Matthew L. Wright
Assistant Professor, St. Olaf College

# Probability Theory

## Math 262 ⋅ Spring 2017

Welcome to Probability Theory! For course info and policies, please see the syllabus. For grades, log into Moodle. If you need help or have questions, please contact Prof. Wright; office hours are Mon. 12:45–1:45, Wed. 9–10, Thurs. 10–11 & 1–2, Fri. 12:45–1:45, or by appointment in RMS 409.

Jump to today
Monday
Feb. 6
Introduction
What is probability?
Do the following before next class:
• Complete the syllabus quiz.
• Read §1.1, Sample Spaces and Events, in the textbook. Come to class knowing the definitions of experiment, sample space, event, complement, intersection, union, and disjoint.
• Read the three axioms of probability at the beginning of §1.2 (page 9 of the text).
• Begin Homework 1.
Wednesday
Feb. 8
Sample spaces and events
Axioms of probability
Do the following before next class:
• Read all of §1.2, Axioms, Interpretations, and Properties of Probability.
• Do Homework 1 (due Friday at 5pm in the homework mailbox).
• Read pages 22–24. Come to class knowing the Fundamental Counting Principle and how to interpret a tree diagram.
Friday
Feb. 10
Counting methods
Permutations and Combinations
Homework 1
due today
Do the following before next class:
• Begin Homework 2 (due 5pm Friday).
• Read §1.3, Counting Methods. Make sure you know how to count the number of ways that k items can be selected from n in each of the following cases: (i) order important, selection with replacement; (ii) order important, selection without replacement; (iii) order unimportant, selection without replacement.
• Think about the question: How many ways can k items can be selected from n if order is unimportant and selection is with replacement?
Monday
Feb. 13
More counting methods
Do the following before next class:
Wednesday
Feb. 15
Counting Extravaganza
Do the following before next class:
• Finish Homework 2 (due 5pm Friday).
• Read pages 36–39, paying close attention to the examples. Come to class knowing the definition of conditional probability.
Friday
Feb. 17
Conditional probability
Bayes' Theorem
Homework 2
due today
Do the following before next class:
• Finish reading §1.4, Conditional Probability.
• Begin Homework 3 (due 5pm Wednesday).
• Read pages 53–54 from §1.5, Independence. Come to class knowing the definition of independent events.
Monday
Feb. 20
Independence
Do the following before next class:
• Finish Homework 3 (due 5pm Wednesday).
• Finish reading §1.5
• Read §1.6, Simulation of Random Events, through page 66.
Wednesday
Feb. 22
Simulation of random events
Homework 3
due today
Do the following before next class:
• Read §2.1. Come to class knowing the definition of random variable, what a Bernoulli random variable is, and what it means for a random variable to be discrete or continuous.
• Begin Homework 4 (due 5pm Monday).
Friday
Feb. 24
Random variables
Discrete distributions
Do the following before next class:
• Finish Homework 4 (due 5pm Monday).
• Read §2.2, paying special attention to the pdfs and cdfs in the examples.
• Read §2.3, through page 103, paying particular attention to the examples. Come to class knowing the definition of the expected value of a discrete random variable.
Monday
Feb. 27
Expected value
Variance and standard deviation
Homework 4
due today
Do the following before next class:
• Finish the worksheet from class on Monday. Come to class ready to discuss part (h).
• Read the rest of §2.3. Take note of Chebyshev's Inequality.
• Read §2.4, through page 120. Pay close attention to examples 2.27, 2.28, and 2.29. Come to class knowing the definitions of binomial experiment and binomial random variable.
• Begin Homework 5 (due 5pm Friday).
Wednesday
Mar. 1
Binomial random variables
Do the following before next class:
• Finish Homework 5 (due 5pm Friday).
• Finish reading §2.4.
Friday
Mar. 3
Binomial random variables
Review for exam
Homework 5
due today
Study for Exam 1.
Monday
Mar. 6
Exam 1
• This exam will cover Sections 1.1 through 1.5 and 2.1 through 2.3.
• Calculators will be permitted, but probably not very helpful, and certainly not necessary.
• Books, notes, and internet-capable devices will not be permitted during the exam.
• A review page, with suggested problems, is available here.
Do the following before next class:
• Read §2.5, pages 130 and 131. Take note of the pmf of the Poisson distribution. Then read subsections 2.5.2 and 2.5.3 (pages 134–136). Can you think of other situations in which you might model something by a Poisson process?
• Begin Homework 6 (due 5pm Friday).
Wednesday
Mar. 8
Poisson distribution
Do the following before next class:
• Review §2.5, The Poisson Distribution.
• Finish Homework 6.
• Read §2.6.1, The Hypergeometric Distribution. Answer the following question: What types of experiments can be modeled by a random variable with a hypergeometric distribution?
Friday
Mar. 10
Hypergeometric distribution
Homework 6
due today
Do the following before next class:
• Read §2.6.2, The Negative Binomial and Geometric Distributions. Answer the following questions: What types of experiments can be modeled by a random variable with a negative binomial distribution? ...with a geometric distribution?
• Begin Homework 7 (due 5pm Wednesday).
Monday
Mar. 13
Negative binomial distribution
Geometric distribution
Do the following before next class:
• Review §2.6, Other Discrete Distributions.
• Finish Homework 7.
• Read from the beginning of §2.8, Simulation of Discrete Random Variables, through page 163. Take note of the inverse cdf method.
Wednesday
Mar. 15
Simulation of discrete random variables
Homework 7
due today
Do the following before next class:
• Read §2.7, Moments and Moment Generating Functions, from the beginning of the section through the theorem on page 155. Pay special attention to the definitions and to Examples 2.46, 2.47, and 2.48.
Friday
Mar. 17
Moment generating functions
Have a great spring break! No class March 20 – 24.
Do the following before next class:
• Read §3.1. Take note of the definitions of probability density function, uniform distribution, and cumulative distribution function (for a continuous random variable).
• Begin Homework 8.
Monday
Mar. 27
Continuous random variables
