Two thermometers
Two thermometers measure the same temperature. Both are unbiased, their errors are independent, and the error standard deviations are 1 degree and 2 degrees.
Among the unbiased weighted averages of the two readings, find the smallest possible variance.
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
The variance is . Minimise over .
The optimal weights are proportional to the inverse variances.
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Weight each reading by its precision, the inverse of its variance. The combined precision is the sum of the precisions.
Step 1: minimise. has derivative , which vanishes at .
Step 2: the variance.
Inverse-variance weighting. The precisions are and , so the weights are and , and the combined variance is . Even the worse thermometer helps: the combination beats the better one alone (variance 1).