(PP 6.8) Marginal distributions of a Gaussian

One of the weird and wonderful things is: the family of Gaussians is preserved under many different operations.

Proposition.(Marginalization)

(PP 6.9) Conditional distributions of a Gaussian

Proposition.(Conditional)

(PP 6.10) Sum of independent Gaussians

Proposition. If $X \sim N(\mu_X, C_X)$ and $X \sim N(\mu_X, C_X)$, then $X+Y \sim N(\mu_X + \mu_Y, C_X + C_y)$.

Remark.

  • $E(X+y) = E(X) + X(Y)$
  • ${\rm Cov}(X+Y) = {\rm Cov}(X) + {\rm Cov}(Y)$