Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Other probability functions ▷ Quantile function in terms of cumulative distribution function

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

\[\label{eq:qf-cdf-qed} Q_X(p) = F_X^{-1}(x) \; .\]
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Metadata: ID: P192 | shortcut: qf-cdf | author: JoramSoch | date: 2020-11-12, 07:48.