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**1. 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]

**2. 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]

**3. 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.

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