Error integral

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The Gaussian error integral (according to Carl Friedrich Gauß ) is the distribution function of the standard normal distribution . It is often referred to as and is the integral from to over the density function of the normal distribution with and . Since the total area below the density curve (also called Gaussian bell) is equal to 1, the value of the error integral for is also 1 (see section Normalization ).

definition

The error integral is through

Are defined.

If the integral is only started at instead of at , one speaks of :

Relation to the Gaussian error function

Through the substitution in the above formulas and through appropriate transformations, the error function can be eliminated or

or.

derive.

application

The error integral indicates the probability that a standard normally distributed random variable assumes a value less than or equal . Conversely, the probability for a value greater than or equal can also be determined by forming.

As an electrotechnical example, assume a Gaussian-distributed interference noise from the scattering that is superimposed on a transmission channel . This channel works without errors as long as the interference is in the range of −5 V ... +5 V. The question of how likely a faulty transmission is:

Probability for a noise value not greater than -5 V:

Probability for a noise value at least equal to +5 V:

The overall probability of a transmission error then results from

Normalization

In order to prove the normalization , we calculate

Even if no antiderivative of the integrand can be expressed as an elementary function , there are still more than half a dozen possible solutions to determine its value, starting with De Moivre's first approximations from 1733 to the work of Laplace and Poisson from around 1800 to towards a completely new approach by SP Evesons from 2005. One of the decisive tricks for his calculation (allegedly by Poisson) is to switch to a higher dimension and to parameterize the resulting 2D integration area differently:

The basis for the first conversion is the linearity of the integral.

Instead of along Cartesian coordinates, now along polar coordinates is integrated, which corresponds to the substitution and from it , and one finally obtains with the transformation set

With this we get:

See also

Individual evidence

  1. Peter M. Lee: The probability integral ; University of York, Department of Mathematics, 2011 , last accessed May 14, 2016.
  2. Denis Bell: Poisson's remarkable calculation - a method or a trick? ; University of North Florida, Department of Mathematics, 2010 (PDF; 248 kB)