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The [probability distribution function]? for the normal distribution with mean μ and standard deviation σ is
One reason that this distribution occurs so often in statistical work is the Central Limit Theorem. Simply stated, this theorem says that if you add up a lot of little things, the resulting distribution will resemble the normal distribution. More precisely: if you have n independent identically distributed random variables with mean 0 and standard deviation 1, then n-1/2 times their sum converges in distribution to the normal distribution with mean 0 and standard deviation 1.
Beware! There are random variables which do not have both a mean and a standard deviation. (The [Cauchy distribution]? is a famous example.) Sums of such unfriendly random variables need not tend to normality.
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