The 2D Gaussian, marginals and conditionals

Move the mean, the spreads and the correlation. The green line conditions on a value of x₁.

A 2D Gaussian N(μ, Σ)

μ₁μ₂ σ₁σ₂ correlation ρ condition on x₁ =

Try thisSet ρ = 0: the green conditional curve is the same as the grey marginal (knowing x₁ tells you nothing about x₂). Push ρ to 0.95: the conditional gets very narrow.