ThmDex – An index of mathematical definitions, results, and conjectures.
Median need not be unique for a random real number
Formulation 0
Let $X \in \text{Bernoulli}(1 / 2)$ be a D3999: Standard Bernoulli random boolean number.
Then \begin{equation} \forall \, x \in (0, 1) : \mathbb{P}(X \leq x) \geq \frac{1}{2} \text{ and } \mathbb{P}(X \geq x) \geq \frac{1}{2} \end{equation}
Proofs
Proof 0
Let $X \in \text{Bernoulli}(1 / 2)$ be a D3999: Standard Bernoulli random boolean number.
Clear. $\square$