Math 431: Probability Theory

Title

Probability Theory.

(3:3:0)

F (odd years)

Description

Axiomatic probability theory, conditional probability, discrete / continuous random variables, expectation, conditional expectation, moments, functions of random variables, multivariate distributions, laws of large numbers, central limit theorem.

Desired Learning Outcomes

This course is a calculus-based first course in probability. It is cross-listed with EC En 370.

Prerequisites

The current prerequisite is linear algebra. Because of the need to work with joint distributions of continuous random variables in Math 431, the department should consider adding multivariable calculus as a prerequisite.

Minimal learning outcomes

Primarily, students should be able to do basic computation of probabilistic quantities, including those involving applications. Students should be able to recall the most common types of discrete and continuous random variables and describe and compute their properties. Students should understand the theory of probability in an elementary context.

1. Basic principles of counting
• Product sets
• Disjoint unions
• Combinations
• Permutations
2. Axiomatic probability
• Outcomes
• Events
• Probability measures
• Continuity
3. Discrete random variables
• Probability mass function
• Cumulative distribution function
• Moments
• Expectation
• Of a function of a random variable
• Variance
• Common types
• Bernoulli
• Binomial
• Poisson

4. Continuous random variables
• Probability density function
• Cumulative distribution function
• Moments
• Expectation
• Of a function of a random variable
• Variance
• Common types
• Uniform
• Exponential
• Normal
5. Conditional probability
• As a probability
• Bayes' Formula
• Independence
• Events
• Random variables
6. Joint distributions
• Covariance
• Conditional distributions
7. Conditional expectation
8. Limit theorems
• Weak Law of Large Numbers
• Strong Law of Large Numbers
• Central Limit Theorem

Textbooks

Possible textbooks for this course include (but are not limited to):

• Sheldon Ross, A First Course in Probability (8th edition), Prentice Hall, 2009.