Mutual information of a bivariate Gaussian

What you are seeing: the joint probability density of two correlated Gaussian random variables $X$ and $Y$. The heatmap is $p(x, y)$. The curves on the top and right are the marginals $p(x)$ and $p(y)$. The number $I(X; Y)$ in the corner measures how much knowing $X$ reduces your uncertainty about $Y$. It is zero when $X$ and $Y$ are independent and grows without bound as they become perfectly correlated.

For a 2D Gaussian with correlation $\rho$ there is a clean closed form: $$I(X; Y) = -\tfrac{1}{2}\,\ln(1 - \rho^2).$$ Slide $\rho$ to see how the heatmap stretches along the diagonal and how $I$ rises. We compute the same quantity numerically by trapezoidal integration on the same grid and report both numbers; they agree to a few percent at the default resolution.