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

In this paper we discuss the representation of the joint probability density function of perfectly correlated continuous random variables, i.e., with correlation coefficients ρ=±1, by Dirac’s δ-function. We also show how this representation allows to define Dirac’s δ-function as the ratio between bivariate distributions and the marginal distribution in the limit ρ → ±1, whenever this limit exists. We illustrate this with the example of the bivariate Rice distribution.