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# Probability and Statistics III: A Gentle Introduction to Statistics

## Overview

This course provides an introduction to basic statistical concepts.

We begin by walking through a library of probability distributions, where we motivate their uses and go over their fundamental properties.
These distributions include such important folks as the Bernoulli, binomial, geometric, Poisson, uniform, exponential, and normal distributions, just to name a few. Particular attention is paid to the normal distribution, because it leads to the Central Limit Theorem (the most-important mathematical result in the universe, actually), which enables us to make probability calculations for arbitrary averages and sums of random variables.

We then discuss elementary descriptive statistics and estimation methods, including unbiased estimation, maximum likelihood estimation, and the method of moments – you gotta love your MoM! Finally, we describe the t, χ2, and F sampling distributions, which will prove to be useful in upcoming statistical applications.

## Syllabus

“FCPS” refers to the free text, A First Course in Probability and Statistics

Module 1: Distributions
• Lesson 1: Bernoulli and Binomial Distributions (FCPS §4.1.1)
• Lesson 2: Hypergeometric Distribution (FCPS §4.1.2)
• Lesson 3: Geometric and Negative Binomial Distributions (FCPS §4.1.3)
• Lesson 4: Poisson Distribution (FCPS §4.1.4)
• Lesson 5: Uniform, Exponential, and Friends (FCPS §4.2.1–4.2.2)
• Lesson 6: Other Continuous Distributions (FCPS §4.2.3)
• Lesson 7: Normal Distribution: Basics (FCPS §4.3.1)
Module 1 (cont’d): Distributions
• Lesson 8: Standard Normal Distribution (FCPS §4.3.2)
• Lesson 9: Sample Mean of Normals (FCPS §4.3.3)
• Lesson 10: The Central Limit Theorem + OPTIONAL Proof (FCPS §4.3.4)
• Lesson 11: Central Limit Theorem Examples (FCPS §4.3.5)
• Lesson 12 [OPTIONAL]: Extensions – Multivariate Normal Distribution (FCPS §4.4.1)
• Lesson 13 [OPTIONAL]: Extensions – Lognormal Distribution (FCPS §4.4.2)
• Lesson 14: Computer Stuff, including OPTIONAL Box-Muller Proof (FCPS §4.5)
Module 2: Getting Started with Statistics
• Lesson 1: Introduction to Descriptive Statistics (FCPS §5.1.1)
• Lesson 2: Summarizing Data (FCPS §5.1.2)
• Lesson 3: Candidate Distributions (FCPS §5.1.3)
• Lesson 4: Introduction to Estimation (FCPS §5.2.1)
• Lesson 5: Unbiased Estimation (FCPS §5.2.2)
• Lesson 6: Mean Squared Error (FCPS §5.2.3)
Module 2 (cont’d): Getting Started with Statistics
• Lesson 7: Maximum Likelihood Estimation (FCPS §5.2.4)
• Lesson 8: Trickier MLE Examples (FCPS §5.2.4)
• Lesson 9: Invariance Property of MLEs (FCPS §5.2.4)
• Lesson 10: Method of Moments Estimation (FCPS §5.2.5)
• Lesson 11: Sampling Distributions (FCPS §5.3)

David Goldsman

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