“Variance”的概念、定义、习题解题方法及翻译-数学辞典

单词

Variance

释义

Variance

For samples of a variate having a distribution with known Mean , the ``populationvariance'' (usually called ``variance'' for short, although the word ``population'' should be added when needed todistinguish it from the Sample Variance) is defined by



			(1)

where

(2)

But since

is an Unbiased Estimator for the Mean

(3)

it follows that the variance

(4)

The population Standard Deviation is then defined as

(5)

A useful identity involving the variance is

(6)

Therefore,




			(7)
			(8)

If the population Mean is not known, using the sample mean instead of the population mean tocompute

(9)

gives a Biased Estimator of the population variance. In such cases, it is appropriate to use a Student's t-Distribution instead of a Gaussian Distribution. However, it turns out (as discussedbelow) that an Unbiased Estimator for the population variance is given by

(10)

The Mean and Variance of the sample standard deviation for a distribution with population mean andVariance are

			(11)
			(12)

The quantity

has a Chi-Squared Distribution.

For multiple variables, the variance is given using the definition of Covariance,






			(13)

A linear sum has a similar form:



	(14)

These equations can be expressed using the Covariance Matrix.

To estimate the population Variance from a sample of elements with a priori unknown Mean (i.e.,the Mean is estimated from the sample itself), we need an Unbiased Estimator for . This is given by the k-Statistic , where

(15)

and

is the Sample Variance

(16)

Note that some authors prefer the definition

(17)

since this makes the sample variance an Unbiased Estimator for the population variance.

When computing numerically, the Mean must be computed before can be determined. This requires storing the set ofsample values. It is possible to calculate using a recursion relationship involving only the last sample asfollows. Here, use to denote calculated from the first samples (not the th Moment)

(18)

and

denotes the value for the sample variance

calculated from the first

samples. The first fewvalues calculated for the Mean are

			(19)
			(20)
			(21)

Therefore, for

, 3 it is true that

(22)

Therefore, by induction,


			(23)
			(24)
			(25)

and

(26)

for

, so



			(27)

Working on the first term,


			(28)

Use (24) to write

(29)

(30)

Now work on the second term in (27),

(31)

Considering the third term in (27),



	(32)

But

(33)



	(34)

Plugging (30), (31), and (34) into (27),






			(35)

(36)

To find the variance of itself, remember that

(37)

and

(38)

Now find



			(39)

Working on the first term of (39),




			(40)

The second term of (39) is known from k-Statistic,

(41)

as is the third term,


			(42)

Combining (39)-(42) gives






			(43)

so plugging in (38) and (43) gives




			(44)

Student calculated the Skewness and Kurtosis of the distribution of

			(45)
			(46)

and conjectured that the true distribution is Pearson Type III Distribution

(47)

where

			(48)
			(49)

This was proven by R. A. Fisher.

The distribution of itself is given by

(50)

(51)

where

(52)

The Moments are given by

(53)

and the variance is


			(54)

An Unbiased Estimator of

. Romanovsky showed that

(55)

See also Correlation (Statistical), Covariance, Covariance Matrix, k-Statistic, Mean, Sample Variance

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Moments of a Distribution: Mean, Variance, Skewness, and So Forth.'' §14.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 604-609, 1992.

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