Category:Probability Measures

This category contains results about Probability Measures.
Definitions specific to this category can be found in Definitions/Probability Measures.

Let $\EE$ be an experiment.

Let $\EE$ be defined as a measure space $\struct {\Omega, \Sigma, \Pr}$.

Then $\Pr$ is a measure on $\EE$ such that $\map \Pr \Omega = 1$.

Let $\Omega$ be the sample space on $\EE$.

Let $\Sigma$ be the event space of $\EE$.

A probability measure on $\EE$ is a mapping $\Pr: \Sigma \to \R$ which fulfils the Kolmogorov axioms:

$(1)$	$:$	$\ds \forall A \in \Sigma:$	$\ds 0 $	$\ds \le $	$\ds \map \Pr A \le 1 $	The probability of an event occurring is a real number between $0$ and $1$
$(2)$	$:$		$\ds \map \Pr \Omega $	$\ds = $	$\ds 1 $	The probability of some elementary event occurring in the sample space is $1$
$(3)$	$:$		$\ds \map \Pr {\bigcup_{i \mathop \ge 1} A_i} $	$\ds = $	$\ds \sum_{i \mathop \ge 1} \map \Pr {A_i} $	where $\set {A_1, A_2, \ldots}$ is a countable (possibly countably infinite) set of pairwise disjoint events
						That is, the probability of any one of countably many pairwise disjoint events occurring
						is the sum of the probabilities of the occurrence of each of the individual events

Let $\Omega$ be the sample space on $\EE$.

Let $\Sigma$ be the event space of $\EE$.

A probability measure on $\EE$ is a mapping $\Pr: \Sigma \to \R$ which fulfils the following axioms:

$(\text I)$	$:$	$\ds \forall A \in \Sigma:$	$\ds \map \Pr A $	$\ds \ge $	$\ds 0 $
$(\text {II})$	$:$		$\ds \map \Pr \Omega $	$\ds = $	$\ds 1 $
$(\text {III})$	$:$	$\ds \forall A \in \Sigma:$	$\ds \map \Pr A $	$\ds = $	$\ds \sum_{\bigcup \set e \mathop = A} \map \Pr {\set e} $	where $e$ denotes the elementary events of $\EE$

Let $\Omega$ be the sample space on $\EE$.

Let $\Sigma$ be the event space of $\EE$.

A probability measure on $\EE$ is an additive function $\Pr: \Sigma \to \R$ which fulfils the following axioms:

$(1)$	$:$			$\ds \forall A, B \in \Sigma: A \cap B = \O:$		$\ds \map \Pr {A \cup B} $	$\ds = $	$\ds \map \Pr A + \map \Pr B $
$(2)$	$:$					$\ds \map \Pr \Omega $	$\ds = $	$\ds 1 $

Pages in category "Probability Measures"

The following 5 pages are in this category, out of 5 total.

\((1)\)	$:$	\(\ds \forall A \in \Sigma:\)	\(\ds 0 \)	\(\ds \le \)	\(\ds \map \Pr A \le 1 \)	The probability of an event occurring is a real number between $0$ and $1$
\((2)\)	$:$		\(\ds \map \Pr \Omega \)	\(\ds = \)	\(\ds 1 \)	The probability of some elementary event occurring in the sample space is $1$
\((3)\)	$:$		\(\ds \map \Pr {\bigcup_{i \mathop \ge 1} A_i} \)	\(\ds = \)	\(\ds \sum_{i \mathop \ge 1} \map \Pr {A_i} \)	where $\set {A_1, A_2, \ldots}$ is a countable (possibly countably infinite) set of pairwise disjoint events
						That is, the probability of any one of countably many pairwise disjoint events occurring
						is the sum of the probabilities of the occurrence of each of the individual events

\((\text I)\)	$:$	\(\ds \forall A \in \Sigma:\)	\(\ds \map \Pr A \)	\(\ds \ge \)	\(\ds 0 \)
\((\text {II})\)	$:$		\(\ds \map \Pr \Omega \)	\(\ds = \)	\(\ds 1 \)
\((\text {III})\)	$:$	\(\ds \forall A \in \Sigma:\)	\(\ds \map \Pr A \)	\(\ds = \)	\(\ds \sum_{\bigcup \set e \mathop = A} \map \Pr {\set e} \)	where $e$ denotes the elementary events of $\EE$

\((1)\)	$:$			\(\ds \forall A, B \in \Sigma: A \cap B = \O:\)		\(\ds \map \Pr {A \cup B} \)	\(\ds = \)	\(\ds \map \Pr A + \map \Pr B \)
\((2)\)	$:$					\(\ds \map \Pr \Omega \)	\(\ds = \)	\(\ds 1 \)