# Cramp function distribution

< List of probability distributions < Cramp function

The (complex) cramp function distribution, also called the error function, is a continuous probability distribution defined as :

The first equality in the formula states that the error function can be expressed as the integral of the Gaussian (normal) function from 0 to x. The second equality states that the error function can also be expressed as 2 times the cumulative distribution function (CDF) of the standard normal distribution, evaluated at x * √(2).

Some authors, such as Whittaker and Watson , define the error function without the leading 2/π.

The function integrates the normal distribution, giving the probability a normally distributed random variable Y (with mean 0 and variance ½), falls into the range [−x, x]. Integration, or the “area under the curve” is used to in calculus to find areas, central points, and volumes. The function is a very important function in statistics and probability, with applications such as:

## Cramp Function vs. Error Function

The term “(complex) cramp function” is seen in literature by Russian or Latvian authors (for example, Mikhailovskiy, 1975 ; Bogdanov et al., 1976 and also see , ), where it is usually denoted as W(x) . Elsewhere, it is called the error function (erf, for short). Other names include the Gauss or Gaussian Error Function.

The CRAMP functions in biology are unrelated.

## References

 Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.

 Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

 Mikhailovskiy, A. B. (1975), Theory of Plasma Instabilities, Atomizdat, in Russian

 Baumjohann, W., and R. A. Treumann (1997), Basic Space Plasma Physics, Imperial College Press, London.

 Zagursky, V. Pilot signal detection in Wireless Sensor Networks (Latvian translation). Online: https://ortus.rtu.lv/science/en/publications/11860/fulltext

 Error Functions (TeX). Online: http://nlpc.stanford.edu/nleht/Science/reference/tex/errorfun/errorfun.tex

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