ThmDex – An index of mathematical definitions, results, and conjectures.

Result R3738 on D2899: Random real number median

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}

Also known as

Random real number may have several medians, Counterexample: random real number with multiple medians, Counterexample: random real number with uncountably many medians

Proofs

Proof 0

Let $X \in \text{Bernoulli}(1 / 2)$ be a D3999: Standard Bernoulli random boolean number.

Clear. $\square$