ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Subset
Power set
Hyperpower set sequence
Hyperpower set
Hypersubset
Subset algebra
Subset structure
Measurable space
Measurable map
Random variable
Random number
Random Euclidean number
Random euclidean real number
Symmetric random euclidean real number
Definition D5208
Symmetric random real number
Formulation 0
A D3161: Random real number $X \in \text{Random}(\mathbb{R})$ is symmetric about $a \in \mathbb{R}$ if and only if \begin{equation} X \overset{d}{=} 2 a - X \end{equation}
Formulation 1
Let $M = (\mathbb{R}, \mathcal{B}(\mathbb{R}))$ be the D5072: Standard real borel measurable space.
A D3161: Random real number $X \in \text{Random}(\mathbb{R})$ is symmetric about $a \in \mathbb{R}$ if and only if \begin{equation} \forall \, B \in \mathcal{B}(\mathbb{R}) : \mathbb{P}(X \in B) = \mathbb{P}(2 a - X \in B) \end{equation}