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

单词

Moment-Generating Function

释义

Moment-Generating Function

Given a Random Variable , if there exists an such that

(1)

for

, then

(2)

is the moment-generating function.


			(3)

where

is the

th Moment about zero. The moment-generating function satisfies


			(4)

is differentiable at zero, then the

th Moments about the Origin are given by

(5)

(6)

(7)

(8)

The Mean and Variance are therefore

			(9)
			(10)

It is also true that

(11)

where

and

is the

th moment about the origin.

It is sometimes simpler to work with the Logarithm of the moment-generating function, which is also called theCumulant-Generating Function, and is defined by

			(12)
			(13)
			(14)

But

, so

			(15)
			(16)

See also Characteristic Function, Cumulant, Cumulant-Generating Function, Moment

References

Kenney, J. F. and Keeping, E. S. ``Moment-Generating and Characteristic Functions,'' ``Some Examples of Moment-Generating Functions,'' and ``Uniqueness Theorem for Characteristic Functions.'' §4.6-4.8 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 72-77, 1951.

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