Inversion Lemma (Quantiles)

Let FF be a CDF and p∈(0,1)p\in(0,1) and t∈Rt\in\mathbb{R}.

F(t)≥p⟺t≥QF−(p),F(t−)≤p⟺t≤QF+(p)F(t) \geq p \Longleftrightarrow t \geq Q_{F}^{-}(p), \quad F(t-) \leq p \Longleftrightarrow t \leq Q_{F}^{+}(p)

If FF is piecewise constant additionally:

F(t)>p⟺t≥QF+(p),F(t−)<p⟺t≤QF−(p)F(t)>p \Longleftrightarrow t \geq Q_{F}^{+}(p), \quad F(t-)<p \Longleftrightarrow t \leq Q_{F}^{-}(p) F(t)=sup⁡{p∈(0,1)∣F(t)≥p}=sup⁡{p∈(0,1)∣t≥QF−(p)}F(t−)=inf⁡{p∈(0,1)∣F(t−)≤p}=inf⁡{p∈(0,1)∣t≤QF+(p)}\begin{aligned} F(t) &=\sup \{p \in(0,1) \mid F(t) \geq p\}=\sup \left\{p \in(0,1) \mid t \geq Q_{F}^{-}(p)\right\} \\ F(t-) &=\inf \{p \in(0,1) \mid F(t-) \leq p\}=\inf \left\{p \in(0,1) \mid t \leq Q_{F}^{+}(p)\right\} \end{aligned}