###### Assumptions Underlying Linear Regression

Assume that we have samples of size \(n\) for dependent variable \(Y\) and... **Read More**

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:

- the individual mean value for each asset (μ
_{1}, μ_{2}, μ_{3}, …, μ_{n-1}, μ_{n}); - a list of variances of return for each asset (\(\sigma_1^2, \sigma_2^2, \sigma_3^2, …, \sigma_{n}^2, \sigma_{n-1}^2)\); and
- pairwise return correlations – \(\cfrac {n(n-1)}{2}\) 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 – \(\frac {2(2 – 1)}{2} = 1.\)

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.