The most widely used family of continuous bivariate distributions is the family of bivariate normal distributions. Their joint probability density function is quite complex, so we will start by examining a special case, the standard bivariate normal distribution.
Definition
Two continuous random variables, \(X\) and \(Y\), are said to have a standard bivariate normal distribution with parameter \(\rho\),
\[ (X,Y) \;\;\sim\;\; \NormalDistn(0, 1, 0, 1, \rho) \]if \(-1 \lt \rho \lt 1\) and their joint pdf is
\[ f(x,y) \;\;=\;\; \frac{1}{2\pi\sqrt{1 - \rho^2}} \exp\left(-\frac{1}{2(1-\rho^2)} \left(x^2 + y^2 - 2\rho x y\right)\right) \]for all \(-\infty \lt x \lt \infty\) and \(-\infty \lt y \lt \infty\).
We will now display this joint pdf graphically.
Shape of joint probability density
The diagram below shows the joint probability density function of the standard bivariate normal distribution. The diagram is 3-dimensional and can be rotated using the buttons underneath or by dragging the centre of the pdf.
Initially the parameter is \(\rho = 0\) so the pdf is
\[ f(x,y) \;\;=\;\; \frac{1}{2\pi} \exp\left(-\frac{1}{2} \left(x^2 + y^2\right)\right) \]This is a function of \(\left(x^2 + y^2\right)\) so any value for the pdf corresponds to values of \((x,y)\) that lie on a circle,
\[\left(x^2 + y^2\right) \;\;=\;\; k \]Drag the black arrow in the coloured band at the left to show these circles — contours of the surface. (You might prefer to rotate the 3-dimensional diagram by dragging its centre to look down on the surface to see these curcular contours.)
Now use the slider to adjust the value of \(\rho\). The contours now correspond to
\[ \left(x^2 + y^2 - 2\rho x y \right) \;\;=\;\; k \]which define ellipses.
Strength of the relationship between X and Y
The value of the parameter \(\rho\) determines how strongly \(X\) and \(Y\) are related to each other.
We will show later that \(\rho\) is actually the correlation coefficient between \(X\) and \(Y\).