Uniform distribution
Do the following before next class:
• Read §3.2. Take note of the definitions of expected value, variance, and moment generating function for continuous random variables.
• Finish Homework 8.
Wednesday
Mar. 29
Expected values and moment generating functions of continuous distributions
Homework 8
due today
Do the following before next class:
• Read §3.3, The Normal (Gaussian) Distribution. Take note of the normal pdf, the standard normal rv, and the Empirical Rule on page 215.
• Begin Homework 9.
Friday
Mar. 31
Normal distribution
Do the following before next class:
• Read §3.4.1, The Exponential Distribution. Take note of the pdf and cdf of the exponential distribution. What sorts of things might you model with an exponential distribution?
• Finish Homework 9.
Monday
Apr. 3
Exponential distribution
Homework 9
due today
Do the following before next class:
• Read pages 229–233 about the gamma distribution. How is the gamma distribution related to the exponential distribution?. What sorts of things might you model with a gamma distribution?
• Begin Homework 10.
Wednesday
Apr. 5
Gamma distribution
Do the following before next class:
• Read §3.7. Take note of the examples. What does it mean to find the distribution of a transformation of a random variable? How can we do this?
• Finish Homework 10.
Friday
Apr. 7
Transformations of continuous random variables
Homework 10
due today
Do the following before next class:
• Re-read §3.7.
• Do problems from the Exam 2 Review to prepare for the exam.
Monday
Apr. 10
Review day
Take-home exam problems distributed
Complete the take-home exam problems before next class.
Wednesday
Apr. 12
Exam 2
Easter break! No class April 14 or 17.
Do the following before next class:
• Read pages 288–296. Come to class knowing the difference between a joint distribution and a marginal distribution, and what it means for two random variables to be independent.
• Begin Homework 11.
Wednesday
Apr. 19
Joint distributions
Do the following before next class:
• Review §4.1. Notice how the concept of joint distribution extends to more than two random variables in §4.1.4.
• Read §4.2, up to and including Example 4.16. Take special note of the definition of expected value of a function of two random variables—how does this generalize the expected value of a function of a single random variable?
• Finish Homework 11.
Friday
Apr. 21
Expected values
Covariance and correlation
Homework 11
due today
Do the following before next class:
• Finish reading §4.2. Take note of the properties of covariance and correlation.
• Read §4.3, up to and including Example 4.20. You may skim the proof, but pay special attention to the examples.
• Begin Homework 12.
Monday
Apr. 24
Linear combinations of random variables
Do the following before next class:
• Read §4.3 through the end of §4.3.1. Take note of the theorem on page 325 and how it is applied in Example 4.21.
• Finish Homework 12.
Wednesday
Apr. 26
Properties of linear combinations
Homework 12
due today
Do the following before next class:
• Read §4.3.2. Take note of how the moment generating function of a sum of random variables relates to the moment generating functions of the individual random variables.
• Read §4.4 through §4.4.1. Take note of the definition of conditional mass/density and how conditional distributions relate to independent random variables.
• Begin Homework 13.
Friday
Apr. 28
Moment generating functions and linear combinations
Conditional distributions
Do the following before next class:
• Read the rest of §4.4. Take note of the definitions of conditional expectation and conditional variance, and how they are used in the examples.
• Finish Homework 13.
Monday
May. 1
No class — attend the events in Tomson Hall and elsewhere on campus.
Do the following before next class:
• Re-read §4.4. Take note of the definitions of conditional expectation and conditional variance, and how they are used in the examples.
• Finish Homework 13.
Wednesday
May. 3
Conditional distributions and conditional expectation
Homework 13
due today
Do the following before next class:
• Read §4.5 through page 357. Take note of the definition of iid random variables, and observe how the central limit theorem is applied in Example 4.35.
• Begin Homework 14.
Friday
May. 5
Limit theorems
Do the following before next class:
• Read the rest of §4.5. Take note of the Law of Large Numbers.
• Finish Homework 14.
Monday
May. 8
Limit theorems
Homework 14
due today
Do the following before next class:
• Re-read §4.5.
• Begin Homework 15: for now, focus on the problems from §4.5.
Wednesday
May. 10
Transformations of random variables
Distribution function method
Do the following before next class:
• Read §4.6 through Example 4.40. Observe how the (bivariate) Transformation Theorem provides a method for computing the joint density of a transformation of two random variables.
• Work on Homework 15.
Friday
May. 12
Transformations of random variables
Bivariate transformation theorem
Do the following before next class:
• Re-read §4.6.
• Read from the beginning of §4.9 through the end of §4.9.1. Observe how the distributions of the maximum and minimum values in a collection of iid random variables can be computed.
• Finish Homework 15.
Monday
May. 15
Order statistics
Homework 15
due today
Complete the take-home portion of the final exam.
Friday
May. 19
Final Exam, 9–11am
• This exam will be cumulative, with emphasis on Chapter 4.
• The exam will consist of an in-class portion and a short take-home portion.
• The take-home portion will be distributed on May 15 and due at the final exam time on May 19. For this portion of the exam you may use your textbook, your notes, the course web site, a calculator, R, Mathematica, and Wolfram Alpha, but not other people, web sites, books, etc. Remember the honor code!
• For the in-class portion: books, notes, and internet-capable devices will not be permitted. Calculators will be permitted, but probably not very helpful, and certainly not necessary.
• Click here for more information and suggested review problems.
• More review problems: transformations of random variables (solutions) and end of semester problems (solutions)
• Lastly, make sure you are familiar with the St. Olaf final exam policies.