We now generalise.

Definition

Two continuous random variables, \(X\) and \(Y\), are said to have a bivariate normal distribution with parameters \(\mu_X, \sigma_X^2, \mu_Y, \sigma_Y^2\) and \(\rho\),

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

if \(-1 \lt \rho \lt 1\) and their joint pdf is

\[ \begin{align} f(x,y) \;\;=\;\; &\frac{1}{2\pi\sigma_X\sigma_Y\sqrt{1 - \rho^2}} \\[0.4em] &\times\exp\left(-\frac{1}{2(1-\rho^2)} \left[\frac{(x-\mu_X)^2}{\sigma_X^2} + \frac{(y-\mu_Y)^2}{\sigma_Y^2} - 2\rho \frac{(x-\mu_X)(y - \mu_Y)}{\sigma_X \sigma_Y}\right]\right) \end{align} \]

for all \(-\infty \lt x \lt \infty\) and \(-\infty \lt y \lt \infty\).

The standard bivariate normal distribution arises when \(\mu_X = \mu_Y = 0\) and \(\sigma_X = \sigma_Y = 1\).

Matrix notation

The formula for the joint pdf of \(X\) and \(Y\) can be simplified if matrix notation is used. If we write

\[ \mathbf{x} = \left[\begin{array}{c} x \\ y\end{array}\right] , \quad \mathbf{\mu} = \left[\begin{array}{c} \mu_X \\ \mu_Y\end{array}\right] , \spaced{and} \mathbf{\Sigma} = \left[\begin{array}{cc} \sigma_X^2 & \rho \sigma_X \sigma_Y\\ \rho \sigma_X \sigma_Y & \sigma_Y^2\end{array}\right] \]

then

\[ f(x,y) \;\;=\;\; \frac{1}{\sqrt{(2\pi)^2 \left| \Sigma \right|}} \exp\left(-\frac{1}{2}(\mathbf{x} - \mathbf{\mu})^T \Sigma^{-1}(\mathbf{x} - \mathbf{\mu})\right) \]

Scale and location parameters

\(\mu_X\) and \(\mu_Y\) are location parameters and \(\sigma_X\) and \(\sigma_Y\) are scale parameters.

Relationship with standard normal distribution

If \((Z_X,Z_Y) \sim \NormalDistn(0, 1, 0, 1, \rho)\), then

\[ X = \mu_X + \sigma_X Z_X \spaced{and} Y = \mu_Y + \sigma_Y Z_Y \]

have the bivariate normal distribution

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