# Math 402: Modeling with Uncertainty and Data 1

## Contents

## Catalog Information

### Title

Probability and Statistics 1

### (Credit Hours:Lecture Hours:Lab Hours)

(3:3:1)

### Offered

The course runs through both Spring and Summer

### Prerequisite

Math 322, Math 346; concurrent with Math 403

### Description

The theory of probability and stochastic processes, emphasizing topics that are used in applications. Random spaces and variables, probability distributions, limit theorems, martingales, diffusion, Markov, Poisson and queuing processes, renewal theory and information theory.

## Desired Learning Outcomes

### Prerequisites

Math 322, Math 346; concurrent with Math 403

### Minimal learning outcomes

Students will have a solid understanding of the concepts listed below. They will be able to prove theorems that are central to this material, including theorems that they have not seen before. They will understand connections between the concepts taught, and will be able to perform the related computations on small, simple problems. They will understand the model specifications for martingales and for diffusion, Markov, Poisson, queuing and renewal theoretic processes, and be able to recognize whether they apply in the context of a given application or not. They will be able to perform the relevant computations on small, simple problems.

- Random Spaces and Variables
- Probability Spaces (including σ-algebras)
- Random Variables (including Measurable Functions)
- Expectation (including Lebesgue Integration)
- Independence
- Conditional Expectation
- Law of Large Numbers

- Distributions
- Generating Functions and Characteristic Functions
- Moments
- Commonly Used Distributions
- Joint and Conditional Distributions

- Limit Theorems
- Weak Convergence
- Central Limit Theorem
- Applications

- Martingales and Diffusion
- Stochastic Processes, Filtrations, Stopping Times
- Martingales
- Doob's Decomposition Theorem
- Doob's Inequality and Convergence Theorems

- Markov Processes
- The Markov Property
- Finite Markov Chains
- Asymptotic Behavior
- Absorbing Markov Chains
- Continuous-Time Markov Chains

- Poisson, Queuing, and Renewal Theory
- Counting Integrals
- Kolomogorov's Forward System
- Poisson Processes
- Queues
- Renewal Processes

- Information Theory
- Entropy
- Conditional and Joint Entropy
- Kullback-Lieber Distance
- Channel Capacity

### Textbooks

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