To find the correlation coefficient of the bivariate normal distribution, we first find the covariance between the two variables.

Covariance

If \(X\) and \(Y\) are bivariate normal,

\[ (X,Y) \;\;\sim\;\; \NormalDistn(\mu_X, \sigma_X^2, \mu_Y, \sigma_Y^2, \rho) \]

then their covariance is

\[ \Covar(X, Y) \;\;=\;\; \rho \sigma_X \sigma_Y \]

(Proved in full version)

We can now find the correlation between \(X\) and \(Y\).

Correlation coefficient

If \(X\) and \(Y\) are bivariate normal,

\[ (X,Y) \;\;\sim\;\; \NormalDistn(\mu_X, \sigma_X^2, \mu_Y, \sigma_Y^2, \rho) \]

then their correlation is \(\rho\).

(Proved in full version)

This explains the use of the symbol \(\rho\) for the bivariate normal distribution's fifth parameter.