Moments of Student's t-distribution: A Unified Approach

TL;DR

This paper derives unified closed-form formulas for the moments of Student's t-distribution in one and higher dimensions, expressed through special functions like Gamma and hypergeometric functions.

Contribution

It provides a novel unified framework to compute moments of Student's t-distribution in multiple dimensions using special functions.

Findings

Closed-form expressions for moments in one dimension.

Extension to higher dimensions.

Formulas involving Gamma and hypergeometric functions.

Abstract

In this note, we derive the closed form formulae for moments of Student's t-distribution in the one dimensional case as well as in higher dimensions through a unified probability framework. Interestingly, the closed form expressions for the moments of Student's t-distribution can be written in terms of the familiar Gamma function, Kummer's confluent hypergeometric function, and the hypergeometric function.