[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

We have two instruments that measure the distance between

two points. The measurements given by the two instruments are random variables [math]X_1[/math] and [math]X_2[/math] that are independent with [math]E(X_1) = E(X_2) = \mu[/math], where [math]\mu[/math] is the true distance. From experience with these instruments, we know the values of the variances [math]\sigma_1^2[/math] and [math]\sigma_2^2[/math]. These variances are not necessarily the same. From two measurements, we estimate [math]\mu[/math] by the weighted average [math]\bar \mu = wX_1 + (1 - w)X_2[/math]. Here [math]w[/math] is chosen in [math][0,1][/math] to minimize the variance of [math]\bar \mu[/math].

• What is [math]E(\bar \mu)[/math]?
• How should [math]w[/math] be chosen in [math][0,1][/math] to minimize the variance of [math]\bar \mu[/math]?