gaussian random vector

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Multivariate Normal

The multivariate normal or Gaussian $N (μ, Σ)$ for parameters $μ \in R^{d}$ and (positive semidefinite) $Σ \in R^{d \times d}$ is the probability measure with density

\frac{1}{\overset{*}{det} (2 π Σ)} \exp (- \frac{1}{2} (x - μ)^{T} Σ^{†} (x - μ))

With respect to the Lebesgue measure on the row space of $Σ$ and where

$Σ^{†}$ is the Moore-Penrose inverse
$\overset{*}{det}$ is the product of all non-zero eigenvalues

Note

When $Σ$ is invertible,

$\overset{*}{det}$ is the ordinary determinant
$Σ^{†} = Σ^{- 1}$
the Lebesgue measure is on all of $R^{d}$

Gaussian Random Vector

If a random vector $x \sim N (μ, Σ)$ , we call $x$ a Gaussian random vector. If $x \sim N (0, I_{d})$ , then we call $x$ a standard Gaussian random vector.

Proposition

Let $x \sim N (μ, Σ)$ be a gaussian random vector. Then $μ$ is the mean vector and $Σ$ is the covariance matrix of $x$ and

μ = E [x] Σ = E [(x - μ) (x - μ)^{T}] = E [x x^{T}] - μ μ^{T}

ie, the law of a gaussian random vector is determined by its mean and covariance (or its linear and quadratic moments)

References

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gaussian random vector

References

See Also

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