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  "body": "\n## Definition\n\nLet $k \\ge 1$ and let $X = (X_1, \\dots, X_k)^{\\mathsf{T}}$ be a random vector in $\\mathbb{R}^k$.\nThen $X$ is multivariate normal if there exist $\\mu \\in \\mathbb{R}^k$, an integer $\\ell \\ge 1$ and a real $k \\times \\ell$ matrix $A$ with this property.\nThere is a vector $Z = (Z_1, \\dots, Z_\\ell)^{\\mathsf{T}}$ of independent $\\mathcal{N}(0, 1)$ random variables with\n$$X = AZ + \\mu.$$\nThen the law of $X$ depends only on $\\mu$ and on $\\Sigma = AA^{\\mathsf{T}}$. It is written $\\mathcal{N}_k(\\mu, \\Sigma)$.\n\n## Remarks\n\nHere $\\mu$ is the mean vector, with $\\mu_i = E[X_i]$, and $\\Sigma$ is the covariance matrix, with $\\Sigma_{ij} = \\operatorname{Cov}(X_i, X_j)$.\nAn equivalent condition: $X$ is multivariate normal if and only if $a^{\\mathsf{T}} X$ has a [[normal distribution|07RXBVTVZB]] for each $a \\in \\mathbb{R}^k$. Here a constant counts as normal with variance $0$.\nIf $\\Sigma$ is positive definite, $X$ has the density\n$$f(x) = \\frac{1}{\\sqrt{(2\\pi)^k \\det \\Sigma}} \\exp\\left(-\\tfrac{1}{2} (x - \\mu)^{\\mathsf{T}} \\Sigma^{-1} (x - \\mu)\\right), \\quad x \\in \\mathbb{R}^k.$$\nThe components of a multivariate normal vector are independent if and only if they are uncorrelated, that is, if $\\Sigma$ is diagonal.\n\n## Explanations\n\n### L1 human\n\nA multivariate normal vector is a list of normal random variables that may be linked to each other.\nThe covariance matrix says how strongly each pair of components moves together. In two dimensions, the density looks like a hill with elliptical contour lines.\n",
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