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Moment Generating Function Tutorial

A tutorial diving into the Moment Generation Function for probability distribution with a complete derivation and code examples in Python

Author(s): Pratik Shukla, Roberto Iriondo

Last updated, January 8, 2021

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This tutorial’s code is available on Github and its full implementation as well on Google Colab.

Table of Contents:

1. Moments in Statistics.
2. Raw Moments.
3. Centered Moments.
4. Standardized Moments.
5. Moment Generating Function.
6. Proof of Moment Generating Function.
7. Derivation of Relationship between Raw and Central Moments.
8. Python Implementation.

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What is a Moment in Statistics?

We generally use moments in statistics, machine learning, mathematics, and other fields to describe the characteristics of a distribution.

Let’s say the variable of our interest is X then, moments are X’s expected values. For example, E(X), E(X²), E(X³), E(X⁴),…, etc.

Figure 1: Moments in Statistics.

Moments in statistics:

1) First Moment: Measure of the central location.

2) Second Moment: Measure of dispersion/spread.

3) Third Moment: Measure of asymmetry.