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Normal Distribution Function


NormalDistributionFunction

A normalized form of the cumulative normal distribution function giving the probability that a variate assumes a value in the range [0,x],

 Phi(x)=Q(x)=1/(sqrt(2pi))int_0^xe^(-t^2/2)dt.
(1)

It is related to the probability integral

 alpha(x)=1/(sqrt(2pi))int_(-x)^xe^(-t^2/2)dt
(2)

by

 Phi(x)=1/2alpha(x).
(3)

Let u=t/sqrt(2) so du=dt/sqrt(2). Then

 Phi(x)=1/(sqrt(pi))int_0^(x/sqrt(2))e^(-u^2)du=1/2erf(x/(sqrt(2))).
(4)

Here, erf is a function sometimes called the error function. The probability that a normal variate assumes a value in the range [x_1,x_2] is therefore given by

 Phi(x_1,x_2)=1/2[erf((x_2)/(sqrt(2)))-erf((x_1)/(sqrt(2)))].
(5)

Neither Phi(z) nor erf can be expressed in terms of finite additions, subtractions, multiplications, and root extractions, and so must be either computed numerically or otherwise approximated.

Note that a function different from Phi(x) is sometimes defined as "the" normal distribution function

N(x)=1/(sqrt(2pi))int_(-infty)^xe^(-t^2/2)dt
(6)
=Phi(-infty,x)
(7)
=1/2+Phi(x)
(8)
=1/2[1+erf(x/(sqrt(2)))]
(9)

(Feller 1968; Beyer 1987, p. 551), although this function is less widely encountered than the usual Phi(x). The notation N(x) is due to Feller (1971).

The value of a for which P(x) falls within the interval [-a,a] with a given probability P is a related quantity called the confidence interval.

For small values x<<1, a good approximation to Phi(x) is obtained from the Maclaurin series for erf,

 Phi(x)=1/(sqrt(2pi))(x-1/6x^3+1/(40)x^5-1/(336)x^7+1/(3456)x^9+...)
(10)

(OEIS A014481). For large values x>>1, a good approximation is obtained from the asymptotic series for erf,

 Phi(x)=1/2-(e^(-x^2/2))/(sqrt(2pi))(x^(-1)-x^(-3)+3x^(-5)-15x^(-7)+105x^(-9)+...)
(11)

(OEIS A001147).

The value of Phi(x) for intermediate x can be computed using the continued fraction identity

 int_0^xe^(-u^2)du=(sqrt(pi))/2-(1/2e^(-x^2))/(x+1/(2x+2/(x+3/(2x+4/(x+...))))).
(12)

A simple approximation of Phi(x) which is good to two decimal places is given by

 Phi_1(x) approx {0.1x(4.4-x)   for 0<=x<=2.2; 0.49   for 2.2<x<2.6; 0.50   for x>=2.6.
(13)

Abramowitz and Stegun (1972) and Johnson et al. (1994) give other functional approximations. An approximation due to Bagby (1995) is

 Phi_2(x)=1/2{1-1/(30)[7e^(-x^2/2)+16e^(-x^2(2-sqrt(2)))+(7+1/4pix^2)e^(-x^2)]}^(1/2).
(14)

The plots below show the differences between Phi and the two approximations.

NormalDistributionFnApprox

The value of t giving 1/4 is known as the probable error of a normally distributed variate.


See also

Berry-Esséen Theorem, Confidence Interval, Erf, Erfc, Fisher-Behrens Problem, Gaussian Integral, Hh Function, Normal Distribution, Owen T-Function, Probability Integral, Tetrachoric Function

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 931-933, 1972.Bagby, R. J. "Calculating Normal Probabilities." Amer. Math. Monthly 102, 46-49, 1995.Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987.Bryc, W. "A Uniform Approximation to the Right Normal Tail Integral." Math. Comput. 127, 365-374, 2002.Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, 1968.Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, p. 45, 1971.Hastings, C. Approximations for Digital Computers. Princeton, NJ: Princeton University Press, 1955.Johnson, N.; Kotz, S.; and Balakrishnan, N. Continuous Univariate Distributions, Vol. 1, 2nd ed. Boston, MA: Houghton Mifflin, 1994.Patel, J. K. and Read, C. B. Handbook of the Normal Distribution. New York: Dekker, 1982.Sloane, N. J. A. Sequences A001147/M3002 and A014481 in "The On-Line Encyclopedia of Integer Sequences."Whittaker, E. T. and Robinson, G. "Normal Frequency Distribution." Ch. 8 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 164-208, 1967.

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Normal Distribution Function

Cite this as:

Weisstein, Eric W. "Normal Distribution Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NormalDistributionFunction.html

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