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.