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 [0, 1] with the probability density f (x) given below, and compute their sum S25.
Repeat this experiment 1000 times, and make up a bar graph of the results. Now plot on the same graph the density φ(x) = normal (x, µ(S25), σ(S25)). How well does the normal density fit your bar graph in each case?
(a) f (x) = 1.
(b) f (x) = 2x.
(c) f (x) = 3x2.
(d) f (x) = 4|x - 1/2|.
(e) f (x) = 2 - 4|x - 1/2|.