Wolfram Demonstrations Project

The Pearson correlation coefficient, denoted

, is a measure of the linear dependence between two random variables, that is, the extent to which a random variable

can be written as

, for some

and some

. This Demonstration explores the following question: what correlation coefficients are possible for a random vector

, where

is a Bernoulli random variable with parameter

and

is a Bernoulli random variable with parameter

? Interestingly, a two-dimensional Bernoulli random vector has a correlation coefficient that is constrained by the choices of

and

Contributed by: Jeff Hamrick

For the sake of simplicity, you are constrained to choose

and

that are slightly bounded away from 0 and 1.

In the upper left-hand corner, there is a possible joint distribution of

that accommodates your choices of

, and

. Additionally, at the bottom is shown (with a red line) the linear correlation coefficients that are attainable for fixed choices of

and

. The basic lesson is clear: it is not possible to couple together two arbitrary Bernoulli random variables in such a way that any conceivable linear correlation coefficient is attained. Notice that choosing

maximizes the range of possible values of

For a two-dimensional random vector

, where both

and

are discrete random variables, finding a joint probability distribution that triggers a particular linear correlation coefficient is equivalent to solving a system of linear equations. It is not surprising, then, that sometimes this system of linear equations has no solution, a unique solution, or infinitely many solutions.

If in the

-dimensional random vector

the

are Gaussian, then finding a joint probability distribution with a particular variance-covariance matrix (and, therefore, particular pairwise linear correlation coefficients between the coordinates of the

-dimensional random vector) also happens to be equivalent to solving a system of linear equations. This fact is true for any elliptical distribution as well, that is, a distribution with a density function whose constant curves are ellipsoids. In general, however, the dependence between two or more random variables cannot be captured by a single number.