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Normalizing constant : ウィキペディア英語版
Normalizing constant

The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics.
==Definition and examples==

In probability theory, a normalizing constant is a constant by which an everywhere non-negative function must be multiplied so the area under its graph is 1, e.g., to make it a probability density function or a probability mass function.〔''Continuous Distributions'' at University of Alabama.〕〔Feller, 1968, p. 22.〕 For example, if we define
:p(x)=e^, x\in(-\infty,\infty)
we have
:\int_^\infty p(x)\,dx=\int_^\infty e^\,dx=\sqrt,
if we define a function \varphi(x) as
:\varphi(x)= \frac
so that
:\int_^\infty \varphi(x)\,dx=\int_^\infty \frac\,dx=1
then the function \varphi(x) is a probability density function.〔Feller, 1968, p. 174.〕 This is the density of the standard normal distribution. (''Standard'', in this case, means the expected value is 0 and the variance is 1.)
And constant \frac^\infty \frac=e^\lambda ,
and consequently
:f(n)=\frac
is a probability mass function on the set of all nonnegative integers.〔Feller, 1968, p. 156.〕 This is the probability mass function of the Poisson distribution with expected value λ.
Note that if the probability density function is a function of various parameters, so too will be its normalizing constant. The parametrised normalizing constant for the Boltzmann distribution plays a central role in statistical mechanics. In that context, the normalizing constant is called the partition function.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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