Why does the average of independent random variables converge to Normal distribution not other distribution? Or in other words, I am looking for an intuitive explanation of the central limit theorem. Why can't the average convert to some other distribution?
2026-04-02 19:22:17.1775157737
Why does the average of independent random variables converge to Normal distribution not other distribution?
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Given that the proof of CLT is not so hard and you can find it anyway... I can give you a very intuitive explanation...
Suppose you have to measure the lenght of a table, and you can choose the instrument you prefer...you can use you hands for a very basic estimation of the lenght, you can use a ruler for a more precise measurement or you can use a 10 tons instrument taken from a Physic Research Lab. In any case, the true value of the lenght of your table in unknown, and it can change according to several factors: external temperature, error in the instrument, errors in the man who takes the measure and so on....
But some fact are surely true:
The probability to do the same error underestimating or overestimating the actual measure is the same
The probabilty do to a big error is lower w.r.t do a little error. (if a table has an actual lenght of 1 meter, the probability do get a measure of 2 meter is very close to zero, as it is zero the probability to get a measure of 1 cm)
The probability to estimate a lenght in the interval of [1;1.01] m is the same as the one to estimate it as [0.99;1] m
Doing a lot of measurement and plotting them on a chart, you will realize that the resulting drawing is a bell function, symmetric around the actual value...it's a Gaussian density....