Introduction to Probability · Problem 9.3.7
Generating Functions — Grinstead & Snell
Mathematics — Probability · central-limit-theorem-continuous
Problem
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 with the probability density given below, and compute their sum . Repeat this experiment 1000 times, and make up a bar graph of the results. Now plot on the same graph the density . How well does the normal density fit your bar graph in each case? (a) . (b) . (c) . (d) . (e) .Intuitive explanation
The Central Limit Theorem predicts that when you sum many independent random variables—even from wildly different distributions—the distribution of that sum converges to a normal distribution centered at with standard deviation . By simulating 1000 repetitions of summing 25 samples from each density, you'll see the empirical histogram of values and can compare it directly to the theoretical normal curve with the matching mean and standard deviation. The key insight is that the shape of the original distribution hardly matters; what matters is that you're adding enough independent pieces that the normal approximation kicks in.
Intuitive explanation
The Central Limit Theorem predicts that when you sum many independent random variables—even from wildly different distributions—the distribution of that sum converges to a normal distribution centered at with standard deviation . By simulating 1000 repetitions of summing 25 samples from each density, you'll see the empirical histogram of values and can compare it directly to the theoretical normal curve with the matching mean and standard deviation. The key insight is that the shape of the original distribution hardly matters; what matters is that you're adding enough independent pieces that the normal approximation kicks in.