Editing Normalizing constant (section)

==Examples==

If we start from the simple [[Gaussian function]]
<math display="block">p(x) = e^{-x^2/2}, \quad x\in(-\infty,\infty) </math>
we have the corresponding [[Gaussian integral]]
<math display="block">\int_{-\infty}^\infty p(x) \, dx = \int_{-\infty}^\infty e^{-x^2/2} \, dx = \sqrt{2\pi\,},</math>

Now if we use the latter's [[reciprocal value]] as a normalizing constant for the former, defining a function <math> \varphi(x) </math> as
<math display="block">\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
<math display="block">\int_{-\infty}^\infty \varphi(x) \, dx = \int_{-\infty}^\infty \frac{1}{\sqrt{2\pi\,}} e^{-x^2/2} \, dx = 1 </math>
then the function <math> \varphi(x) </math> is a probability density function.<ref>{{harvnb|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 display="inline"> \frac{1}{\sqrt{2\pi}} </math> is the '''normalizing constant''' of function <math>p(x)</math>.

Similarly,
<math display="block">\sum_{n=0}^\infty \frac{\lambda^n}{n!} = e^{\lambda} ,</math>
and consequently
<math display="block">f(n) = \frac{\lambda^n e^{-\lambda}}{n!} </math>
is a probability mass function on the set of all nonnegative integers.<ref>{{harvnb|Feller|1968|p=156}}</ref>  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 (statistical mechanics)|partition function]].