Univariate Vs. Multivariate Distribution

A univariate distribution refers to the distribution of a single random variable. Note that the above characteristics we saw of a normal distribution are for the distribution of one normal random variable, representing a univariate distribution.

On the other hand, a multivariate distribution refers to the probability distribution of a group of random variables. For example, a multivariate normal distribution is used to specify the probabilities of returns of a group of n stocks. This has relevance because the returns of different stocks in the group influence each other’s behaviour, that is, the behaviour of one random variable in the group is influenced by the behaviour of another variable.

How to Construct Multivariate Distribution?

For discrete random variables, joint probabilities are used to describe the multivariate distribution

For continuous random variables, if each random variable follows a normal distribution, a multivariate normal distribution is created. Remember that a linear combination of 2 or more normally distributed random variables is also normally distributed.

If we want to describe the multivariate normal distribution of the returns of a group of stocks, we need the following three parameters:

  • List of means returns of each stock
  • List of variances of returns of each stock
  • List of correlations between each pair of stocks.

A univariate normal distribution is described using just the two variables namely mean and variance. For a multivariate distribution we need a third variable, i.e., the correlation between each pair of random variables. This is what distinguishes a multivariate distribution from a univariate distribution. If there are n random variables in the group, we will have n*(n-1)/2 pairs of correlations.

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