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# Normal distribution

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The normal or Gaussian probability distribution is actually a family of distributions of the same general form, differing only in their location and scale parameters, commonly called the mean and standard deviation. The shape consists of a central bulge centered on the mean with about 99% of the area under the density curve between the mean plus and minus three standard deviations.

The [probability distribution function]? for the normal distribution with mean μ and standard deviation σ is

  p(x) = exp(-(x-μ)2/2σ2) / (2π)1/2σ

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.

Links:

- http://ce597n.www.ecn.purdue.edu/CE597N/1997F/students/michael.a.kropinski.1/project/tutorial
- http://www.math.csusb.edu/faculty/stanton/m262/central_limit_theorem/clt.html

back to Probability -- Statistics

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Last edited July 25, 2001 12:43 pm (diff)
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