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].