We now generalise our earlier definition of the standard bivariate normal distribution.

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

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

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

This is a standard bivariate normal distribution if \(\mu_X = \mu_Y = 0\) 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

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

(This matrix notation gives a hint about how the bivariate normal distribution can be generalised further to the joint distribution of more than two random variables.)

Scale and location parameters

The parameters \(\mu_X\) and \(\mu_Y\) are location parameters of the bivariate normal distribution — they shift the joint pdf along the x- and y-axes. The parameters \(\sigma_X\) and \(\sigma_Y\) are scale parameters — they change the spread of x- and y-values but leave other aspects of the distribution's shape unchanged.

We simply state the following result without proof.

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

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

The following diagram shows how the five parameters affect the shape of the bivariate normal distribution's pdf.

Parameters of the bivariate normal distribution

The sliders below the joint normal probability density function adjust the distribution's five parameters. Observe how