Discussing the article: "Uncertainty as a Model (Part 2): Dependence Among Random Variables — From Correlation to Copulas"

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Check out the new article: Uncertainty as a Model (Part 2): Dependence Among Random Variables — From Correlation to Copulas.

The second part of the series examines the mathematical framework for multivariate random variables, which is necessary for analyzing the dependence and joint behavior of market assets. This section describes joint distribution functions, the concepts of marginal and conditional distributions, and the conditions for dependence and independence of variables. The theoretical material is based on extending the analogy between probability and mass to multidimensional space. Particular attention is given to measures of association: from classical linear covariance and correlation to modern tools such as copulas and Shannon mutual information.

A one-dimensional random variable is a function that maps outcomes ω from the space Ω to points x on the real line ℝ. Now we move on to a situation in which an entire family of variables coexists simultaneously in our probability space Ω: X1, X2, ..., Xn.

This transition can be compared to the evolution from analyzing individual points on a line to working with multidimensional vectors.

This is where the main pitfall for novice statisticians lies: in general, the distribution of a set of random variables cannot be described by a simple set of their individual distribution functions. Even if we know how asset A and asset B behave individually, we still know nothing about how they behave together. To fill this gap, we need the machinery of multivariate distribution functions.

Author: Aleksey Nikolayev