Mathematic Review
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Multivariables Normal distribution
The multivariate normal distribution of a k-dimensional random vector \(X = (X_1,...,X_2)^\intercal\) can be written in the following notation:
\[X \sim \mathcal{N}(\mu,\Sigma)\]
With
\(\mu = E[X] = E[X_1]\): k-dimensional mean vector
\(\Sigma = Cov[X_i,X_j]\): k x k covariance matrix
The probability density function (PDF)
\[\begin{aligned} f(x|Y=k) = \frac{1}{(2 \pi)^\frac{k}{2}|\Sigma|^{\frac{1}{2}}}exp\{ -\frac{1}{2}(x-\mu)^\intercal \sum\nolimits^{-1}(x-\mu)\} \end{aligned}\]An example of a multivariate normal distribution with
\[\begin{aligned} \mu = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \Sigma = \begin{bmatrix} 1 & 3/5 \\ 3/5 & 2 \end{bmatrix} \end{aligned} \]