A pair of independent unbiased estimates of the same quantity have variances 4 and 12. You combine them as w × A + (1 − w) × B, choosing w to minimise the variance. What is the variance of the combined estimate? Give a number.

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Merging a pair of unbiased estimates

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A pair of independent unbiased estimates of the same quantity have variances $4$ and $12$ . You combine them as $w \times A + (1 - w) \times B$ , choosing $w$ to minimise the variance. What is the variance of the combined estimate? Give a number.

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Hint

Precisions add: weight each estimate in proportion to one over its variance.

Solution

The precisions are $\frac{1}{4}$ and $\frac{1}{12}$ .
They add to $\frac{1}{4} + \frac{1}{12} = \frac{1}{3}$ .
The combined variance is $\frac{1}{\frac{1}{3}} = 3$ .

Answer: $3$

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