Covariance when two parameters have relative scaling error

I was reading error propagation from "Introduction to Statistics and Data Analysis for Physicists" by G. Bohm and G. Zech, and I stumbled upon this example,

Given are the sides a, b with a reading error $\delta_1$ and a relative scaling error $\delta_2$, caused by a possible extension or shrinkage of the measuring tape. We want to calculate the error $\delta$F of the area F = ab. We find, $$(\delta_a)^{2} = (\delta_1)^{2} + (a\delta_2)^{2}\\(\delta_b)^{2} = (\delta_1)^{2} + (b\delta_2)^{2}\\C_{ab} = ab(\delta_2)^{2}$$ where $C_{ab}$ is covariance

I'm not able to understand how this covariance comes about.

• I'll answer this in the next days, meanwhile have a look at this answer to better understand the framework. – Massimo Ortolano Oct 28 '16 at 11:09

So, while the variance is given by $$\sigma^2 = E\{(x-\mu)^2\} = E\{x^2\} - \mu^2 ,$$ the covariance (in your notation) is given by $$C_{ab} = E\{(a-\delta_a)(b-\delta_b)\} = E\{ab\} - \delta_a\delta_b .$$ So what the covariance is telling us is that the quantities are not statistically independent, because statistical independence would have meant that $$E\{ab\} = E\{a\}E\{b\} = \delta_a\delta_b .$$ So the reason why you are given a nonzero value for the covariance is to take into account the fact that $a$ and $b$ are not statistically independent.