probability distribution function

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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.

$f$ is $\mu$ -measurable
2.

$f$ is nonnegative $\mu$ -almost everywhere.
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, even 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 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\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}}}.

Defines:

cumulative distribution function

Keywords:

distribution, random variable

Synonym:

probability density function, distribution

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

Editing Group for Distribution

Mathematics Subject Classification

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probability distribution function

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Primary tabs

probability distribution function

1 Definition

2 Examples

2.1 Discrete case

2.2 Continuous case

Mathematics Subject Classification

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probability distribution function

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Primary tabs

probability distribution function

1 Definition

2 Examples

2.1 Discrete case

2.2 Continuous case

Mathematics Subject Classification

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