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Student,s z-Distribution

المؤلف:  Kenney, J. F. and Keeping, E. S.

المصدر: 

الجزء والصفحة:  pp. 174-175

14-4-2021

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Student's z-Distribution

The probability density function for Student's -distribution is given by

(1)

Now define

(2)

then the cumulative distribution function is given by

{d_n(z) for z<=0; 1-d_n(z) for z>=0 " src="https://mathworld.wolfram.com/images/equations/Studentsz-Distribution/NumberedEquation3.gif" style="height:41px; width:178px" />

(3)

The mean is 0, so the moments are

			(4)
			(5)
			(6)
			(7)

The mean, variance, skewness, and kurtosis excess are

			(8)
			(9)
			(10)
			(11)

The characteristic function is

(12)

where  is a modified Bessel function of the second kind.

Letting

(13)

where  is the sample mean and  is the population mean gives Student's t-distribution.

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REFERENCES:

Kenney, J. F. and Keeping, E. S. "'Student's' -Distribution." §7.11 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 174-175, 1951.