Weighted least-squares fit error propagation

[mbigras]

Homework Statement

Suppose we measure N pairs of values (xi, yi) of two variables x and y that are supposed to statisfy a linear relation y = A + Bx suppose the xi have negligible uncertainty and the yi have different uncertainties [itex]\sigma_{i}[/itex]. We can define the weight of the ith measurement as [itex]w_{i} = 1/\sigma_{i}[/itex]. Then the best estimates of A and B are:

[tex]
A = \frac{\Sigma w x^{2}\Sigma w y - \Sigma w x \Sigma w x y}{\Delta}\\
B = \frac{\Sigma w \Sigma w x y - \Sigma w x \Sigma w y}{\Delta}\\
\Delta = \Sigma w \Sigma x^{2} - \left(\Sigma w x \right)^{2}
[/tex]

Use error propagation to prove that the uncertainties in the constants A and B are given by

[tex]
\sigma_{A} = \sqrt{\frac{\Sigma w x^{2}}{\Delta}}\\
\sigma_{B} = \sqrt{\frac{\Sigma w}{\Delta}}
[/tex]

Homework Equations

rules for sums and differences
[tex]
q = x \pm z\\
\delta q = \sqrt{(\delta x)^{2} + (\delta z)^{2}}\\
[/tex]
rules for products and quotients
[tex]
\delta q = \sqrt{(\delta x/x)^{2} + (\delta z/z)^{2}}\\
[/tex]

The Attempt at a Solution

What I'm thinking is because the uncertainty in x is negligible it will be treated like a constant. I'm not sure how to deal with all these sums. I feel like I don't know how to approach this question.

 

[haruspex]

Try calculating E(A²) etc. The negligible uncertainty in the x allows you to treat the Δ denominators as constant.