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Countable Random Variable Riemann Multiple Integral Equation Cardinality Continuation Of Exponent Set Real Number Relation Theory Measure Borel Space Almost Surely Lebesgue Measure Normal Random Variable Radon Nikodym Theorem Poisson Random Variable Counting Measure

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Distribution

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

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

1	Random Variable
1	Riemann Multiple Integral
1	Measure
1	Real Number
1	Countable
1	Set
1	Cardinality
1	Relation Theory
1	Continuation Of Exponent
1	Equation
2	Almost Surely
2	Borel Space
3	Lebesgue Measure
4	Normal Random Variable
5	Counting Measure
7	Poisson Random Variable
9	Radon Nikodym Theorem