I understand Central limit theorem, but cannot understand the result when it applies to diverse situation.
As far as I understand, Central limit theorem states follows:
- No matter what random variable has its distribution, if we pick n samples and mean (or just sum) them and do this many time, those values will follow normal distribution (if n is big enough).
And there is a good example for this: a Brownian motion. If we measure the distance that particles moved suspending in a fluid, distance will follow normal distribution because sum of random hit(movement) of fluid particle will follow normal distribution according to Central limit theorem.
However, for central limit theorem applies, random variables which are merged should follow all the same means and variances. In upper example, all random particle movement will follow same mean and variances, because they have same temperature.
But as far as I understand, there are many cases where noise follows normal distribution but there components are not necessarily follow same mean.
(S_n = (X_1 + X_2 + ... X_n) / n and S_n follows normal distribution but X_i s have all DIFFERENT means)
For example, electrical noise will be composed of many other things which are not necessarily follow equal to each mean. But why electrical noise can be approximated as gaussian noise? Not only electrical noise, why many other things are treated as normal distribution(S_n follows normal distribution) although their components don't have equal mean?(X_i s have all DIFFERENT mean and variances)
Thank you for reading!