Understanding the Bivariate Normal Distribution

Irene Markelic, PhD

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4 days ago

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A Mathematical Derivation of its Probability Density Function

1 Introduction

Have you ever wondered where the cryptic formula of the multivariate normal distribution (MVN) comes from and what all its components mean? In this article, which is also available at markelic.de, you’ll learn exactly that by using the bivariate normal distribution (BVN) as an example. The normal distribution is a cornerstone of various disciplines including statistics, machine learning, and finance; therefore, understanding this distribution is crucial for many applications in these fields. We can categorize the normal distribution into three cases: The

1. univariate case: The probability density function (pdf) of a single normally distributed random variable (r.v.).
2. multivariate case: The pdf over the joint distribution of multiple r.v.s that fulfill specific criteria, which will be stated later.
3. bivariate case: A special subcase of the multivariate, involving exactly two r.v.s.

The formula for the normal distribution may seem complex at first, especially compared to the more straightforward univariate case. I have explored this topic in depth here.

In this article, we will understand the formula for the bivariate normal distribution by deriving it mathematically, which will also enhance our understanding of the general multivariate case.

I aimed to make this article as accessible and clearly structured as possible, but it assumes some prior knowledge of probability, basic linear algebra, and multivariate analysis (partial derivatives). Additionally, there are two topics that I will explain as detailed as necessary for this text, but if you seek a deeper understanding, you might want to look at:

• ”How to construct a BVN from two independent normal random variables”. In fact I wrote the latter as a pre-work for this text.
• The ”change of variables” method, which we will use to derive the BVN. An excellent description can be found in [1], p. 369.

This text is meant to be gentle in the sense that I’ll walk you through all the involved steps very explicitly. If you are looking for a more concise text you can…