ThmDex – An index of mathematical definitions, results, and conjectures.
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Definition D2900
Quantile function
Formulation 0
Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that
(i) $F : \mathbb{R} \to [0, 1]$ is a D205: Probability distribution function for $X$
The quantile function of $X$ is the D4364: Real function \begin{equation} [0, 1] \to \mathbb{R}, \quad x \mapsto \inf \left\{ t \in \mathbb{R} : x \leq F(t) \right\} \end{equation}
Formulation 1
Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number.
The quantile function of $X$ is the D4364: Real function \begin{equation} [0, 1] \to \mathbb{R}, \quad x \mapsto \inf \left\{ t \in \mathbb{R} : x \leq \mathbb{P}(X \leq t) \right\} \end{equation}