Procedure for the Experiments

This section contains three sub-experiments designed to enhance the student's understanding of Gaussian Random Variables/Vectors and their properties.

Experiment 1: 1D Gaussian PDF Visualization and Realizations

1. Use the "Mean (μ)" and "Variance (σ²)" sliders to shape the Gaussian PDF curve.
2. Calculate the peak height for the current variance, enter it in the input box, and click "Check Answer" to verify.
3. Select the desired number of data points with the "Number of Samples" slider.
4. Click "Generate Samples" to draw random realizations from your configured distribution on the chart.
5. Observe the percentage of samples that fall within the two-sigma range ([μ - 2σ, μ + 2σ]), indicated by the green lines, and compare it to the theoretical value of 95.45%.

Experiment 2: 2D Gaussian PDF Visualization

1. Input the desired center of the distribution into the two fields of the "Mean Vector".
2. Input the four values for the 2x2 "Covariance Matrix". Note that this matrix must be symmetric and positive-definite for a valid PDF.
3. Click the "Update 2D Gaussian" button to render the distribution.
4. Two plots are generated: a 3D surface plot and a 2D contour plot. The user can click and drag to rotate the 3D plot and use the scroll wheel to zoom.
5. Observe how changing the mean vector repositions the entire distribution, while adjusting the covariance matrix values changes its shape, spread, and orientation.
6. Any input errors or parameter observations will be displayed in the "Observations" section.