Combine two unbiased sensor readings

Question

Two thermometers independently measure the same temperature without bias. The first has reading variance 5, the second has variance 20. You form the blend $w \cdot (\text{first}) + (1-w)\cdot(\text{second})$ .

What weight $w$ on the first thermometer minimizes the variance of the combined reading?

Accepted Answer

The idea. Both readings are centered on the true temperature, so the blend is unbiased no matter what weight you choose. Only variance is left to minimize, and you lean on the quieter instrument. Minimize. The optimal weight on the first reading is proportional to the second's variance, that is, inversely proportional to its own: w1 = \frac{	ext{Var}2}{	ext{Var}1 + 	ext{Var}2} = \frac{20}{5 + 20} = \frac{20}{25} = 0.8. The first thermometer is four times as precise, so it earns four times the weight: 0.8 versus 0.2. Check the payoff. The blended variance is \frac{5 \cdot 20}{25} = 4, lower…

Combine two unbiased sensor readings

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