probability distribution function
1 Definition
Let $(\mathrm{\Omega},\U0001d505,\mu )$ be a measure space^{}. A probability distribution function on $\mathrm{\Omega}$ is a function $f:\mathrm{\Omega}\u27f6\mathbb{R}$ such that:

1.
$f$ is $\mu $measurable

2.
$f$ is nonnegative $\mu $almost everywhere.
 3.
The main feature of a probability distribution function is that it induces a probability measure $P$ on the measure space $(\mathrm{\Omega},\U0001d505)$, given by
$$P(A):={\int}_{A}f(x)\mathit{d}\mu ={\int}_{\mathrm{\Omega}}{1}_{A}f(x)\mathit{d}\mu ,$$ 
for all $A\in \U0001d505$. The measure $P$ is called the associated probability measure of $f$. Note that $P$ and $\mu $ are different measures, though they both share the same underlying measurable space^{} $(\mathrm{\Omega},\U0001d505)$.
2 Examples
2.1 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 $f:I\u27f6\mathbb{R}$ satisfying the equation
$$\sum _{i\in I}f(i)=1.$$ 
One example is the Poisson distribution^{} ${P}_{r}$ on $\mathbb{N}$ (for any real number $r$), which is given by
$${P}_{r}(i):={e}^{r}\frac{{r}^{i}}{i!}$$ 
for any $i\in \mathbb{N}$.
Given any probability space $(\mathrm{\Omega},\U0001d505,\mu )$ and any random variable^{} $X:\mathrm{\Omega}\u27f6I$, 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 $I$.
2.2 Continuous case
Suppose $(\mathrm{\Omega},\U0001d505,\mu )$ equals $(\mathbb{R},{\U0001d505}_{\lambda},\lambda )$, the real numbers equipped with Lebesgue measure^{}. Then a probability distribution function $f:\mathbb{R}\u27f6\mathbb{R}$ is simply a measurable, nonnegative almost everywhere function such that
$${\int}_{\mathrm{\infty}}^{\mathrm{\infty}}f(x)\mathit{d}x=1.$$ 
The associated measure has Radon–Nikodym derivative^{} (http://planetmath.org/RadonNikodymTheorem) with respect to $\lambda $ equal to $f$:
$$\frac{dP}{d\lambda}=f.$$ 
One defines the cumulative distribution function^{} $F$ of $f$ by the formula^{}
$$F(x):=P(\{X\le x\})={\int}_{\mathrm{\infty}}^{x}f(t)dt,$$ 
for all $x\in \mathbb{R}$. A well known example of a probability distribution function on $\mathbb{R}$ is the Gaussian distribution, or normal distribution^{}
$$f(x):=\frac{1}{\sigma \sqrt{2\pi}}{e}^{{(xm)}^{2}/2{\sigma}^{2}}.$$ 
Title  probability distribution function 
Canonical name  ProbabilityDistributionFunction 
Date of creation  20130322 12:37:25 
Last modified on  20130322 12:37:25 
Owner  Mathprof (13753) 
Last modified by  Mathprof (13753) 
Numerical id  11 
Author  Mathprof (13753) 
Entry type  Definition 
Classification  msc 60E99 
Synonym  probability density function 
Synonym  distribution 
Related topic  Measure 
Related topic  Stochastic 
Related topic  DiscreteDensityFunction 
Related topic  DistributionFunction 
Related topic  DensityFunction 
Related topic  AreaUnderGaussianCurve 
Defines  cumulative distribution function 