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Normal distribution

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A random vector X of dimension n is said to have a Normal distribution with mean μ and covariance matrix Σ if Σ if 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 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 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 satisfies X 2 ∼N(μ 2 ,Σ 22 ), then the joint distribution of (X 1 ,X 2 ) is given by (2).

See Also

category: Statistics

Revised on August 8, 2007 14:23:35 by Jason Blevins (152.3.158.22)
Edit | Back in time (4 revisions) | See changes | Views: Print | TeX | Linked from: Kalman filter, Formatting rules, Bivariate normal distribution, Truncated normal distribution, Brownian motion