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A note on the bivariate distribution representation of two perfectly correlated random variables by Dirac's $δ$-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 $ρ=pm1$, 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 $ρ\rightarrow \pm1$, whenever this limit exists. We illustrate this with the example of the bivariate Rice distribution