Math 431 – Probability
Notes on Math 431: Introduction to the Theory of Probability Your comments and criticism are greatly welcomed.
Textbook
Introduction to Probability, by David F. Anderson, Timo Seppäläinen, Benedek Valkó
Calendar
September
| Monday | Wednesday | Friday |
| - | 9/6 | 9/8 |
| 9/11 | 9/13 | 9/15 |
| 9/18 | 9/20 | 9/22 |
| 9/25 | 9/27 | 9/29 |
October
| Monday | Wednesday | Friday |
| 10/2 | 10/4 | 10/6 |
| 10/9 | 10/11 | 10/13 |
| Review | Midterm 1 | 10/20 |
| 10/23 | 10/25 | 10/27 |
| 10/30 | 11/1 | 11/3 |
November
| Monday | Wednesday | Friday |
| 11/1 | 11/3 | |
| 11/6 | 11/8 | 11/10 |
| 11/13 | 11/15 | 11/17 |
| 11/20 | 11/22 | Holiday |
| Review | Midterm 2 | 12/1 |
December
| Monday | Wednesday | Friday |
| Review | Midterm 2 | 12/1 |
| 12/4 | 12/6 | 12/8 |
| 12/11 | 12/13 | Final |
Lecture Outlines
Chapter 1: Experiment with Random Variables
Week 01 - 9/6: Introduction, 431 Card Game
Week 01 - 9/8: Probability Space, Equally Likely Outcome
Week 02 - 9/11: Equally Likely Outcome, Different Types of Random Experiments
Week 02 - 9/13: Equally Likely Outcome, Infinite Sample Spaces
Week 02 - 9/15: Three Rules of Sample Spaces
Week 03 - 9/18: Random Variables, Conditional Probability
Chapter 2: Conditional Probability and Independence
Week 03 - 9/20: Conditional Probability, Multiplication Rule, Partition
Week 03 - 9/22: Conditional Probability, Bayes’ Rule
Week 04 - 9/25: Independence, Bernoulli Distribution, Binomial Distribution
Week 04 - 9/27: Independence, Bernoulli Distribution, Binomial Distribution, Geometric Distribution, Conditional Independence
Chapter 3: Random Variables
Week 04 - 9/29: Birthday Paradox, Probability Mass Function, Probability Density Function
Week 05 - 10/2: Cumulative Distribution Function
Week 05 - 10/4: Conversion Between PMF/PDF and CDF
Week 05 - 10/6: Expectation
Week 06 - 10/9: Median, Variance, Standard Deviation, Affine Transformation of Random Variable
Week 06 - 10/11: Normal Distribution
Week 06 - 10/13: Normal Distribution
Week 07 - 10/16: Practice Midterm 1
Week 07 - 10/18: Midterm 1
Chapter 4: Approximations of the Binomial Distribution
Week 07 - 10/20: Expectation and Variance of Binomial Distribution, Normal Approximation of Binomial Distribution
Week 08 - 10/23: Normal Approximation of Binomial Distribution
Week 08 - 10/25: Central Limit Theorem, Law of Large Numbers, Confidence Intervals
Week 08 - 10/27: Poisson Distribution, Law of Rare Events, Exponential Distribution
Chapter 5: Transforms and Transformations
Week 09 - 10/30: Distribution of Function of Random Variables
Chapter 6: Joint Distribution of Random Variables
Week 09 - 11/1: Joint Distribution of Discrete Random Variables, Marginal Probability Mass Function, Multinomial Distribution
Week 09 - 11/3: Joint Distribution of Continuous Random Variables, Uniform Distribution in Two Dimensions
Week 10 - 11/6: Uniform Distribution in Two Dimensions, Joint Distribution and Independence
Chapter 7: Sums and Symmetry
Week 10 - 11/8: Independent and Identically Distributed Random Variables, Sum of Independent Random Variables
Week 10 - 11/10: Negative Binomial Distribution, Convolution of Normal Random Variables, Linearity of Expectation
Chapter 8: Expectation and Variance in the Multivariate Setting
Week 11 - 11/13: Linearity of Expectation, Expectation and Independence, Expectation and Variance of Sample Mean, Mean of Sample Variance
Week 11 - 11/15: Covariance
Week 11 - 11/17: Independence and Covariance, Bilinearity of Covariance
Week 12 - 11/20: Correlation
Section 6.4 & Section 8.5: Bivariate Normal Distribution
Week 12 - 11/22: Standard Bivariate Normal Distribution, Bivariate Normal Distribution
Week 12 - 11/24: Thanksgiving Break
Week 13 - 11/27: Practice Midterm 2
Week 13 - 11/29: Midterm 2
Chapter 10: Conditional Distribution
Week 13 - 12/1: Conditional Distribution (Discrete Case)
Week 14 - 12/4: Conditional Distribution (Discrete Case)
Week 14 - 12/6: Conditional Distribution (Continuous Case)
Chapter 9: Tail Bounds and Limit Theorems
Week 14 - 12/8: Conditional Distribution (Continuous Case), Law of Large Numbers, Central Limit Theorem
Review
Week 15 - 12/11: Probability Space, Basic Rules of Probability, Conditional Probability, Independence, Random Variables