# Student-t distribution

(35 minutes to learn)

## Summary

The student-t distribution is a continuous probability distribution motivated by estimating the mean of a Gaussian population with unknown variance.

## Context

This concept has the prerequisites:

- Gaussian distribution (The student-t distribution is motivated in terms of the Gaussian distribution.)

## Goals

- Understand how the student-t distribution is defined in terms of samples drawn from a normal distribution.

- Know how the shape of the distribution depends on the degrees of freedom (few degrees of freedom gives a heavy-tailed distribution, many degrees of freedom means approximately normal)

## Core resources (read/watch one of the following)

## -Paid-

→ Probability and Statistics

An introductory textbook on probability theory and statistics.

Location:
Section 8.4, "The t distributions," pages 480-484

→ Mathematical Statistics and Data Analysis

An undergraduate statistics textbook.

Location:
Section 6.2, "Chi-squared, t, and F distributions," pages 192-194

## See also

- The student-t distribution is the predictive distribution for Bayesian parameter estimation of Gaussian distributions with unknown variance .
- The distribution is related to the t-test from frequentist statistics.
- This is an example of a heavy-tailed distribution .
- In the multivariate case, the t-distribution can be generalized to Gaussian scale mixtures .