Interactive Bivariate Gaussian Distribution

Mathematical Framework

Mean Vector (μ): Defines the central location of the distribution in the 2D space. Geometrically, this represents the center of mass of the distribution.

Covariance Matrix (Σ): A symmetric positive-definite matrix that characterizes the shape, spread, and orientation of the distribution.

• Diagonal elements (σ_xx, σ_yy) represent the variances along the principal axes
• Off-diagonal elements (σ_xy = σ_yx) quantify the covariation between dimensions

Correlation Coefficient (ρ): A normalized measure of linear dependency between the variables, defined as ρ = σ_xy/(σ_xσ_y).

• ρ = 0: The variables are linearly independent (principal axes align with coordinate axes)
• ρ > 0: Positive correlation (elliptical contours oriented along the diagonal from lower left to upper right)
• ρ < 0: Negative correlation (elliptical contours oriented along the diagonal from upper left to lower right)
• |ρ| → 1: As |ρ| approaches 1, the distribution becomes increasingly concentrated along a line

Probability Density Function

This function defines a surface over the X-Y plane where the height at any point (x,y) represents the probability density. The volume under the entire surface equals 1.

Conditional Distribution

For bivariate Gaussian distributions, the conditional distribution of Y given X=x is also Gaussian. This represents a vertical slice through the joint distribution at a specific value of X.

Note that the conditional mean depends linearly on x, while the conditional variance is reduced by a factor of (1-ρ²) compared to the marginal variance of Y. This variance reduction quantifies the information gained about Y when X is known.

Marginal Distributions

The marginal distributions represent the probability distributions of each variable individually, obtained by integrating out the other variable from the joint distribution.

For a bivariate Gaussian, the marginal distributions are univariate Gaussians with means and variances corresponding to the respective components of the mean vector and the diagonal elements of the covariance matrix.

Geometrical Interpretation

The contours of constant density for a bivariate Gaussian distribution form ellipses. The principal axes of these ellipses are determined by the eigenvectors of the covariance matrix, while their lengths are proportional to the square roots of the corresponding eigenvalues.

As the correlation coefficient ρ approaches ±1, these ellipses become increasingly elongated. When ρ = 0, the principal axes align with the coordinate axes, indicating statistical independence between the variables.

This interactive visualization allows you to explore how the distribution's parameters affect its shape, orientation, and statistical properties.