Chapter 7 The Multivariate Normal Distribution

The multivariate normal distribution (MVN) generalises the univariate normal distribution from scalar to vector random variables. It is important for a number of reasons:

  1. It is entirely defined by its mean vector \({\boldsymbol{\mu}}\) and its covariance matrix \(\boldsymbol{\Sigma}\).
  2. Zero correlation implies independence.
  3. Linear functions of multivariate normal vectors are also multivariate normal vectors.
  4. The multivariate version of the Central Limit Theorem means that it appears naturally throughout statistics.
  5. It has simple geometric properties, and is easy to work with mathematically.

In this chapter we’ll define the MVN and look at some its properties. We’ll then look at multivariate analogues of the t-test for comparing the mean of different populations. This will involve us defining two important the distributions:

  • Wishart distribution which we can think of as the multivariate \(\chi^2\) distrbution, and which gives the distribution of sample covariance matrices.

  • Hotelling’s \(T^2\) distribution, which is the multivariate version of Student’s t-distribution.

The videos for this chapter, in the order I recommend you watch, are available at