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\begin{document}
This is a contributed topic on probability distribution functions and their
applications in physics, mostly in spectroscopy, \htmladdnormallink{quantum mechanics}{http://planetphysics.us/encyclopedia/QuantumParadox.html}, \htmladdnormallink{statistical mechanics}{http://planetphysics.us/encyclopedia/ThermodynamicLaws.html} and the theory of extended \htmladdnormallink{QFT}{http://planetphysics.us/encyclopedia/HotFusion.html} \htmladdnormallink{operator algebras}{http://planetphysics.us/encyclopedia/Groupoid.html} (extended symmetry, \htmladdnormallink{quantum groupoids}{http://planetphysics.us/encyclopedia/QuantumGroupoids.html} with \htmladdnormallink{Haar measure}{http://planetphysics.us/encyclopedia/HigherDimensionalQuantumAlgebroid.html} and \htmladdnormallink{quantum algebroids}{http://planetphysics.us/encyclopedia/LongRangeCoupling.html}).
\subsection{Probability Distribution Functions in Physics}
\subsubsection{Physical Examples}
\begin{example}
{\bf \htmladdnormallink{Fermi-Dirac distribution}{http://planetphysics.us/encyclopedia/FermiDiracDistribution.html}}
This is a widely used probability distribution function (pdf) applicable to all \htmladdnormallink{fermion}{http://planetphysics.us/encyclopedia/QuarkAntiquarkPair.html} \htmladdnormallink{particles}{http://planetphysics.us/encyclopedia/Particle.html} in \htmladdnormallink{quantum statistical mechanics}{http://planetphysics.us/encyclopedia/QuantumStatisticalTheories.html}, and is defined as:
\[
f_{D-F}(\epsilon) = \frac{1}{1+exp(\frac{\epsilon - \mu}{kT})},
\]
where $\epsilon$ denotes the \htmladdnormallink{energy}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} of the fermion \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} and $\mu$ is the {\em chemical potential} of the fermion system at an \htmladdnormallink{absolute temperature}{http://planetphysics.us/encyclopedia/ThermodynamicLaws.html} T.
\end{example}
\begin{example}
A classical example of a continuous probability distribution function on $\reals$ is the {\em Gaussian distribution}, or {\em normal distribution}
$$ f(x) := \frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-m)^2/2\sigma^2},$$
where $\sigma^2$ is a \htmladdnormallink{parameter}{http://planetphysics.us/encyclopedia/Parameter.html} related to the width of the distribution (measured for example at half-heigth).
\end{example}
In high-resolution spectroscopy, however, similar but much narrower continuous distribution \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} called {\em Lorentzians} are more common; for example, high-resolution $^1H$ \htmladdnormallink{NMR}{http://planetphysics.us/encyclopedia/SpectralImaging.html} \htmladdnormallink{absorption}{http://planetphysics.us/encyclopedia/FluorescenceCrossCorrelationSpectroscopy.html} spectra of neat liquids consist of such Lorentzians whereas rigid \htmladdnormallink{solids}{http://planetphysics.us/encyclopedia/CoIntersections.html} exhibit often only Gaussian peaks resulting from both the overlap as well as the marked broadening of Lorentzians.
\subsection{General definitions of probability distribution functions}
\begin{definition}
One needs to introduce first a \htmladdnormallink{Borel space}{http://planetphysics.us/encyclopedia/BorelSpace.html} $\borel$, then consider a {\em measure space} $S_M:= (\Omega, \borel, \mu)$, and finally define a real function that is measurable `almost everywhere' on its \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html} $\Omega$ and is also normalized to unity. Thus, consider $(\Omega, \borel, \mu)$ to be a measure space $S_M$. A {\em probability distribution function (pdf)} on (the domain) $\Omega$ is a function $f_p: \Omega \longrightarrow \reals$ such that:
\begin{enumerate}
\item $f_p$ is $\mu$-measurable
\item $f_p$ is nonnegative $\mu$-measurable-almost everywhere.
\item $f_p$ satisfies the equation
$$
\int_{\Omega} f_p(x)\ d\mu = 1.
$$
\end{enumerate}
\end{definition}
Thus, a probability distribution function $f_p$ induces a {\em probability measure} $M_P$ on the measure space $(\Omega, \borel)$, given by
$$M_P(X) := \int_X f_p(x)\ d\mu = \int_{\Omega} 1_X f_p(x)\ d\mu,$$
for all $x \in \borel$. The measure $M_P$ is called the {\em associated probability measure} of $f_p$. $M_P$ and $\mu$ are different measures although both have the same underlying \htmladdnormallink{measurable space}{http://planetphysics.us/encyclopedia/InvariantBorelSet.html} $S_M := (\Omega, \borel)$.
\begin{definition}
\textbf{The discrete distribution (dpdf)}
Consider a countable set $I$ with a counting measure imposed on $I$, such that $\mu(A) := |A|$, is the cardinality of $A$, for any subset $A \subset I$. A {\em discrete probability distribution function (\bf dpdf)} $f_d$ on $I$ can be then defined as a nonnegative function $f_d : I \longrightarrow \reals$ satisfying the equation
$$\sum_{i \in I} f_d(i) = 1.$$
\end{definition}
A simple example of a $dpdf$ is any Poisson distribution $P_r$ on $\naturals$ (for any real number $r$), given by the \htmladdnormallink{formula}{http://planetphysics.us/encyclopedia/Formula.html} $$ P_r(i) := e^{-r} \frac{r^i}{i!}, $$
for any $i \in \naturals$.
Taking any probability (or measure) space $S_M$ defined by the triplet $(\Omega, \borel, \mu)$ and a {\em random variable} $X: \Omega \longrightarrow I$, one can construct a distribution function on $I$ by defining
$$f(i) := \mu(\{X = i\}).$$ The resulting $\Delta$ function is called the {\em distribution of $X$ on $I.$}
\begin{definition}
\textbf{The continuous distribution (cpdf)}
Consider a measure space $S_M$ specified as the triplet
$(\reals, \borel_\lambda, \lambda)$, that is, the set of real numbers equipped with a {\em Lebesgue measure}. Then, one can define a {\em continuous probability distribution function} ({\em cpdf}) $f_c : \reals \longrightarrow \reals$ is simply a measurable, nonnegative almost everywhere function such that
$$ \int_{-\infty}^\infty f_c(x)\ dx = 1.$$
\end{definition}
The associated measure has a \htmladdnormallink{Radon--Nikodym derivative}{RadonNikodymTheorem} with respect to $\lambda$ equal to $f_c$:
$$ \frac{dP}{d\lambda} = f_c.$$
\begin{definition}
One defines the {\em cummulative distribution function, or {\bf cdf},} $F$ of $f_c$ by the formula
$$F(x) := P(\{X \leq x\}) = \int_{-\infty}^x f(t)\ dt, $$
for all $x \in \reals.$
\end{definition}
\begin{thebibliography}{9}
\bibitem{BA91}
B. Aniszczyk. 1991. A rigid Borel space., {\em Proceed. AMS.}, 113 (4):1013-1015.,
\htmladdnormallink{available online}{http://www.jstor.org/pss/2048777}.
\bibitem{AC79}
A. Connes.1979. Sur la th\'eorie noncommutative de l' integration, {\em Lecture Notes in
Math.}, Springer-Verlag, Berlin, {\bf 725}: 19-14.
\end{thebibliography}
\end{document}
