# Math 404: Modeling with Uncertainty and Data 2

### Title

Probability and Statistics 2

(3:3:1)

W

### Prerequisite

Math 402; concurrent with Math 405

### Description

A first course in mathematical statistics, focusing on the mathematical aspects of the subject. Specific topics include estimation, inference, analysis of variance, regression, multivariate statistics, Bayesian statistics, state estimation, Kalman filtering, time series, GARCH models.

## Desired Learning Outcomes

### Prerequisites

Math 402; concurrent with Math 405

### Minimal learning outcomes

This course is a survey-level examination of the mathematical and computational issues surrounding the statistical concepts and methods listed below. Students will be able to prove many of the theorems that are central to this material. They will be able to recognize whether the models and methods mentioned apply in the context of a given application or not, and identify the questions of practical relevance that these methods would be able to answer. This course is NOT a replacement for a statistics degree, those who want to understand how to better implement these concepts should consider the statistics minor.

1. Estimation
• Method of Moments
• Maximum Likelihood Estimates
• Bayes Estimators
• The Expectation Maximization Algorithm
2. Inference
• Sampling Distributions
3. Analysis of Variance and Regression
• One-Way Analysis of Variance
• Simple Linear Regression
• Logistic Regression
4. Multivariate Statistics
• Multivariate Normal Distributions
• Correlation and Covariance Matrices
• Principle Component Analysis
• Canonical Correlation
• Factor Analysis
5. Bayesian Statistics
• Overview of Bayesian Statistics
• Bayes Theorem and Subjective Probability
• Conjugate Bayesian Inference
• Non-Conjugate Bayesian Inference
• Markov Chain Monte Carlo (MCMC)
6. State Estimation
• Recursive Least Squares
• Kalman Filtering
• Exponentially Weighted Averaging
7. Time Series Analysis
• Stationary Time Series (ARMA)
• Non-Stationary Time Series (ARIMA)
• Forecasting
• Seasonality

### Textbooks

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