I was being curious of how the probability density function formulacame up or was derived, for normal distribution?Can someone help?Thanks

asd, here is a concrete way of seeing it for a simple case. Let's say you have a coin. After N flips, the chanceof getting n heads is you would expect(N choose n)/ 2^Nokay now as you take N -> infinity, you can use the stirling approximation,n! ~ (2 \pi n)^(1/2) n^n exp[-n + 1/(12 n)]to see that the P(n) ~ Normal distribution.Now you can clearly generalize this to brownian motion in 1D. Here you can either go to left or right with 50/50 probability. Then getting n heads is like going n - (N-n) = 2 n - N to the right. Doing a simple transformation gives you a normaldistribution for brownian motion. If you have a unfair coin with p << 1, then a similar argument gives Poisson distribution I think.How does Cauchy distribution arise in nature? The closed-form density function of stable random variables are: normal, Cauchy and Bernoulli distribution. (Stable meaning sum of stable random variablesis also stable random variable). I've read that research of option pricing under Cauchy distributionwas popular some time ago because it has fat tails but was shot down because it wasn't analytically very pleasant. Do people still consider them in practice?

I believe it was Abraham de Moivre, a french mathematician, who first introduced the normal distribution sometime in the early 18th century. He was using it to approximate the binomial distribution for a large number of independent bernoulli trials. damn, I am always in second place.

Plee and JabairuStork - Thanks a lot for your replies!

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If you are interested in this sort of thing, I recommend Stephen Stigler's wonderful book, The History of Statistics.Gauss is usually given credit for "discovering" the Normal distribution, although Laplace and De Moivre have important claims as well. There are really two important results: assuming Normality justified a lot of common sense procedures (like taking means and least squares) and the Central Limit Theorem. The three mathematicians above, and others, figured out parts of the answers; but Gauss was the one who really brought both results together and put them on a solid mathematical basis.Before 1750, there was no good theoretical justification for techniques. There was serious debate about whether you should averaging a number of noisy observations led to a better or worse conclusion than just picking one of them at random. Most sensible people realized averaging was better, but they couldn't explain why. And no one had a good answer for whether the median or a trimmed mean or geometric mean or something else might be better.By 1830, we had full modern proofs that averaging and least squares methods were optimal for observations from a Normal distribution and that large numbers of independent increments of finite variance resulted in a Normal distribution. This gave theoretical support to the common sense things people were doing, and showed how to evaluate alternatives.One body of work without the other was not very useful. Knowing the optimal procedures under a Normal distribution wasn't relevant if you had no reason to believe your data were Normal. Knowing data tended toward Normality under certain assumptions didn't help you if you didn't know what to do with Normal data.In many ways, the Normal distribution has been bad for statistics. It is such a special case, both in the sense of special mathematical properties and the sense of unlikely to appear in real data, that it often blinds people to common sense.