Normalizing constant: Difference between revisions

Revision as of 20:55, 11 January 2011

Revision as of 09:04, 15 January 2011

Line 3:

==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 havedefine

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

we have

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

if we define function <math> \varphi(x) </math> as

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

so that

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

Function <math> \varphi(x) </math> is a probability density function.<ref>Feller, 1968, p. 174.</ref> 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 <math> \frac{1}{\sqrt{2\pi\,}} </math> is the '''normalizing constant''' of function <math>p(x)</math>.

is a probability density function.<ref>Feller, 1968, p. 174.</ref> This is the density of the standard [[normal distribution]]. (''Standard'', in this case, means the [[expected value]] is 0 and the [[variance]] is 1.)

Similarly,