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Normal distribution

A random vector X of dimension n is said to have a Normal distribution with mean μ and covariance matrix Σ if Σ is positive definite and its density function is given by

(1)f(x)=1(2π) n/2Σ 1/2e 12(xμ) Σ 1(xμ).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

XN(μ,Σ).X \sim N(\mu,\,\Sigma).

Properties

Regression

Suppose that X is a random vector with a multivariate Normal distribution and partition the vector as X=(X 1,X 2) so that

(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 given X 2=x 2 is

(3)X 1X 2=x 2N(μ 1+Σ 12Σ 22 1(x 2μ 2),Σ 11Σ 12Σ 22 1Σ 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 satisfies X 2N(μ 2,Σ 22), then the joint distribution of (X 1,X 2) is given by (2).

See Also