Univariate Distribution, Multivariate Distribution, and Correlation

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.

Univariate Distributions

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. Let us now look at multivariate distributions.

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 case, the strength of the relationship between the variables (correlation) is very important. As you will recall, a linear combination of two normal random variables results in another normal random variable.

The Role of Correlation in Multivariate Normal Distributions

We could be interested, for instance, in the distribution of returns in a group of assets. Correlation defines the strength of the linear relationship between any two random variables. For us to define a multivariate distribution with \(n\) variables, we need the following:

1. The individual mean value for each asset (\(μ_1, μ_2, μ_3, …, μ_n\)).
2. A list of variances of return for each asset (\(\sigma_1^2, \sigma_2^2, \sigma_3^2, …, \sigma_{n}^2\).
3. Pairwise return correlations, so \(\cfrac {n(n-1)}{2}\) correlations in total.

Correlation is the distinguishing feature between univariate and multivariate normal distributions.

Example: Multivariate Normal Distributions

Suppose we wish to model the distribution of two asset returns so as to describe the return multivariate distribution; we will need two means, two variances, and just one correlation, \(\frac {2(2 – 1)}{2} = 1.\)

On the other hand, if we had 5 assets, we would need to establish 5 means, 5 variances, and 10 distinct correlation values, \(\frac {5(5 – 1)}{2} = 10.\)

In conclusion, if we have individual security returns that are jointly distributed normally, the return of a portfolio made up of such assets will also be distributed normally. In our analysis of such a portfolio, we can come up with a multivariate distribution provided we have the three items that we have listed above (mean, variance, and correlation).

Question

Suppose we wish to model the distribution of three asset returns so as to describe the return multivariate distribution; the number of distinct correlations needed is closest to:

1. 6.
2. 10.
3. 40.

Solution

The correct answer is B.

Recall that to define a multivariate distribution with \(n\) variables, we need:

$$\cfrac {n(n-1)}{2} \ \text{distinct correlation values}$$

In this case we need:

$$\frac {3(3 – 1)}{2} = 6\ \text{distinct correlation values}.$$