Lesson 4: Multivariate Normal Distribution

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Introduction

This lesson is concerned with the multivariate normal distribution. Just as the univariate normal distribution tends to be the most important statistical distribution in univariate statistics, the multivariate normal distribution is the most important distribution in multivariate statistics.

The question one might ask is, "Why is the multivariate normal distribution so important?" There are three reasons why this might be so:

  1. Mathematical Simplicity. It turns out that this distribution is relatively easy to work with, so it is easy to obtain multivariate methods based on this particular distribution.
  2. Multivariate version of the Central Limit Theorem. You might recall in the univariate course that we had a central limit theorem for the sample mean for large samples of random variables. A similar result is available in multivariate statistics that says if we have a collection of random vectors X1, X2,...,Xn that are independent and identically distributed, then the sample mean vector, \(\bar{x}\), is going to be approximately multivariate normally distributed for large samples.
  3. Many natural phenomena may also be modeled using this distribution, just as in the univariate case.

Learning objectives & outcomes

Upon completion of this lesson, you should be able to:

  • Understand the definition of the multivariate normal distribution;
  • Compute eigenvalues and eigenvectors for a 2 × 2 matrix;
  • Determine the shape of the multivariate normal distribution from the eigenvalues and eigenvectors of the multivariate normal distribution.