< List of probability distributions < Cramp function
The (complex) cramp function distribution, also called the error function, is a continuous probability distribution defined as [1]:
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 [2], 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:
- Estimating the mean and standard deviation of a population.
- Calculating the probability a random variable will fall within a certain range.
- Testing hypotheses.
Cramp Function vs. Error Function
The term “(complex) cramp function” is seen in literature by Russian or Latvian authors (for example, Mikhailovskiy, 1975 [3]; Bogdanov et al., 1976 and also see [4], [5]), where it is usually denoted as W(x) [6]. 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
[1] Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.
[2] Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.
[3] Mikhailovskiy, A. B. (1975), Theory of Plasma Instabilities, Atomizdat, in Russian
[4] Baumjohann, W., and R. A. Treumann (1997), Basic Space Plasma Physics, Imperial College Press, London.
[5] Zagursky, V. Pilot signal detection in Wireless Sensor Networks (Latvian translation). Online: https://ortus.rtu.lv/science/en/publications/11860/fulltext
[6] Error Functions (TeX). Online: http://nlpc.stanford.edu/nleht/Science/reference/tex/errorfun/errorfun.tex