuesday April 7, in class.
Final Exam:     Thursday May 5, 11:30am - 2:30 pm, Morrill 323       Spring 2020 final exam schedule

A 95 - 100, A- 90 - 95, B+ 87 - 90, B 83 - 87, B- 80 - 83, C+ 77 - 80, C 73 - 77, C- 70 - 73, D+ 67 - 70, D 60 - 67, F < 60.

Evaluation:   Your grade will be based on quizzes, two in-class exams, and a final. We will have bi-weekly quizzes. The quizzes will be based on the material covered in class and suggested problems which will not be collected. (Special assignments, which might include use of Maple, will be collected.) Quizzes cannot be made up, but your lowest quiz grade will be dropped. Makeup exams will be given only for serious illness or emergency, and these must be documented.
Exams will draw from material covered in class, that is, any theorem, proof, or example that we cover in class and any suggested problem is a possible material for the tests. There will be two in-class exams and a comprehensive final. Dates see above.