**M 362K. PROBABILITY I.**

*Unique #54575, Spring 2015*

### Instructor

Prof. Todd Arbogast

E-Mail: arbogast@ices.utexas.edu

Office: RLM 11.162, Phone: 512-471-0166

Office hours: MTWTh 8:30-9:30 a.m.

### Prerequisite

M 408D, M 408L, or M 408S with a grade of at least C-.
### Meeting

TTh 12:30-2:00 p.m. in RLM 5.114.
Attendance is required at all class meetings.
### Textbook

A First Course in Probability, 9th edition, by Shelden Ross, Pearson, 2012.
### Class Web Site

We will use the University's CLIPS and
Canvas web sites. Please check that your scores are recorded
correctly in Canvas.
### Course Description

This is an introductory course in the mathematical theory of probability. While both theorem
proving and problem solving are required, we emphasize problem solving and intuition. Some advanced
concepts will be presented without proof.
### Course Outline

We plan to cover the following textbook chapters and sections.
1 Combinatorial Analysis [3 hours]

1.1 Introduction

1.2 The Basic Principle of Counting

1.3 Permutations

1.4 Combinations

2 Axioms of Probability [4 hours]

2.1 Introduction

2.2 Sample Space and Events

2.3 Axioms of Probability

2.4 Some Simple Propositions

2.5 Sample Spaces Having Equally Likely Outcomes

2.7 Probability as a Measure of Belief

3 Conditional Probability and Independence [5 hours]

3.1 Introduction

3.2 Conditional Probabilities

3.3 Bayes's Formula

3.4 Independent Events

3.5 *P(.|F)* Is a Probability

4 Random Variables [7 lectures]

4.1 Random Variables

4.2 Discrete Random Variables

4.3 Expected Value

4.4 Expectation of a Function of a Random Variable

4.5 Variance

4.6 The Bernoulli and Binomial Random Variables (omit 4.6.2)

4.7 The Poisson Random Variable (omit 4.7.1)

4.8 Other Discrete Probability Distributions (4.8.1 Geometric Random Variable only)

5 Continuous Random Variables [7 hours]

5.1 Introduction

5.2 Expectation and Variance of Continuous Random Variables

5.3 The Uniform Random Variable

5.4 Normal Random Variables

5.5 Exponential Random Variables (omit 5.5.1)

5.7 The Distribution of a Function of a Random Variable

6 Jointly Distributed Random Variables [5 hours]

6.1 Joint Distribution Functions

6.2 Independent Random Variables

6.3 Sums of Independent Random Variables

6.4 Conditional Distributions: Discrete Case

6.5 Conditional Distributions: Continuous Case

7 Properties of Expectation [4 hours]

7.1 Introduction

7.2 Expectation of Sums of Random Variables (omit 7.2.1 and 7.2.2)

7.4 Covariance, Variance of Sums, and Correlations

7.7 Moment Generating Functions (omit 7.7.1; if time permits)

8 Limit Theorems [4 hours]

8.1 Introduction

8.2 Chebyshev's Inequality and the Weak Law of Large Numbers

8.3 The Central Limit Theorem

8.4 The Strong Law of Large Numbers

### Computer Accounts

A computer account on the Mathematics Department network can be obtained in the Undergraduate
Computer Lab, RLM 7.122.
### Homework and Quizzes

Homework will be assigned weekly, with only a portion to be fully graded, and due generally on
Tuesdays. It is acceptable for groups of students to help each other with the homework exercises;
however, each student must write up his or her own work. Late homework will not be accepted for
credit (unless there is a valid health issue), and homework must be turned in during class. The
textbook has answers to selected exercises. Quizzes may be given periodically in class.
### Exams

Two exams will be given on Thursdays, February 26 and April 9. A comprehensive final exam will be
given during finals week on Monday, May 18, 9:00-12:00 noon.
### Final Grade

Grades on the two midterm exams will count 20% each, the homework and quizzes will count 20%, and
the final exam will count 40% in determining the final grade on the letter plus/minus scale. The
two lowest homework and quiz scores will be dropped in determining the homework grade.
### Student Honor Code

"As a student of The University of Texas at Austin, I shall abide by the core values of the
University and uphold academic integrity."
### Code of Conduct

The core values of The University of Texas at Austin are learning, discovery, freedom, leadership,
individual opportunity, and responsibility. Each member of the university is expected to uphold
these values through integrity, honesty, trust, fairness, and respect toward peers and community.
### Students with Disabilities

The University provides upon request appropriate academic accommodations for qualified students with
disabilities. Contact the Office of the Dean of Students at 512-471-6259, 512-471-4641 TTY, and
notify your instructor early in the semester.
### Religious Holidays

Appropriate academic accommodation for major religious holidays is provided upon request.
### Emergency Classroom Evacuation

Occupants of University of Texas buildings are required to evacuate when a fire alarm is activated.
Alarm activation or announcement requires exiting and assembling outside. Familiarize yourself with
all exit doors of each classroom and building you may occupy. Remember that the nearest exit door
may not be the one you used when entering the building. Do not re-enter a building unless given
instructions by the Austin Fire Department, the University Police Department, or the Fire Prevention
Services office.