# PlanetPhysics/Probability Distribution Functions in Physics

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This is a contributed topic on probability distribution functions and their


applications in physics, mostly in spectroscopy, quantum mechanics, statistical mechanics and the theory of extended QFT operator algebras (extended symmetry, quantum groupoids with Haar measure and quantum algebroids).

### Probability Distribution Functions in Physics

#### Physical Examples

{\mathbf Fermi-Dirac distribution}

This is a widely used probability distribution function (pdf) applicable to all fermion particles in quantum statistical mechanics, and is defined as:

${\displaystyle f_{D-F}(\epsilon )={\frac {1}{1+exp({\frac {\epsilon -\mu }{kT}})}},}$

where ${\displaystyle \epsilon }$ denotes the energy of the fermion system and ${\displaystyle \mu }$ is the chemical potential of the fermion system at an absolute temperature T.

A classical example of a continuous probability distribution function on ${\displaystyle \mathbb {R} }$ is the Gaussian distribution , or normal distribution ${\displaystyle f(x):={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-(x-m)^{2}/2\sigma ^{2}},}$ where ${\displaystyle \sigma ^{2}}$ is a parameter related to the width of the distribution (measured for example at half-heigth).

In high-resolution spectroscopy, however, similar but much narrower continuous distribution functions called Lorentzians are more common; for example, high-resolution ${\displaystyle ^{1}H}$ NMR absorption spectra of neat liquids consist of such Lorentzians whereas rigid solids exhibit often only Gaussian peaks resulting from both the overlap as well as the marked broadening of Lorentzians.

### General definitions of probability distribution functions

One needs to introduce first a Borel space $\displaystyle \borel$ , then consider a measure space $\displaystyle S_M:= (\Omega, \borel, \mu)$ , and finally define a real function that is measurable `almost everywhere' on its domain ${\displaystyle \Omega }$ and is also normalized to unity. Thus, consider $\displaystyle (\Omega, \borel, \mu)$ to be a measure space ${\displaystyle S_{M}}$. A probability distribution function (pdf) on (the domain) ${\displaystyle \Omega }$ is a function ${\displaystyle f_{p}:\Omega \longrightarrow \mathbb {R} }$ such that:

1. ${\displaystyle f_{p}}$ is ${\displaystyle \mu }$-measurable
2. ${\displaystyle f_{p}}$ is nonnegative ${\displaystyle \mu }$-measurable-almost everywhere.
3. ${\displaystyle f_{p}}$ satisfies the equation

${\displaystyle \int _{\Omega }f_{p}(x)\ d\mu =1.}$

Thus, a probability distribution function ${\displaystyle f_{p}}$ induces a probability measure ${\displaystyle M_{P}}$ on the measure space $\displaystyle (\Omega, \borel)$ , given by ${\displaystyle M_{P}(X):=\int _{X}f_{p}(x)\ d\mu =\int _{\Omega }1_{X}f_{p}(x)\ d\mu ,}$ for all $\displaystyle x \in \borel$ . The measure ${\displaystyle M_{P}}$ is called the associated probability measure of ${\displaystyle f_{p}}$. ${\displaystyle M_{P}}$ and ${\displaystyle \mu }$ are different measures although both have the same underlying measurable space $\displaystyle S_M := (\Omega, \borel)$ .

The discrete distribution (dpdf)

Consider a countable set ${\displaystyle I}$ with a counting measure imposed on ${\displaystyle I}$, such that ${\displaystyle \mu (A):=|A|}$, is the cardinality of ${\displaystyle A}$, for any subset ${\displaystyle A\subset I}$. A discrete probability distribution function (\mathbf dpdf) ${\displaystyle f_{d}}$ on ${\displaystyle I}$ can be then defined as a nonnegative function ${\displaystyle f_{d}:I\longrightarrow \mathbb {R} }$ satisfying the equation ${\displaystyle \sum _{i\in I}f_{d}(i)=1.}$

A simple example of a ${\displaystyle dpdf}$ is any Poisson distribution ${\displaystyle P_{r}}$ on $\displaystyle \naturals$ (for any real number ${\displaystyle r}$), given by the formula ${\displaystyle P_{r}(i):=e^{-r}{\frac {r^{i}}{i!}},}$ for any $\displaystyle i \in \naturals$ .

Taking any probability (or measure) space ${\displaystyle S_{M}}$ defined by the triplet $\displaystyle (\Omega, \borel, \mu)$ and a random variable ${\displaystyle X:\Omega \longrightarrow I}$, one can construct a distribution function on ${\displaystyle I}$ by defining ${\displaystyle f(i):=\mu (\{X=i\}).}$ The resulting ${\displaystyle \Delta }$ function is called the distribution of ${\displaystyle X}$ on ${\displaystyle I.}$

The continuous distribution (cpdf)

Consider a measure space ${\displaystyle S_{M}}$ specified as the triplet $\displaystyle (\reals, \borel_\lambda, \lambda)$ , that is, the set of real numbers equipped with a Lebesgue measure . Then, one can define a continuous probability distribution function (cpdf ) ${\displaystyle f_{c}:\mathbb {R} \longrightarrow \mathbb {R} }$ is simply a measurable, nonnegative almost everywhere function such that ${\displaystyle \int _{-\infty }^{\infty }f_{c}(x)\ dx=1.}$

The associated measure has a [RadonNikodymTheorem Radon--Nikodym derivative] with respect to ${\displaystyle \lambda }$ equal to ${\displaystyle f_{c}}$: ${\displaystyle {\frac {dP}{d\lambda }}=f_{c}.}$

One defines the cummulative distribution function, or {\mathbf cdf ,} ${\displaystyle F}$ of ${\displaystyle f_{c}}$ by the formula ${\displaystyle F(x):=P(\{X\leq x\})=\int _{-\infty }^{x}f(t)\ dt,}$ for all ${\displaystyle x\in \mathbb {R} .}$

## References

1. B. Aniszczyk. 1991. A rigid Borel space., Proceed. AMS. , 113 (4):1013-1015., available online.
2. A. Connes.1979. Sur la th\'eorie noncommutative de l' integration, {\em Lecture Notes in Math.}, Springer-Verlag, Berlin, {\mathbf 725}: 19-14.