Probability and Statistics for Finance

Chapter 14. Joint Probability Distributions

In previous chapters, we explained the properties of a probability distribution of a single random variable; that is, the properties of a univariate distribution. Univariate distributions allow us to analyze the random behavior of individual assets, for example. In this chapter, we move from the probability distribution of a single random variable (univariate distribution) to the probability law of two or more random variables, which we call a multivariate distribution. Understanding multivariate distributions is important in financial applications such as portfolio selection theory, factor modeling, and credit risk modeling, where the random behavior of more than one quantity needs to be modeled simultaneously. For example, Markowitz portfolio theory builds on multivariate randomness of assets in a portfolio.

We begin this chapter by introducing the concept of joint events and joint probability distributions with the joint density function. For the latter, we present the contour lines of constant density. From there, we proceed to the marginal probability distribution, followed by the immensely important definition of stochastic dependence. As common measures of joint random behavior, we will give the covariance and correlation parameters along with their corresponding matrices.

In particular, we present as continuous distributions the multivariate normal and multivariate Student's t-distribution, as well the elliptical distributions. All ...