exercise:088f67773d: Difference between revisions
From Stochiki
(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...") |
No edit summary |
||
| Line 1: | Line 1: | ||
The Central Limit Theorem says the sums of independent random variables tend to look normal, no matter what crazy distribution the | |||
random variables tend to look normal, no matter what crazy distribution the | |||
individual variables have. Let us test this by a computer simulation. Choose | 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 | independently 25 numbers from the interval <math>[0,1]</math> with the probability | ||
| Line 13: | Line 6: | ||
same graph the density <math>\phi(x) = \mbox {normal \,\,\,}(x,\mu(S_{25}),\sigma(S_{25}))</math>. | 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? | How well does the normal density fit your bar graph in each case? | ||
<ul><li> <math>f(x) = 1</math>. | <ul style="list-style-type:lower-alpha"><li> <math>f(x) = 1</math>. | ||
</li> | </li> | ||
<li> <math>f(x) = 2x</math>. | <li> <math>f(x) = 2x</math>. | ||
Latest revision as of 23:25, 14 June 2024
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].