Problem 2: Weighted averaging
Suppose ozone in Atlanta is measured by m different instruments simultaneously. The
measurements are [X1, X2, ..., Xm] and the instrument uncertainties are [?1, ?2,... ?m].
(1) Derive equation to calculate a weighted arithmetic mean (wam) such that the cost function,
$\sum_{i=1}^{m} \left(\frac{wam - X_i}{?_i}\right)^2$, is minimized.
(2) How is the uncertainty (variance) for the wam calculated? Derive the expression from a basic
property of variance.