L19.3 Discussion of the CLT
The central limit theorem is absolutely remarkable. It is a very deep result, and highly nontrivial and non intuitive. There's no apparent reason why this random variable here, a standardized version of the sum of random variables, should have an approximately normal distribution. Furthermore, it is very useful, and one key reason is that it is universal. It doesn't matter what the distribution of the X's is. No matter what the distribution is, still in the limit, this standardized version of the sum is going to behave like a normal random variable. And if we wish to apply it to particular examples or models, the only thing that we need to know about the distribution of the X's are the corresponding means and variances, as we're going to see in multiple examples. When we apply it, it turns out to be very accurate, and it is also a very nice computational shortcut. Even if we knew, in detail, the distribution of the X's, in order to calculate the di...