AMS 311 | Applied Mathematics & Statistics (2024)

AMS 311, Probability Theory

Required Textbook for Summer 2024:
"Introduction to Probability" by Mark Daniel Ward and Ellen Gundlach, second printing, W.H. Freeman and Company; 2016; ISBN: 978-0-7167-7109-8

Required Textbook for Fall 2024 and Spring 2025:
"A First Course in Probability" by Sheldon Ross, 10th Edition, 2020, Pearson Publishing; ISBN 978-0134753119

Topics
1. Probability Spaces – 3 class hours.
2. Conditional probability and independence – 4 class hours.
3. Random Variables; Special Distributions – 6 class hours.
4. Expectation – 4 class hours.
5. Joint Distributions – 4 class hours.
6. Conditional Distributions – 3 class hours.
7. Covariance and correlations – 2 class hours.
8. Moment Generating Functions – 3 class hours.
9. Transformation of variables – 4 class hours.
10. Order Statistics – 2 class hours.
11. Law of Large Numbers – 3 class hours.

Learning Outcomes for AMS 311, Probability Theory

1.) Demonstrate an understanding of core concepts of probability theory and their use in applications:
* experiments, outcomes, sample spaces, events, and the role of set theory in probability;
* the axioms of probability and the theorems and their consequences;
* using counting principles to calculate probabilities of events in sample spaces of equally likely outcomes;
* independence and disjointness;
* conditional probability;
* the law of total probability and Bayes. law;
* the method of conditioning to solve problems;
* Markov chains and associated conditioning arguments.

2.) Demonstrate an understanding of the theory of random variables and their applications:
* the difference between discrete random variables, continuous random variables, and random variables with hybrid distributions;
* cumulative distribution functions and their properties;
* probability mass functions for discrete random variables and computations to evaluate probabilities;
* properties of commonly used discrete distributions, such as binomial, geometric, Poisson, and hypergeometric distributions;
* probability density functions, computing them from cumulative distribution functions, and vice versa;
properties of commonly used density functions, such as uniform, exponential, gamma, beta, and normal densities;
means, variances, and higher moments of random variables, and their properties;
connections and differences between different distribution functions, e.g., normal approximation to binomial, Poisson approximation to binomial, and the difference between binomial and hypergeometric;
* Markov and Chebyshev inequalities and utilizing them to give bounds and estimates of probabilities.

3.) Demonstrate an understanding of the theory of jointly distributed random variables and their applications:
computations with joint distributions, both for discrete and continuous random variables;
computations with joint density functions and conditional density functions;
conditional expectation and conditioning arguments in computations involving two or more random variables;
computations with the bivariate normal distribution, the t-distribution, and chi-squared distributions, order statistics;
applying indicator random variables to compute expectations;
using moment generating functions in solving problems with sums of independent random variables;
the weak and strong laws of large numbers;
applying the central limit theorem in estimating probabilities.