2D Gaussian distribution explorer

Adjust the mean and covariance parameters to explore the shape of a bivariate normal distribution.

Probability density function $$p(\mathbf{x}) = \frac{1}{(2\pi)|\boldsymbol{\Sigma}|^{1/2}} \exp\!\left(-\tfrac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1}(\mathbf{x}-\boldsymbol{\mu})\right), \qquad \boldsymbol{\mu} = \begin{pmatrix} \mu_1 \\ \mu_2 \end{pmatrix}, \qquad \boldsymbol{\Sigma} = \begin{pmatrix} \sigma_{11} & \sigma_{12} \\ \sigma_{12} & \sigma_{22} \end{pmatrix}$$

3D surface

2D heatmap & contours

Mean $\boldsymbol{\mu}$

$\mu_1$

$\mu_2$

Covariance matrix $\boldsymbol{\Sigma}$

$\sigma_{11}$ (variance of $x_1$)

$\sigma_{22}$ (variance of $x_2$)

$\sigma_{12} = \sigma_{21}$ (covariance)

Created by Adarsh Pyarelal with assistance from Claude · INFO 521, Spring 2026 · University of Arizona