Editing Propagation of uncertainty (section)

=== Example ===
Any non-linear differentiable function, <math>f(a,b)</math>, of two variables, <math>a</math> and <math>b</math>, can be expanded as
<math display="block">f\approx f^0+\frac{\partial f}{\partial a}a+\frac{\partial f}{\partial b}b.</math>
If we take the variance on both sides and use the formula<ref>{{Cite web| last=Soch|first=Joram| date=2020-07-07| title=Variance of the linear combination of two random variables|url=https://statproofbook.github.io/P/var-lincomb.html| access-date=2022-01-29| website=The Book of Statistical Proofs|language=en}}</ref> for the variance of a linear combination of variables
<math display="block">\operatorname{Var}(aX + bY) = a^2\operatorname{Var}(X) + b^2\operatorname{Var}(Y) + 2ab \operatorname{Cov}(X, Y),</math>
then we obtain
<math display="block">\sigma^2_f\approx\left| \frac{\partial f}{\partial a}\right| ^2\sigma^2_a+\left| \frac{\partial f}{\partial b}\right|^2\sigma^2_b+2\frac{\partial f}{\partial a}\frac{\partial f} {\partial b}\sigma_{ab},</math>
where <math>\sigma_{f}</math> is the standard deviation of the function <math>f</math>, <math>\sigma_{a}</math> is the standard deviation of <math>a</math>, <math>\sigma_{b}</math> is the standard deviation of <math>b</math> and <math>\sigma_{ab} = \sigma_{a}\sigma_{b} \rho_{ab}</math> is the covariance between <math>a</math> and <math>b</math>.

In the particular case that {{nowrap|<math>f = ab</math>,}} {{nowrap|<math>\frac{\partial f}{\partial a} = b</math>,}} {{nowrap|<math>\frac{\partial f}{\partial b} = a</math>.}} Then
<math display="block">\sigma^2_f \approx b^2\sigma^2_a+a^2 \sigma_b^2+2ab\,\sigma_{ab}</math>
or
<math display="block">\left(\frac{\sigma_f}{f}\right)^2 \approx \left(\frac{\sigma_a}{a} \right)^2 + \left(\frac{\sigma_b}{b}\right)^2 + 2\left(\frac{\sigma_a}{a}\right)\left(\frac{\sigma_b}{b}\right)\rho_{ab}</math>
where <math>\rho_{ab}</math> is the correlation between <math>a</math> and <math>b</math>.

When the variables <math>a</math> and <math>b</math> are uncorrelated, <math>\rho_{ab}=0</math>. Then
<math display="block">\left(\frac{\sigma_f}{f}\right)^2 \approx \left(\frac{\sigma_a}{a} \right)^2 + \left(\frac{\sigma_b}{b}\right)^2.</math>