Gaussian integral

The Gaussian error integral (after Carl Friedrich Gauss ) is the distribution function of the standard normal distribution. It is often referred to and is the integral of the density function to the normal distribution with and. Since the entire surface is less than the density curve (also referred to as Gaussian bell ) is equal to 1, the value of the error function for well 1 (see section normalization).

Definition

The error integral is by

Defined.

Leaving the integral only when instead start at, then one speaks of:

Related to the Gaussian error function

By substituting in the above formulas, and by appropriate transformations can be made ​​or the error function

Or

Derived.

Application

The error function indicates to the probability that a value in a Gaussian distribution stochastic process ( with, ) is included. Conversely, the probability value can be determined by forming.

As electrotechnical example a Gaussian noise contained the dispersion, it is assumed, which is superimposed on a transmission channel. This channel work flawlessly as long as the disturbances in the area are -5V ... 5 V. It clears up quickly now the question is how likely is a faulty transmission:

Probability for a noise level no greater than 5V:

Probability of a noise value at least equal to 5 V:

The total probability of a transmission error is then given by

Standardization

To prove the normalized awareness, we calculate

This integral is not elementary to integrate as before. The crucial trick for calculation ( allegedly by Poisson ), is now resorting to a higher dimension and to parameterize the resulting 2D region of integration differently:

Basis for the first conversion, the linearity of the integral.

Instead of Cartesian coordinates along is integrated over now along polar coordinates, corresponding to the substitution and of it, and finally obtained with the transform set

Thus we obtain:

See also: Table standard normal distribution