Normalizing constant: Difference between revisions

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The concept of a '''normalizing constant''' arises in [[probability theory]] and a variety of other areas of [[mathematics]]. The normalizing constant is used to reduce any probability function to a probability density function with total probability of one.

==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]].<ref>''Continuous Distributions'' at University of Alabama.</ref><ref>Feller, 1968, p. 22.</ref> For example, if we define

==Examples==

If we start from the simple [[Gaussian function]]

:<math>p(x)=e^{-x^2/2}, x\in(-\infty,\infty) </math>

we have the corresponding [[Gaussian integral]]

we have

:<math>\int_{-\infty}^\infty p(x)\,dx=\int_{-\infty}^\infty e^{-x^2/2}\,dx=\sqrt{2\pi\,},</math>

Now if we defineuse the latter's [[reciprocal value]] as a normalizing constant for the former, defining a function <math> \varphi(x) </math> as

:<math>\varphi(x)= \frac{1}{\sqrt{2\pi\,}} p(x) = \frac{1}{\sqrt{2\pi\,}} e^{-x^2/2} </math>

so that its [[integral of a Gaussian function|integral]] is unit

so that

:<math>\int_{-\infty}^\infty \varphi(x)\,dx=\int_{-\infty}^\infty \frac{1}{\sqrt{2\pi\,}} e^{-x^2/2}\,dx=1 </math>