This experiment visualizes the effect of correlation on the joint probability distribution of a noise signal.

The 2D Joint Distribution

• The main plot shows the 2D joint probability distribution of two random variables, \(X_1\) and \(X_2\).
• Uncorrelated (\(\rho \approx 0\)): When two variables are statistically independent, their joint distribution is circularly symmetric. The value of \(X_1\) does not depend on the value of \(X_2\).
• Positively Correlated (\(\rho > 0\)): The distribution becomes an ellipse tilted to the right. This shows that a high value for \(X_1\) makes a high value for \(X_2\) more likely.
• Negatively Correlated (\(\rho < 0\)): The distribution is an ellipse tilted to the left. This shows that a high value for \(X_1\) makes a low value for \(X_2\) more likely (and vice versa).

The Covariance Matrix

The relationship between \(n\) random variables is described by an \(n \times n\) covariance matrix. The circular and elliptical contours you see are 2D representations of this matrix, which is displayed in the control panel.

How to Interact

• Use the "Correlation Strength" slider to change the correlation coefficient \(\rho\) from -0.95 to +0.95.
• Observe how the contours change from circles to ellipses and how the tilt of the ellipses changes based on the sign of \(\rho\).
• Use the Start/Stop button to pause the visualization.