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  "body": "\n## Definition\n\nLet $X = (X_1, \\ldots, X_n)$ be a random vector with [[mean|0WQNRCNY3M]] $\\mu \\in \\R^n$ and [[covariance matrix|0V7G6WMFK0]] $\\Sigma_X$.\nLet $f \\colon \\R^n \\to \\R^m$ be differentiable at $\\mu$, with [[Jacobian matrix|047NDTEPCV]] $J = Df(\\mu)$.\nThe *linear propagation of uncertainty* assigns to $Y = f(X)$ the approximate mean $f(\\mu)$ and the approximate covariance matrix\n$$\\Sigma_Y = J \\, \\Sigma_X \\, J^{T}.$$\nFor $m = 1$ and uncorrelated $X_i$ with [[variances|0477R9PYAE]] $\\sigma_i^2$, this reads\n$$\\sigma_Y^2 = \\sum_{i=1}^{n} \\left( \\frac{\\partial f}{\\partial x_i}(\\mu) \\right)^2 \\sigma_i^2.$$\nIt is the covariance of the first-order Taylor approximation $f(\\mu) + J(X - \\mu)$.\n\n## Remarks\n\nExample: for $f(x_1, x_2) = x_1 x_2$ and uncorrelated inputs, $\\partial f/\\partial x_1 = \\mu_2$ and $\\partial f/\\partial x_2 = \\mu_1$, so $\\sigma_Y^2 = \\mu_2^2 \\sigma_1^2 + \\mu_1^2 \\sigma_2^2$, that is $(\\sigma_Y/\\mu_1\\mu_2)^2 = (\\sigma_1/\\mu_1)^2 + (\\sigma_2/\\mu_2)^2$ when $\\mu_1 \\mu_2 \\ne 0$.\nIf $f(x) = Ax + b$ is affine, the formula is exact: $\\mathrm{Cov}(AX + b) = A \\Sigma_X A^T$.\nNon-example: for $f(x) = x^2$ and $X$ normal with mean $0$ and variance $\\sigma^2$, the formula gives $\\sigma_Y^2 = 0$, while the true variance of $X^2$ is $2\\sigma^4$; the linearization fails because $f'(\\mu) = 0$.\n",
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