Coefficient of Variation
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. Let us 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 another normal random variable.
We could be interested, for instance, 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:
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 – \(\frac {2(2 – 1)}{2} = 1.\)
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, normally distributed, the return of a portfolio made up of such assets will also be normally distributed. As such, we can come up with a multivariate distribution provided we have the three items that we have listed above.