Univariate and multivariate normal distributions are very robust and useful in most statistical procedures. Understanding their form and function will help you learn a lot about most statistical routines.
A univariate distribution is defined as a distribution that involves just one random variable. For instance, suppose we wish to model the distribution of returns on an asset, such as a holding of stocks; such a model would be a univariate distribution. In previous learning outcome statements, we have been focusing on univariate distributions such as the binomial, uniform, and normal distributions. We now look at multivariate distributions:
A multivariate distribution describes the probabilities for a group of continuous random variables particularly if the individual variables follow a normal distribution. Each variable has its own mean and variance. In this regard, the strength of the relationship between the variables (correlation) is very important. As you will recall, a linear combination of 2 normal random variables results in the creation of another normal random variable.
The Role of Correlation in Multivariate Normal Distributions
We could be interested in the distribution of returns on a group of assets. Correlation defines the strength of the linear relationship between any 2 random variables. For us to define a multivariate distribution (n variables), we need the following:
- The individual mean value for each asset (μ1, μ2, μ3, …, μn-1, μn)
- A list of variances of return for each asset (σ12, σ22, σ32, …, σn-12, σn-12)
- Pairwise return correlations – n(n-1)/2 correlations in total.
Correlation is the distinguishing feature between univariate and multivariate normal distributions.
Suppose we wish to model the distribution of two asset returns: To describe the return multivariate distribution, we will need two means, two variances and just one correlation – 2(2 – 1)/2 = 1.
On the other hand, if we had 5 assets, we would need to establish 5 means, 5 variances, and 10 distinct correlation values – 5(5 – 1)/2 = 10.
In conclusion, if we have individual security returns that are jointly, normally distributed, the return of a portfolio made up of such assets will also be normally distributed and we can come up with a multivariate distribution provided we have the three items that we have listed above.
Reading 10 LOS 10j:
Distinguish between a univariate and a multivariate distribution, and explain the role of correlation in the multivariate normal distribution.