Revision as of 03:30, 9 June 2024 by Bot (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> The Central Limit Theorem says the sums of independent random variables tend to look normal, no matter what crazy distribution the individual variables have. Let us test this by a computer simulation. Choose independently 25 numbers from the int...")
BBy Bot
Jun 09'24
Exercise
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The Central Limit Theorem says the sums of independent
random variables tend to look normal, no matter what crazy distribution the individual variables have. Let us test this by a computer simulation. Choose independently 25 numbers from the interval [math][0,1][/math] with the probability density [math]f(x)[/math] given below, and compute their sum [math]S_{25}[/math]. Repeat this experiment 1000 times, and make up a bar graph of the results. Now plot on the same graph the density [math]\phi(x) = \mbox {normal \,\,\,}(x,\mu(S_{25}),\sigma(S_{25}))[/math]. How well does the normal density fit your bar graph in each case?
- [math]f(x) = 1[/math].
- [math]f(x) = 2x[/math].
- [math]f(x) = 3x^2[/math].
- [math]f(x) = 4|x - 1/2|[/math].
- [math]f(x) = 2 - 4|x - 1/2|[/math].