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probability distribution function

(Definition)

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:

is $\mu$ -measurable
is nonnegative $\mu$ -almost everywhere.
satisfies the equation
$\displaystyle \int_{\Omega} f(x)\ d\mu = 1$

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

$\displaystyle P(X) := \int_X f(x)\ d\mu = \int_{\Omega} 1_X f(x)\ d\mu,$

for all $X \in \mathfrak{B}$ . The measure

is called the associated probability measure of

. Note that

and $\mu$ are different measures, even though they both share the same underlying measurable space $(\Omega, \mathfrak{B})$ .

Examples

Discrete case

Let

be a countable set, and impose the counting measure on

( $\mu(A) := \vert A\vert$ , the cardinality of

, for any subset $A \subset I$ ). A probability distribution function on

is then a nonnegative function $f: I \longrightarrow \mathbb{R}$ satisfying the equation

$\displaystyle \sum_{i \in I} f(i) = 1.$

One example is the Poisson distribution on $\mathbb{N}$ (for any real number ), which is given by

$\displaystyle P_r(i) := e^{-r} \frac{r^i}{i!}$

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 by taking $f(i) := \mu(\{X = i\})$ . The resulting function is called the distribution of on .

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

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

The associated measure has Radon-Nikodym derivative with respect to $\lambda$ equal to

$\displaystyle \frac{dP}{d\lambda} = f.$

One defines the cumulative distribution function

by the formula

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

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

$\displaystyle f(x) := \frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-m)^2/2\sigma^2}.$