“Normal Distribution Function”的概念、定义、习题解题方法及翻译-数学辞典

单词

Normal Distribution Function

释义

Normal Distribution Function

A normalized form of the cumulative Gaussian Distribution function giving the probability that a variateassumes a value in the range ,

(1)

It is related to the Probability Integral

(2)

(3)

Let

. Then

(4)

Here, Erf is a function sometimes called the error function. The probability that a normal variate assumes a value in the range

is therefore given by

(5)

Neither

nor Erf can be expressed in terms of finite additions, subtractions, multiplications, and rootextractions, and so must be either computed numerically or otherwise approximated.

Note that a function different from is sometimes defined as ``the'' normal distribution function

(6)

(Beyer 1987, p. 551), although this function is less widely encountered than the usual

The value of for which falls within the interval with a given probability is a related quantitycalled the Confidence Interval.

For small values , a good approximation to is obtained from the Maclaurin Series for Erf,

(7)

For large values

, a good approximation is obtained from the asymptotic series for Erf,

(8)

The value of

for intermediate

can be computed using the Continued Fraction identity

(9)

A simple approximation of

which is good to two decimal places is given by

(10)

Abramowitz and Stegun (1972) and Johnson and Kotz (1970) give other functional approximations. An approximation due to Bagby(1995) is

(11)

The plots below show the differences between

and the two approximations.

The first Quartile of a standard Normal Distribution occurs when

(12)

The solution is

. The value of

giving

is known as the Probable Error of a normallydistributed variate.

See also Confidence Interval, Erf, Erfc, Fisher-Behrens Problem, Gaussian Distribution,Gaussian Integral, Hh Function, Normal Distribution, Probability Integral, TetrachoricFunction

References

Abramowitz, M. and Stegun, C. 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.

Johnson, N.; Kotz, S.; and Balakrishnan, N. Continuous Univariate Distributions, Vol. 1, 2nd ed. Boston, MA: Houghton Mifflin, 1994.

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