xbeta
A random vector X n Normal distribution with mean μ Σ Σ positive definite and its density function is given by
(1) f ( x ) = 1 ( 2 π ) n / 2 ∣ Σ ∣ 1 / 2 e − 1 2 ( x − μ ) ⊤ Σ − 1 ( x − μ ) . f(x) = \frac{1}{(2\pi)^{n/2} \left\vert\Sigma \right\vert^{1/2}}
e^{-\frac{1}{2}(x - \mu)^\top \Sigma^{-1} (x - \mu)}.
In this case, we write
X ∼ N ( μ , Σ ) . X \sim N(\mu,\,\Sigma).
Properties
Regression
Suppose that X Normal distribution and partition the vector as X = ( X 1 , X 2 )
(2) ( X 1 X 2 ) ∼ N [ ( μ 1 μ 2 ) , ( Σ 11 Σ 12 Σ 21 Σ 22 ) ] . \begin{pmatrix} X_1 \\ X_2 \end{pmatrix} \sim N\left[ \begin{pmatrix} \mu_1 \\ \mu_2 \end{pmatrix}, \begin{pmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \end{pmatrix} \right].
The conditional distribution of X 1 X 2 = x 2
(3) X 1 ∣ X 2 = x 2 ∼ N ( μ 1 + Σ 12 Σ 22 − 1 ( x 2 − μ 2 ) , Σ 11 − Σ 12 Σ 22 − 1 Σ 21 ) . X_1 \vert X_2 = x_2 \sim N\left(\mu_1 + \Sigma_{12}\Sigma_{22}^{-1}(x_2 - \mu_2),\, \Sigma_{11} - \Sigma_{12}\Sigma_{22}^{-1} \Sigma_{21} \right).
Conversely, whenever (3) holds and when the marginal distribution of X 2 X 2 ∼ N ( μ 2 , Σ 22 ) ( X 1 , X 2 ) (2) .
See Also