MAT2313

Foundations of Mathematics (3-0) 3 Credit Hours

Course Catalog

Corequisite: MAT1224.

Content: Permutations, combinations, multinational coefficients, inclusion/exclusion principle, axioms of probability, conditional probability, Bayes formula, independent events, discrete random variables, expected value,m variance, discrete random variables (Bernoulli, Binomial, Poisson, geometric, hypergeometric and Zeta random variables), continuous random variables (uniform, normal and other distributions), joint distributions, properties of expectations, limit theorems (Chebyshev's inequality, Central Limit Theorem, Law of Large Numbers)) Generally offered: Fall, Spring, Summer.

Description

Introduction to the theory of probability, through the study of discrete and continuous random variables.

Sample textbooks

• Modern Mathematical Statistics with Applications (Springer Texts in Statistics). Jay L. Devore and Kenneth N. Berk. Second Edition.
• A Probability Course for the Actuaries: A Preparation for Exam P/1, by Marcel B. Finan. Freely available online.

Topics List

Week	Topic	Sections from Finan's book	Subtopics
1-2	Populations and Samples	Chapters 1-2	The Fundamental Principle of Counting Permutations and Combinations Permutations and Combinations with Indistinguishable Objects
3-4	Probability: Definitions and Properties	Chapter 3	Basic Definitions and Axioms of Probability Properties of Probability Probability and Counting Techniques
5	Conditional Probability and Independence	Chapter 4	Conditional Probabilities Posterior Probabilities: Bayes’ Formula Independent Events Odds and Conditional Probability
6-8	Discrete Random Variables	Chapter 5	Random Variables Probability Mass Function and Cumulative Distribution Function119 Expected Value of a Discrete Random Variable Expected Value of a Function of a Discrete Random Variable Variance and Standard Deviation Binomial and Multinomial Random Variables Poisson Random Variable Other Discrete Random Variables (Geometric, Hypergeometric, etc)
9-10	Continuous Random Variables	Chapter 6	Distribution Functions Expectation, Variance and Standard Deviation The Uniform Distribution Function Normal Random Variables Exponential Random Variables Gamma and Beta Distributions The Distribution of a Function of a Random Variable
11	Joint Distributions	Chapter 7	Jointly Distributed Random Variables Independent Random Variables Sum of Two Independent Random Variables
12	Properties of Expectation	Chapter 8	Expected Value of a Function of Two Random Variables . . . . . 351 Covariance, Variance of Sums, and Correlations . . . . . . . . . 362 Conditional Expectation . . . . . . . . . . . . . . . . . . . . . . 376 Moment Generating Functions
13-14	Limit Theorems	Chapter 9	The Law of Large Numbers The Central Limit Theorem

See also

• UTSA Undergraduate Mathematics Course Descriptions