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∣.