---
tag: ZQKA90X325
kind: definition
name: Propagation of uncertainty
msc: [65G99, 62P99]
defines: WS6GCG3YF8
uses: [0V7G6WMFK0, 047NDTEPCV, 0WQNRCNY3M, 0477R9PYAE]
sources:
  - url: https://en.wikipedia.org/wiki/Propagation_of_uncertainty
---

## Definition

Let $X = (X_1, \ldots, X_n)$ be a random vector with [[mean|0WQNRCNY3M]] $\mu \in \R^n$ and [[covariance matrix|0V7G6WMFK0]] $\Sigma_X$.
Let $f \colon \R^n \to \R^m$ be differentiable at $\mu$, with [[Jacobian matrix|047NDTEPCV]] $J = Df(\mu)$.
The *linear propagation of uncertainty* assigns to $Y = f(X)$ the approximate mean $f(\mu)$ and the approximate covariance matrix
$$\Sigma_Y = J \, \Sigma_X \, J^{T}.$$
For $m = 1$ and uncorrelated $X_i$ with [[variances|0477R9PYAE]] $\sigma_i^2$, this reads
$$\sigma_Y^2 = \sum_{i=1}^{n} \left( \frac{\partial f}{\partial x_i}(\mu) \right)^2 \sigma_i^2.$$
It is the covariance of the first-order Taylor approximation $f(\mu) + J(X - \mu)$.

## Remarks

Example: 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 $\mu_1 \mu_2 \ne 0$.
If $f(x) = Ax + b$ is affine, the formula is exact: $\mathrm{Cov}(AX + b) = A \Sigma_X A^T$.
Non-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 $f'(\mu) = 0$.