= Minimum-variance two-cue estimate
{title2=$w_1=\sigma_2^2/(\sigma_1^2+\sigma_2^2)$}
Let unbiased independent estimates $X_1,X_2$ have <variances> $a,b>0$. The <variance> of $wX_1+(1-w)X_2$ is $w^2a+(1-w)^2b$. Its derivative is $2w(a+b)-2b$ and its second derivative is $2(a+b)>0$, proving the unique optimum $w=b/(a+b)$ and <variance> $ab/(a+b)$. Independent Gaussian likelihoods give the same estimate by differentiating $(x_1-x)^2/a+(x_2-x)^2/b$. With <covariance> $c$, the <variance> becomes $w^2a+(1-w)^2b+2w(1-w)c$, and the unconstrained optimum is $(b-c)/(a+b-2c)$ when the denominator is positive; it need not lie between zero and one.
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