AIM:  Distinguish between univariate and multivariate probability density functions

A single variable (univariate) probability distribution is concerned with only a single random variable; e.g., roll of a die, default of a single obligor. A multivariate probability density function concerns the outcome of an experiment with more than one random variable. This includes, the simplest case, two variables (i.e., a bivariate distribution).

Note that because the events above are mutually exclusive and cumulatively exhaustive, the sum of all probabilities must equal 100%. In mathematical terms, this can be expressed as follows for the discrete and continuous cases:

Examples of joint distributions

For example, consider two stocks. Assume that both Stock (S) and Stock (T) can each only reach three price levels. Stock (S) can achieve: $10, $15, or $20. Stock (T) can achieve: $15, $20, or $30.

Historically, assume we witnessed 26 outcomes and they were distributed as follows. Note S = S_$10/15/20 and T = T_$15/20/30 :

A joint probability is the probability that both random variables will have a certain outcome; e.g., given the above distribution, the joint probability P(S=$20, T=$30) = 3/26.

The above is an example of discrete random variables. But the variables could also be continuous.

• Tip:  A copula is a multivariate probability distribution that transforms several univariate distributions.

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