Definition:Random Variable/Real-Valued/Notation
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Notation
As an abuse of notation, we may write:
- $\set {\omega \in \Omega : \map X \omega \le x}$ as $\set {X \le x}$
- $\set {\omega \in \Omega : \map X \omega \ge x}$ as $\set {X \ge x}$
- $\set {\omega \in \Omega : \map X \omega < x}$ as $\set {X < x}$
- $\set {\omega \in \Omega : \map X \omega > x}$ as $\set {X > x}$
- $\set {\omega \in \Omega : \map X \omega = x}$ as $\set {X = x}$
- $\set {\omega \in \Omega : \map X \omega \in A}$ as $\set {X \in A}$
Generally, we write:
- $\set {\omega \in \Omega : \map P {\map X \omega} }$ as $\set {\map P X}$
for any propositional function of $\map X \omega$ such that:
- $\set {\omega \in \Omega : \map P {\map X \omega} }$ is $\Sigma$-measurable.
We may therefore write, for example:
- $\map \Pr {\set {\omega \in \Omega : \map X \omega = x} }$
as:
- $\map \Pr {\set {X = x} }$
Usually the curly brackets are dropped and we write:
- $\map \Pr {\set {\omega \in \Omega : \map X \omega = x} } = \map \Pr {X = x}$
Sources
- 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $10$: Probability: $10.1$: Basics