Proof: Quantile function is inverse of strictly monotonically increasing cumulative distribution function
Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Other probability functions ▷ Quantile function in terms of cumulative distribution function
Metadata: ID: P192 | shortcut: qf-cdf | author: JoramSoch | date: 2020-11-12, 07:48.
Theorem: Let $X$ be a continuous random variable with the cumulative distribution function $F_X(x)$. If the cumulative distribution function is strictly monotonically increasing, then the quantile function is identical to the inverse of $F_X(x)$:
\[\label{eq:qf-cdf} Q_X(p) = F_X^{-1}(x) \; .\]Proof: The quantile function $Q_X(p)$ is defined as the function that, for a given quantile $p \in [0,1]$, returns the smallest $x$ for which $F_X(x) = p$:
\[\label{eq:qf} Q_X(p) = \min \left\lbrace x \in \mathbb{R} \, \vert \, F_X(x) = p \right\rbrace \; .\]If $F_X(x)$ is continuous and strictly monotonically increasing, then there is exactly one $x$ for which $F_X(x) = p$ and $F_X(x)$ is an invertible function, such that
\[\label{eq:qf-cdf-qed} Q_X(p) = F_X^{-1}(x) \; .\]∎
Sources: - Wikipedia (2020): "Quantile function"; in: Wikipedia, the free encyclopedia, retrieved on 2020-11-12; URL: https://en.wikipedia.org/wiki/Quantile_function#Definition.
Metadata: ID: P192 | shortcut: qf-cdf | author: JoramSoch | date: 2020-11-12, 07:48.