Statistics - (Normal|Gaussian) Distribution - Bell Curve

Thomas Bayes


A normal distribution is one of underlying assumptions of a lot of statistical procedures.

In nature, every outcome that depends on the sum of many independent events will approximate the Gaussian distribution after some time, if respected the assumptions of the Central limit theorem.

Data from physical processes typically produce a normal distribution curve.

Because of the Central limit theorem, the normal distribution plays a fundamental role in probability theory and statistics.

The normal distribution is commonly denoted as <math>N(0,1)</math> .


Normal Distribution Z Scale

The properties of a normal distribution are well-known:

See density for the function


Considering the classic bean machine (Galtonboard, Galtonbrett Simulation, quincunx or Galton).

The Galtonboard is a device invented by Sir Francis Galton to demonstrate the central limit theorem, in particular that the normal distribution is a good approximate to the binomial distribution.

Gaussian Column First There's only one way for a ball to reach the first column
Gaussian Column Second There are four ways to reach the second column
Gaussian Column Third There are six ways to reach the third column
Gaussian Total Because the machine is symetrical, after some time it will look like a gaussian distribution



The Gaussian function (density) has the form:

<MATH> f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{\displaystyle -\frac{1}{2} \left (\frac{x-\mu}{\sigma} \right )^2} </MATH>


Pdf Normal Distribution

As notated on the figure, the probabilities of intervals of values correspond to the area under the curve.


The cumulative distribution function (CDF) is noted <math>\Phi(z)</math> .

Normal Distribution Cdf



Trigonometry - (Cosine|Cosinus) <MATH> f(x) = \frac{1+cos(x)}{2\pi} </MATH> This approximation can be integrated in closed form

Documentation / Reference

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