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
\[ \Covar(X, Y) \;\;=\;\; \rho \sigma_X \sigma_Y \]To prove this, we will use the earlier result that
\[ E[g(X,Y)] \;\;=\;\; E \big[E[g(X,Y) \mid X] \big] \]where the inner expected value is from the conditional distribution of \(Y\) given \(X\). For variables with a bivariate normal distribution, we showed that this marginal distribution is
\[ \NormalDistn\left(\mu_Y + \frac{\sigma_Y}{\sigma_X}\rho(x-\mu_X),\; (1-\rho^2)\sigma_Y^2\right) \]The covariance is therefore
\[ \begin{align} \Covar(X, y) \;\;&=\;\; E\left[(X - \mu_X)(Y - \mu_Y)\right] \\[0.4em] &=\;\; E \big[E[(X - \mu_X)(Y - \mu_Y) \mid X] \big] \\[0.4em] &=\;\; E \big[(X - \mu_X) \times E[(Y - \mu_Y) \mid X] \big] \\ &=\;\; E \Big[(X - \mu_X) \times \frac{\sigma_Y}{\sigma_X}\rho(X-\mu_X) \Big] \\ &=\;\; \frac{\sigma_Y}{\sigma_X}\rho \times E \big[(X - \mu_X)(X-\mu_X) \big] \\[0.4em] &=\;\; \frac{\sigma_Y}{\sigma_X}\rho \times \sigma_X^2 \\[0.4em] &=\;\; \rho \sigma_X \sigma_Y \end{align} \]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 \(\rho\).
This explains the use of the symbol \(\rho\) for the bivariate normal distribution's fifth parameter.