▶One variable
For a scalar Gaussian, the variance σ² controls spread and σ has the same physical unit as X.
Changing μ moves the curve. Changing σ changes its width and peak while the total area remains one.
Gaussian Distributions
A probability model determined by center and shape
A Gaussian distribution is special because its mean and covariance determine the entire distribution, including exact marginal and conditional forms.
meanμ
+covarianceΣ
→Gaussian modelN(μ, Σ)
The essential idea: for a Gaussian model, center, spread, and dependence are enough to determine the full probability distribution.
One Concept · One Example
What to notice
- The mean μ sets the center.
- The covariance Σ controls spread and dependence.
- Gaussian marginals and conditionals are still Gaussian.
One model, two levels
Several variables
For a Gaussian vector, μ is a vector and Σ is a covariance matrix.
The diagonal entries of Σ are variances. Off-diagonal entries describe how components vary together.
Gaussian closure: select components or condition on known ones, and the resulting distribution remains Gaussian.
How do μ and σ shape a Gaussian?
Move the center, change the spread, and place a threshold. The curve and its tail probability update together.
The red line marks the mean. Blue markers show one and two standard deviations. The shaded region is a probability, not merely a curve height.
What makes a joint Gaussian especially useful?
Marginalization only selects the relevant entries of μ and Σ. Conditioning also has an exact formula, so known information changes the center and reduces the remaining variance.
Condition on one variable, update the other
Use the numerical example from the video: μ = (1, 2) and Σ = [[1, 1.4], [1.4, 4]]. Move the known value x₂ and watch the conditional distribution of x₁ respond.
The blue band selects samples with x₂ close to the observed value. The analytic conditional distribution does not depend on the finite sample count.
Marginal distribution of x₁
variance = 1.00
Conditional x₁ | x₂ = 3.0
variance = 0.51
Selected sample band25 of 650 pointsConditional mean1.35Conditional variance0.51
Mean moves with the observation1 + (1.4/4)(x₂ − 2) = 1.35
Variance becomes smaller1 − 1.4²/4 = 0.51
What to notice: the marginal distribution of x₁ never changes. The conditional mean follows x₂ because the variables covary, while the conditional variance stays at 0.51 and is smaller than the marginal variance 1.
Why the closed form matters
Marginalize
Keep the required components of the mean vector and covariance matrix. No numerical integration is needed.
drop unused components → Gaussian marginalCondition
Use covariance to shift the mean and reduce the variance after another component is known.
known value + dependence → Gaussian conditionalFor joint Gaussians, marginal and conditional distributions remain Gaussian. Their means and covariances are available through exact formulas, so the effects of selecting variables or learning a value remain visible and computable.
joint modelN(μ, Σ)
→marginalselect entries
+conditionalupdate with Σ