ProbabilityDistributionFunctionsInPhysics 2009-04-22 12:45:27 2009-04-22 12:45:27 Topic probability distribution function probability measure associated probability measure %%Planet physics followed by special preamble \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % define commands here \newcommand{\md}{d} \newcommand{\mv}[1]{\mathbf{#1}} % matrix or vector \newcommand{\mvt}[1]{\mv{#1}^{\mathrm{T}}} \newcommand{\mvi}[1]{\mv{#1}^{-1}} \newcommand{\mderiv}[1]{\frac{\md}{\md {#1}}} %d/dx \newcommand{\mnthderiv}[2]{\frac{\md^{#2}}{\md {#1}^{#2}}} %d^n/dx \newcommand{\mpderiv}[1]{\frac{\partial}{\partial {#1}}} %partial d^n/dx \newcommand{\mnthpderiv}[2]{\frac{\partial^{#2}}{\partial {#1}^{#2}}} %partial d^n/dx \newcommand{\borel}{\mathfrak{B}} \newcommand{\integers}{\mathbb{Z}} \newcommand{\rationals}{\mathbb{Q}} \newcommand{\reals}{\mathbb{R}} \newcommand{\complexes}{\mathbb{C}} \newcommand{\naturals}{\mathbb{N}} \newcommand{\defined}{:=} \newcommand{\var}{\mathrm{var}} \newcommand{\cov}{\mathrm{cov}} \newcommand{\corr}{\mathrm{corr}} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\powerset}[1]{\mathcal{P}(#1)} \newcommand{\bra}[1]{\langle#1 \vert} \newcommand{\ket}[1]{\vert \hspace{1pt}#1\rangle} \newcommand{\braket}[2]{\langle #1 \ket{#2}} \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\norm}[1]{\left|\left|#1\right|\right|} \newcommand{\esssup}{\mathrm{ess\ sup}} \newcommand{\Lspace}[1]{L^{#1}} \newcommand{\Lone}{\Lspace{1}} \newcommand{\Ltwo}{\Lspace{2}} \newcommand{\Lp}{\Lspace{p}} \newcommand{\Lq}{\Lspace{q}} \newcommand{\Linf}{\Lspace{\infty}} \newcommand{\sequence}[1]{\{#1\}} This is a contributed topic on probability distribution functions and their applications in physics (mostly in quantum mechanics and statistical mechanics). \section{Probability Distribution Functions in Physics} One needs to introduce first a Borel space $\borel$, then a measure space $(\Omega, \borel, \mu)$, and finally define a real function that is measurable `almost everywhere' and is also normalized to unity. Let $(\Omega, \borel, \mu)$ be a measure space. A {\em probability distribution function} on (the domain) $\Omega$ is a function $f: \Omega \longrightarrow \reals$ such that: \begin{enumerate} \item $f$ is $\mu$-measurable \item $f$ is nonnegative $\mu$-almost everywhere. \item $f$ satisfies the equation $$ \int_{\Omega} f(x)\ d\mu = 1 $$ \end{enumerate} Thus, a probability distribution function $f$ induces a {\em probability measure} $P$ on the measure space $(\Omega, \borel)$, given by $$ P(X) := \int_X f(x)\ d\mu = \int_{\Omega} 1_X f(x)\ d\mu, $$ for all $X \in \borel$. The measure $P$ is called the {\em associated probability measure} of $f$. Note that $P$ and $\mu$ are different measures, \PMlinkescapetext{even} though they both share the same underlying measurable space $(\Omega, \borel)$. \section{Examples} \subsection{Discrete case} Let $I$ be a countable set, and impose the counting measure on $I$ ($\mu(A) := |A|$, the cardinality of $A$, for any subset $A \subset I$). A probability distribution function on $I$ is then a nonnegative function $f: I \longrightarrow \reals$ satisfying the equation $$ \sum_{i \in I} f(i) = 1. $$ One example is the Poisson distribution $P_r$ on $\naturals$ (for any real number $r$), which is given by $$ P_r(i) := e^{-r} \frac{r^i}{i!} $$ for any $i \in \naturals$. Given any probability space $(\Omega, \borel, \mu)$ and any random variable $X: \Omega \longrightarrow I$, we can form a distribution function on $I$ by taking $f(i) := \mu(\{X = i\})$. The resulting function is called the distribution of $X$ on $I$. \subsection{Continuous case} Suppose $(\Omega, \borel, \mu)$ equals $(\reals, \borel_\lambda, \lambda)$, the real numbers equipped with Lebesgue measure. Then a probability distribution function $f: \reals \longrightarrow \reals$ is simply a measurable, nonnegative almost everywhere function such that $$ \int_{-\infty}^\infty f(x)\ dx = 1. $$ The associated measure has \PMlinkname{Radon--Nikodym derivative}{RadonNikodymTheorem} with respect to $\lambda$ equal to $f$: $$ \frac{dP}{d\lambda} = f. $$ One defines the {\em cumulative distribution function} $F$ of $f$ by the formula $$ F(x) := P(\{X \leq x\}) = \int_{-\infty}^x f(t)\ dt, $$ for all $x \in \reals$. A well known example of a probability distribution function on $\reals$ is the Gaussian distribution, or normal distribution $$ f(x) := \frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-m)^2/2\sigma^2}. $$