Probability integral

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error integral

The function

In probability theory one mostly encounters not the probability integral, but the normal distribution function

which is the so-called Gaussian probability integral. For a random variable having the normal distribution with mathematical expectation 0 and variance , the probability that is equal to . For real , the probability integral takes real values; in particular,

Figure: p074920a

The graph of the probability integral and its derivatives are illustrated in the figure. Regarded as a function of the complex variable , the probability integral is an entire function of .

The asymptotic representation for large , , is given by:

In a neighbourhood of the probability integral can be represented by the series

The probability integral is related to the Fresnel integrals and by the formulas

The derivative of the probability integral is given by:

The following notations are sometimes used:


[1] H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953)
[2] E. Jahnke, F. Emde, "Tables of functions with formulae and curves" , Dover, reprint (1945) (Translated from German)
[3] A. Krazer, W. Franz, "Transzendente Funktionen" , Akademie Verlag (1960)
[4] N.N. Lebedev, "Special functions and their applications" , Prentice-Hall (1965) (Translated from Russian)


The series representation of the probability integral around takes the form of a confluent hypergeometric function:

How to Cite This Entry:
Probability integral. A.B. Ivanov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098