Quantile function

The quantile function or (generalized) inverse distribution function of a real random variable with cumulative distribution function $𝐹 : ℝ \to [0, 1]$ is defined as #m/def/prob

𝐹 - 1 : (0, 1) \to ℝ 𝑝 \mapsto i n f {𝑥 \in ℝ : 𝐹 (𝑥) \geq 𝑝}

and is thus a left inverse of $𝐹$ .

Proof of left-inverse

Note that since $𝐹$ is monotone increasing, $𝐹 (𝑦) \geq 𝐹 (𝑥)$ implies $𝑦 \geq 𝑥$ , so

𝐹 - 1 𝐹 (𝑥) = i n f {𝑦 \in ℝ : 𝐹 (𝑦) \geq 𝐹 (𝑥)} = 𝑥

as required.

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