# probability distribution function

## 1 Definition

Let $(\Omega,\mathfrak{B},\mu)$ be a measure space. A probability distribution function on $\Omega$ is a function $f:\Omega\longrightarrow\mathbb{R}$ such that:

1. 1.

$f$ is $\mu$-measurable

2. 2.

$f$ is nonnegative $\mu$-almost everywhere.

3. 3.

$f$ satisfies the equation

 $\int_{\Omega}f(x)\ d\mu=1$

The main feature of a probability distribution function is that it induces a probability measure $P$ on the measure space $(\Omega,\mathfrak{B})$, given by

 $P(A):=\int_{A}f(x)\ d\mu=\int_{\Omega}1_{A}f(x)\ d\mu,$

for all $A\in\mathfrak{B}$. 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 $(\Omega,\mathfrak{B})$.

## 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 $f:I\longrightarrow\mathbb{R}$ 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 $(\Omega,\mathfrak{B},\mu)$ 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 $I$.

### 2.2 Continuous case

Suppose $(\Omega,\mathfrak{B},\mu)$ equals $(\mathbb{R},\mathfrak{B}_{\lambda},\lambda)$, the real numbers equipped with Lebesgue measure. Then a probability distribution function $f:\mathbb{R}\longrightarrow\mathbb{R}$ is simply a measurable, nonnegative almost everywhere function such that

 $\int_{-\infty}^{\infty}f(x)\ dx=1.$

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 $F$ of $f$ by the formula

 $F(x):=P(\{X\leq x\})=\int_{-\infty}^{x}f(t)\ dt,$

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

 $f(x):=\frac{1}{\sigma\sqrt{2\pi}}e^{-(x-m)^{2}/2\sigma^{2}}.$
 Title probability distribution function Canonical name ProbabilityDistributionFunction Date of creation 2013-03-22 12:37:25 Last modified on 2013-03-22 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