exercise:70f525f222: Difference between revisions
(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> In general, the Central Limit Theorem gives a better estimate than Chebyshev's inequality for the average of a sum. To see this, let <math>A_{25}</math> be the average calculated in Exercise Exercise, and let <math>N</ma...") |
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In general, the Central Limit Theorem gives a better estimate than Chebyshev's inequality for the average of a sum. To see this, let <math>A_{25}</math> be the average calculated in [[exercise:234ec6829d |Exercise]], and let <math>N</math> be the normal | |||
approximation for <math>A_{25}</math>. Modify your program in [[exercise:234ec6829d |Exercise]] | |||
Chebyshev's inequality for the average of a sum. To see this, let <math>A_{25}</math> be | |||
the average calculated in | |||
approximation for <math>A_{25}</math>. Modify your program in | |||
to provide a table of the function <math>F(x) = P(|A_{25} - 10| \geq x) = | to provide a table of the function <math>F(x) = P(|A_{25} - 10| \geq x) = | ||
{}</math> fraction of the total of 1000 trials for which <math>|A_{25} - 10| \geq x</math>. Do | {}</math> fraction of the total of 1000 trials for which <math>|A_{25} - 10| \geq x</math>. Do | ||
the same for the function <math>f(x) = P(|N - 10| \geq x)</math>. (You can use the normal | the same for the function <math>f(x) = P(|N - 10| \geq x)</math>. (You can use the normal | ||
table, | table, [[guide:146f3c94d0#tabl 9.1|Table]], or the procedure ''' NormalArea''' for this.) | ||
Now plot on the same axes the graphs of <math>F(x)</math>, <math>f(x)</math>, and the Chebyshev | Now plot on the same axes the graphs of <math>F(x)</math>, <math>f(x)</math>, and the Chebyshev | ||
function <math>g(x) = 4/(3x^2)</math>. How do <math>f(x)</math> and <math>g(x)</math> compare as estimates for | function <math>g(x) = 4/(3x^2)</math>. How do <math>f(x)</math> and <math>g(x)</math> compare as estimates for | ||
<math>F(x)</math>? | <math>F(x)</math>? | ||
Latest revision as of 23:24, 14 June 2024
In general, the Central Limit Theorem gives a better estimate than Chebyshev's inequality for the average of a sum. To see this, let [math]A_{25}[/math] be the average calculated in Exercise, and let [math]N[/math] be the normal approximation for [math]A_{25}[/math]. Modify your program in Exercise to provide a table of the function [math]F(x) = P(|A_{25} - 10| \geq x) = {}[/math] fraction of the total of 1000 trials for which [math]|A_{25} - 10| \geq x[/math]. Do the same for the function [math]f(x) = P(|N - 10| \geq x)[/math]. (You can use the normal table, Table, or the procedure NormalArea for this.) Now plot on the same axes the graphs of [math]F(x)[/math], [math]f(x)[/math], and the Chebyshev function [math]g(x) = 4/(3x^2)[/math]. How do [math]f(x)[/math] and [math]g(x)[/math] compare as estimates for [math]F(x)[/math]?