gaussian random vector

[[concept]]

Topics

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Multivariate Normal

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

$\frac{1}{{\det }^{\ast }(2\pi \Sigma )}\exp \left(-\frac{1}{2}(x-\mu {)}^T{\Sigma }^{†}(x-\mu )\right)$ With respect to the Lebesgue measure on the row space of $\Sigma$ and where

• ${\Sigma }^{†}$ is the Moore-Penrose inverse
• ${\det }^{\ast }$ is the product of all non-zero eigenvalues

NOTE

When $\Sigma$ is invertible,

• ${\det }^{\ast }$ is the ordinary determinant
• ${\Sigma }^{†}={\Sigma }^{-1}$
• the Lebesgue measure is on all of ${\mathbb{R}}^d$

Gaussian Random Vector

If a random vector $x\sim \mathcal{N}(\mu ,\Sigma )$, we call $x$ a Gaussian random vector. If $x\sim \mathcal{N}(0,I_d)$, then we call $x$ a standard Gaussian random vector.

Proposition

Let $x\sim \mathcal{N}(\mu ,\Sigma )$ be a gaussian random vector. Then $\mu$ is the mean vector and $\Sigma$ is the covariance matrix of $x$ and $\mu =\mathbb{E}[x]\Sigma =\mathbb{E}[(x-\mu )(x-\mu {)}^T]=\mathbb{E}[x{x}^T]-\mu {\mu }^T$

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

References

References

See Also

Mentions

Mentions

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