exercise:436d269017: Difference between revisions
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(Created page with "<div class="d-none"><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></div> 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 <ma...") |
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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 | |||
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 | <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 | distance. From experience with these instruments, we know the values of the | ||
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= wX_1 + (1 - w)X_2</math>. Here <math>w</math> is chosen in <math>[0,1]</math> to minimize the variance of | = 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>. | <math>\bar \mu</math>. | ||
<ul><li> What is <math>E(\bar \mu)</math>? | <ul style="list-style-type:lower-alpha"><li> What is <math>E(\bar \mu)</math>? | ||
</li> | </li> | ||
<li> How should <math>w</math> be chosen in <math>[0,1]</math> to minimize the variance of | <li> How should <math>w</math> be chosen in <math>[0,1]</math> to minimize the variance of | ||
Latest revision as of 22:15, 14 June 2024
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]?