Conditional information using copulas with an application to decision making

Abstract

We focus on the task of calculating conditional probabilities of the form Prob(U≤x|V≤y). We point out that in this case the relevant probabilities, Prob(U≤x) and Prob(V≤y), have the nature of a cumulative distribution. This enables us to use the Sklar theorem to directly calculate the required joint probability as a simple binary aggregation of these marginals using a copula. Here the choice of copula reflects the type of correlation between U and V. We study in considerable detail the effects of using different copulas. We also show that this enables us to simply and directly calculate the probability that U=x conditioned on the knowledge of the Prob(V≤y). We use this result to aid in decision-making where we compare alternative's expected payoffs based on the conditioned probabilities.