Mathematical Statistics Lecture May 2026

A set ( X_1, X_2, \dots, X_n ) is a random sample if the RVs are:

Set sample moments equal to population moments and solve for parameters.

Example (Normal ( N(\mu, \sigma^2) )):

Idea: Equate the population moments to the sample moments and solve for the parameters. mathematical statistics lecture

Procedure: If you have $k$ parameters to estimate, set the first $k$ population moments equal to the first $k$ sample moments and solve the system of equations.

Mathematical Statistics is the branch of applied mathematics that provides the theoretical underpinning for data analysis. Unlike descriptive statistics (which simply summarizes data), mathematical statistics develops methods for inference—drawing conclusions about a population based on a sample.

The core question: Given observed data, what can we say about the unknown process that generated it? A set ( X_1, X_2, \dots, X_n )

You cannot transcribe the board. You must filter.

What happens when the sample size ( n \to \infty )?

Idea: What value of $\theta$ makes the data we actually observed most probable? This is the "gold standard" of estimation. Procedure: If you have $k$ parameters to estimate,

Procedure:

Properties of MLE: