probability distribution function

1 Definition

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

1. $f$ is $\mu$-measurable

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

3. $f$ satisfies the equation

   $${\int }_{\mathrm{\Omega }}f(x)𝑑\mu =1$$

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

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

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

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 $f:I⟶ℝ$ 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 $(\mathrm{\Omega },𝔅,\mu )$ and any random variable $X:\mathrm{\Omega }⟶I$, 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 $(\mathrm{\Omega },𝔅,\mu )$ equals $(ℝ,{𝔅}_{\lambda },\lambda )$, the real numbers equipped with Lebesgue measure. Then a probability distribution function $f:ℝ⟶ℝ$ is simply a measurable, nonnegative almost everywhere function such that

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)𝑑x=1.$$

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 cumulative distribution function $F$ of $f$ by the formula

$$F(x):=P(\{X\le x\})={\int }_{-\mathrm{\infty }}^{x}f(t)dt,$$

for all $x\in ℝ$. A well known example of a probability distribution function on $ℝ$ 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