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

Research paper by Andrés Alayón Glazunov, Jie Zhang

Indexed on: 04 May '12Published on: 04 May '12Published in: Computer Science - Networking and Internet Architecture

#### 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