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

单词

Chi-Squared Distribution

释义

Chi-Squared Distribution

A distribution is a Gamma Distribution with and , where is thenumber of Degrees of Freedom. If have NormalIndependent distributions with Mean 0 and Variance 1, then

(1)

is distributed as

with

Degrees of Freedom. If

are independentlydistributed according to a

distribution with

, ...,

Degrees of Freedom, then

(2)

is distributed according to

with

Degrees of Freedom.

(3)

The cumulative distribution function is then


			(4)

where

is a Regularized Gamma Function. The Confidence Intervals can befound by finding the value of

for which

equals a given value. The Moment-Generating Function of the

distribution is

			(5)
			(6)
			(7)
			(8)

			(9)
			(10)
			(11)
			(12)

The

th Moment about zero for a distribution with

Degrees of Freedom is

(13)

and the moments about the Mean are

			(14)
			(15)
			(16)

The

th Cumulant is

(17)

The Moment-Generating Function is



			(18)

(19)

so for large

(20)

is approximately a Gaussian Distribution with Mean

and Variance

. Fishershowed that

(21)

is an improved estimate for moderate

. Wilson and Hilferty showed that

(22)

is a nearly Gaussian Distribution with Mean

and Variance

In a Gaussian Distribution,

(23)

let

(24)

Then

(25)

(26)

But

(27)

(28)

This is a

distribution with

, since

(29)

If are independent variates with a Normal Distribution having Means andVariances for , ..., , then

(30)

is a Gamma Distribution variate with

(31)

The noncentral chi-squared distribution is given by

(32)

where

(33)

is the Confluent Hypergeometric Limit Function and

is the Gamma Function. The Mean,Variance, Skewness, and Kurtosis are

			(34)
			(35)
			(36)
			(37)

See also Chi Distribution, Snedecor's F-Distribution

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 940-943, 1972.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 535, 1987.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Incomplete Gamma Function, Error Function, Chi-Square Probability Function, Cumulative Poisson Function.'' §6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 209-214, 1992.

Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 115-116, 1992.

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