Procedure for the experiments in the following section

In the experiments section there are 3 sub-experiments which enhance the students understanding of Gaussian Random Vectors and their properties.

Sub Experiment (1D and 2D Gaussian Visualization) :

In this subexperiment, there are 2 parts, one for 1D Gaussian Random Variable and the other for Bi-variate Gaussiang Random vector. The subexperiments aim to help visualize the nature of the Probability Density Function (PDF) of both 1D and 2D gaussian random variables and how the parameters affect the nature of the curve.

• The first part presents the user with sliders for mean and variance for 1D Gaussian Random Variable and observe how the nature of the PDF changes in real-time.

• The user is also given an input field for the height of the gaussian, which can be calculated by obtaining the value of the PDF at x=0. An error margin of 5% is allowed in the answer.

• The result of the experiment entered is displayed in the Observation section.

• Similar to the first part, the 2nd part takes input the mean vector (consisting of 2 values) and the corelation matrix (2x2 symmetric matrix consisting of 4 values). Based on the values entered, 2D gaussian is visualized in both 2D and contour plots.

• The user can zoom in/out and move around the 2D Gaussian Plots and observe how changeing the mean vector affects the position of the pdf and the covariance matrix affects the shape of the pdf.

• Any errors in input fields or values are displayed in the observation section.

Sub Experiment (Standard Normal Realizations) :

This sub experiment helps users to build their understanding of the Standard Normal Distribution and how most of the Random values generated lie around the mean, within $-2\sigma$ to $2\sigma$ interval on the x-axis.

• The user is given a plot of the 1D-Standard Normal Distribution (Gaussian pdf mean 0 and variance 1), and two lines signifying the $-2\sigma$ and $2\sigma$ boundaries.

• A slider allows the user to control the number of samples to be generated and plotted on the graph. The user can plot the samples using the Generate Samples button.

• Ideally, the percentage of samples lying in the $-2\sigma$ to $2\sigma$ range should converge to 95.4%.

• As the number of samples increase, the approximation of fraction of samples lying in the $-2\sigma$ to $2\sigma$ range becomes more accurate.