A note on the bivariate distribution representation of two perfectly correlated random variables by Dirac's $\delta$-function

TL;DR

This paper explores representing the joint distribution of perfectly correlated continuous random variables using Dirac's delta function, providing a new perspective on their mathematical relationship and limits.

Contribution

It introduces a novel representation of perfectly correlated variables' joint distribution via Dirac's delta, linking it to the ratio of bivariate and marginal distributions as correlation approaches ±1.

Findings

Representation of perfect correlation using Dirac's delta function.

Limit of the ratio of bivariate to marginal distribution as correlation approaches ±1.

Application to the bivariate Rice distribution.

Abstract

In this paper we discuss the representation of the joint probability density function of perfectly correlated continuous random variables, i.e., with correlation coefficients $\rho=pm1$, by Dirac's $\delta$-function. We also show how this representation allows to define Dirac's $\delta$-function as the ratio between bivariate distributions and the marginal distribution in the limit $\rho\rightarrow \pm1$, whenever this limit exists. We illustrate this with the example of the bivariate Rice distribution