# MATH 697

*Sahir Rai Bhatnagar*

*2017-12-05*

# Syllabus

## General Information

- Instructor(s): Sahir Bhatnagar and Dr. Alexandra M. Schmidt
- Email: sahir.bhatnagar@mail.mcgill.ca,
- Website: http://sahirbhatnagar.com/MATH697/
- Lectures: Tuesdays 9am - 12pm
- Office: TBD
- Office Hours: By appointment only
- Prerequisite(s): Calculus and Algebra
- Texts:
*Modern Mathematical Statistics with Applications*, 2nd Edition by Jay L. Devore and Kenneth N. Berk

## Course Description

The main learning outcomes of this course are to get a broad idea about some frequently used probability models and to learn basic results and techniques in probability theory and statistical inference. Most of the materials for the course will be drawn from the first seven chapters of the textbook. The book does not, however, contain all the materials we intend to cover in this course. Some extra notes will therefore be given on those topics not in the text book. We will also introduce computational methods in statistics with the statistical software program `R`

.

## Grade Distribution

Assignments | 10% |

Quizzes | 40% |

Final Exam | 50% |

## Target Syllabus

### Overview and Descriptive Statistics (Weeks 1-4)

- 1.1 Populations and Samples

### Probability (Weeks 1-4)

- 2.1 Sample Spaces and Events
- 2.2 Axioms, Interpretations, and Properties of Probability
- 2.3 Counting Techniques
- 2.4 Conditional Probability
- 2.5 Independence

### Discrete Random Variables and Probability Distributions (Weeks 1-4)

- 3.1 Random Variables
- 3.2 Probability Distributions for Discrete Random Variables
- 3.3 Expected Values of Discrete Random Variables
- 3.4 Moments and Moment Generating Functions
- 3.5 The Bernoulli/Binomial Probability Distribution
- 3.6 The Geometric/Negative Binomial Probability Distribution
- 3.7 The Poisson Probability Distribution

### Continuous Random Variables and Probability Distributions (Weeks 5-8)

- 4.1 Probability Density Functions and Cumulative Distribution Functions

- 4.2 Expected Values and Moment Generating Functions

- 4.3 The Uniform Distribution

- 4.4 The Exponential Distribution

- 4.5 The Gamma Distribution

- 4.6 The Normal Distribution

- 4.7 One-Dimensional Change of Variable (Discrete and Continuous)

### Joint Probability Distributions (Weeks 5-8)

- 5.1 Jointly Distributed Random Variables
- 5.2 Expected Values, Covariance, and Correlation
- 5.3 Conditional Distributions
- 5.4 Multidimensional Change of Variable (Discrete and Continuous)

### Sampling Distributions and Limits (Weeks 5-8)

- 6.1 Sampling Distributions
- 6.2 Convergence in Probability, Weak Law of Large Numbers
- 6.3 Convergence with Probability 1, Strong Law of Large Numbers

- 6.4 Convergence in Distribution, Central Limit Theorem

### Statistical Inference (Weeks 9-12)

- 7.1 Inference Using a Probability Model

- 7.2 Statistical Models

- 7.3 Data Collection (Finite Populations, Simple Random Sampling, Histograms)

- 7.4 Basic Inferences (Descriptive, Plots, Types of Inferences)

### Likelihood Inference (Weeks 9-12)

- 8.1 The Likelihood Function, Sufficient Statistics

- 8.2 Maximum Likelihood Estimation

- 8.3 Inferences Based on the MLE (Standard Errors, Bias, Consistency, Confidence Intervals, Hypotheses and Test Procedures, P-values, Inferences for the Variance)

- 8.4 Distribution-Free Methods (Method of Moments, Bootstrapping)

### Regression and Correlation (Weeks 9-12)

- 9.1 The Simple Linear and Logistic Regression Models

- 9.2 Estimating Model Parameters

- 9.3 Inferences About the Regression Coefficient

- 9.4 Inferences Concerning Prediction of Future \(Y\) Values

- 9.5 Correlation

- 9.6 Model Checking (\(\chi^2\) Goodness of Fit Test, Cross-Validation)

- 9.7 Multiple Regression Analysis

- 9.8 Regression with Matrices