Mathematical Statistics For Data Science

An introduction to mathematical statistics for data science, covering method of moments, maximum likelihood, and more

What you'll learn

Learn how to estimate statistical parameters using the method of moments and maximum likelihood

Learn how to evaluate and compare different estimators using notions such as bias, variance, and mean squared error.

Learn about the Cramer-Rao lower bound and how to know if we have found the best possible estimator

Learn to evaluate asymptotic properties of estimators, including consistency and the central limit theorem.

Learn to create confidence intervals

Requirements

High school algebra, including manipulating functions with variables

Basic knowledge of calculus (integration and differentiation) is recommended for some chapters.

Prior experience with probability or statistics will be useful, but we cover everything assuming no previous knowledge!

Description

This course teaches the foundations of mathematical statistics, focusing on methods of estimation such as the method of moments and maximum likelihood estimators (MLEs), evaluating estimators by their bias, variance, and efficiency, and an introduction to asymptotic statistics including the central limit theorem and confidence intervals.The course includes:Over four hours of video lectures, using the innovative lightboard technology to deliver face-to-face lecturesSupplementary lecture notes with each lesson covering important vocabulary, examples and explanations from the video lessonsEnd of chapter practice problems to reinforce your understanding and develop skills from the courseYou will learn about:Three common probability distributions, the Bernoulli distribution, uniform distribution, and normal distributionExpected value and its relation to the sample meanThe method of moments for creating estimatorsExpected value of estimators and unbiased estimatorsVariance of random variables and variance of estimatorsFisher information and the Cramer-Rao Lower BoundThe central limit theoremConfidence intervalsThis course is ideal for many types of students:Students who have taken an introductory statistics class and who would like to dive into the mathematical detailsData science professionals who would like to refresh or expand their statistics knowledge to prepare for job interviewsAnyone who wants to learn how to think like a statisticianPre-requisitesThe course requires a good understanding of high school algebra and manipulating equations with variables.Some chapters use concepts from introductory calculus like differentiation or integration.  If you do not know calculus but otherwise have strong math skills, you can still follow along while only missing a few mathematical details.

Overview

Section 1: Introduction

Lecture 1 Course Introduction

Section 2: Probability Distributions

Lecture 2 Random variables, PMFs and PDFs

Lecture 3 The Bernoulli Distribution

Lecture 4 The Uniform Distribution

Lecture 5 The Normal Distribution

Lecture 6 Probability Distribution Recap

Section 3: Expected Values

Lecture 7 Sample mean and Expected Value

Lecture 8 Bernoulli Distribution Expected Value

Lecture 9 Uniform Distribution Expected Value

Lecture 10 Normal Distribution Expected Value

Lecture 11 Expected Value Recap

Lecture 12 Expected Value Practice Problems and Solutions

Section 4: Estimators and the Method of Moments

Lecture 13 Estimators and the Method of Moments

Lecture 14 Bernoulli Distribution MOM

Lecture 15 Uniform Distribution MOM

Lecture 16 Normal Distribution MOM

Lecture 17 Method of Moments Recap

Lecture 18 Method of Moments Practice and Solutions

Section 5: Unbiased Estimators

Lecture 19 Sampling Distribution, Evaluating Estimators, Bias

Lecture 20 Properties of Expected Values

Lecture 21 Bernoulli MOM Bias

Lecture 22 Uniform MOM Bias

Lecture 23 Normal MOM Bias

Lecture 24 Bias Recap

Lecture 25 Unbiased Estimators Practice and Solutions

Section 6: Variance

Lecture 26 Variance

Lecture 27 Bernoulli Distribution Variance

Lecture 28 Uniform Distribution Variance

Lecture 29 Normal Distribution Variance

Lecture 30 Variance of Estimators and Properties of Variance

Lecture 31 Bernoulli MOM Variance

Lecture 32 Uniform MOM Variance

Lecture 33 Normal MOM Variance

Lecture 34 Variance Recap

Lecture 35 Variance Practice and Solutions

Section 7: Maximum Likelihood Estimation

Lecture 36 Likelihood Function and Maximum Likelihood Estimation - Motivation

Lecture 37 Joint pdf, joint likelihood

Lecture 38 Log-likelihood and finding the MLE

Lecture 39 Properties of logarithms

Lecture 40 Bernoulli MLE

Lecture 41 Uniform MLE

Lecture 42 Mean Squared Error

Lecture 43 Normal MLE

Lecture 44 MLE Recap

Lecture 45 MLE Practice and Solutions

Section 8: Fisher Information and the Cramer-Rao Lower Bound

Lecture 46 The Cramer-Rao Lower Bound (CRLB) and Fisher Information

Lecture 47 Bernoulli CRLB

Lecture 48 Uniform CRLB

Lecture 49 Normal CRLB

Lecture 50 Efficiency

Lecture 51 CRLB Recap

Lecture 52 CRLB Practice and Solutions

Section 9: Central Limit Theorem

Lecture 53 Distribution of Estimators and Convergence in Distribution

Lecture 54 Bernoulli MOM/MLE Distribution

Lecture 55 Uniform MOM Distribution

Lecture 56 Normal MOM/MLE Distribution

Lecture 57 Consistency

Lecture 58 CLT Recap

Section 10: Confidence Intervals

Lecture 59 Confidence Intervals

Lecture 60 Bernoulli Confidence Interval

Lecture 61 Uniform Confidence Interval based on MOM

Lecture 62 Normal Confidence Interval

Lecture 63 Confidence Interval Recap, Link to Hypothesis Testing

Lecture 64 Confidence Interval Practice and Solutions

Anyone who has taken a basic statistics class and wants to dive into more mathematical detail,Data scientists looking to learn some basics of mathematical statistics,Undergraduate and graduate students looking for help in mathematical statistics courses,Academics and professionals wanting a strong foundation for further study in statistics