STAT 415 | Introduction to Mathematical Statistics

About this course

Welcome to the course notes for STAT 415: Introduction to Mathematical Statistics. These notes are designed and developed by Penn State’s Department of Statistics and offered as open educational resources. These notes are free to use under Creative Commons license CC BY-NC 4.0.

This course is part of the Online Master of Applied Statistics program offered by Penn State’s World Campus.

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If you are a current student in this course, please see Canvas for your syllabus, assignments, lesson videos, and communication from your instructor.

Lessons

0: Review Concepts from STAT 414
Distributions
Bayes'
Probability
Central Limit Theorem
Many of the concepts from STAT 414: Introduction to Probability Theory will carry over in this course. In this unit, some of the essential formulas and theorems are presented here for review.
6/11/25

Order Statistics
Order Statistics
Probability Density Functions
We typically don’t pay particular attention to the order of a set of random variables \(X_1, X_2, \ldots, X_n\). But, what if we did? Suppose, for example, we needed to know the probability that the third largest value was…
7/15/25

Estimation (Part I)
Point Estimation
Unbiased Estimation
Bias
Variance and Mean Square
Factorization
Sufficiency
Method of Moments
This lesson introduces the fundamental concepts of point estimation, a cornerstone of statistical inference. You will learn how to estimate unknown population parameters, such as means and proportions, using sample data.…
7/15/25

Estimation (Part II)
Point Estimation
Unbiased Estimation
Bias
Variance and Mean Square
Factorization
Sufficiency
Method of Moments
Building on the foundational concepts of point estimation from Lesson 2, this lesson delves deeper into the critical concept of sufficiency and introduces the method of moments as a technique for deriving estimators. You…
7/15/25

Maximum Likelihood Estimation (MLE) (Part I)
MLE
R
In the last lesson, we introduced a way of finding an estimator using the Method of Moments. In this lesson, we will introduce another method to find an estimator, finding an estimator by maximizing the likelihood.
7/15/25

Maximum Likelihood Estimation (MLE) (Part II)
Point Estimation
Unbiased Estimation
Bias
Variance and Mean Square
Factorization
Sufficiency
Method of Moments
This lesson advances your understanding of maximum likelihood estimation (MLE) by focusing on its practical implementation, particularly through numerical methods in R. You will learn to compute MLEs for single and multiple…
7/15/25

Confidence Intervals
Confidence Intervals
In this lesson, we are going to take a brief break from MLEs. Don’t worry! We will go back to maximum likelihood estimation.
7/15/25

Asymptotic Distribution of MLE (Part I)
MLEs
Asymptotic Confidence Intervals
Numerical Approximations
Bootstrap
Let’s revisit maximum likelihood estimators (MLEs) to explore their asymptotic properties and their application in constructing confidence intervals. This lesson introduces the theoretical foundations of MLEs, including…
7/15/25

Asymptotic Distribution of MLE (Part II)
MLEs
Asymptotic Confidence Intervals
Numerical Approximations
Bootstrap
This lesson introduces the bootstrap method—both parametric and nonparametric—for estimating confidence intervals when analytical solutions are infeasible. Additionally, you will explore the Delta method and bootstrap…
7/15/25

Hypothesis Tests (Part I)
Hypothesis Tests
Tests about Proportions
Tests about one Mean
P-Value
In the previous sections, we developed statistical methods, primarily in the form of confidence intervals, for answering the question “what is the value of the parameter \(\theta\)?” In this section, we’ll learn how to…
7/15/25

Hypothesis Tests (Part II)
Power
Confidence Intervals
Wald Test
Hypothesis Tests
This lesson picks up where our first look at hypothesis testing left off. We move beyond single‑sample tests to compare two populations, explore lesser‑known examples, and show how confidence intervals and hypothesis tests…
7/15/25

Bayesian Methods
Bayesian
Subjective Probability
Credible Intervals
All of the methods we have developed and used thus far in this course have been developed using what statisticians would call a “frequentist” approach. In this section, we revisit some of those methods using what…
7/15/25

Simple Linear Regression
Regression
Simple linear regression is a way of evaluating the relationship between two continuous variables. One variable is regarded as the predictor variable, explanatory variable, or independent variable ( ). The other variable is…
7/15/25
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