Probability integral
error integral
The function
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In probability theory one mostly encounters not the probability integral, but the normal distribution function
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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,
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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:
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In a neighbourhood of the probability integral can be represented by the series
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The probability integral is related to the Fresnel integrals and
by the formulas
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The derivative of the probability integral is given by:
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The following notations are sometimes used:
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References
[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) |
Comments
The series representation of the probability integral around takes the form of a confluent hypergeometric function:
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Probability integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Probability_integral&oldid=18624