added proof "cuni-qf"

I/Table_of_Contents.md

@@ -60,6 +60,7 @@ Proofs by **[Number](/I/Proof_by_Number.html)** and **[Topic](/I/Proof_by_Topic.
    &emsp;&ensp; 3.1.1. *[Definition](/D/cuni.html)* <br>
    &emsp;&ensp; 3.1.2. **[Probability density function](/P/cuni-pdf.html)** <br>
    &emsp;&ensp; 3.1.3. **[Cumulative distribution function](/P/cuni-cdf.html)** <br>
+   &emsp;&ensp; 3.1.4. **[Quantile function](/P/cuni-cdf.html)** <br>
 
    3.2. Normal distribution <br>
    &emsp;&ensp; 3.2.1. *[Definition](/D/norm.html)* <br>

P/cuni-qf.md

@@ -0,0 +1,62 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2020-01-02 18:27:00
+
+title: "Quantile function of the continuous uniform distribution"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "Continuous uniform distribution"
+theorem: "Quantile function"
+
+sources:
+
+proof_id: "P39"
+shortcut: "cuni-qf"
+username: "JoramSoch"
+---
+
+
+**Theorem:** Let $X$ be a random variable following a continuous uniform distribution:
+
+$$ \label{eq:cuni}
+X \sim \mathcal{U}(a, b) \; .
+$$
+
+Then, the quantile function of $X$ is
+
+$$ \label{eq:cuni-qf}
+Q_X(p) = bp + a(1-p) \; .
+$$
+
+
+**Proof:** The [cumulative distribution function of the continuous uniform distribution](/P/cuni-cdf.html) is:
+
+$$ \label{eq:cuni-cdf}
+F_X(x) =
+\begin{cases}
+\;\; 0 & , \text{if} \; x < a \\
+\frac{x-a}{b-a} & , \text{if} \; a \leq x \leq b \\
+\;\; 1 & , \text{if} \; x > b \; .
+\end{cases}
+$$
+
+Thus, the [quantile function](/D/qf.html) is:
+
+$$ \label{eq:cuni-qf-s1}
+Q_X(p) = F_X^{-1}(x) \; .
+$$
+
+This can be derived by rearranging equation \eqref{eq:cuni-cdf}:
+
+$$ \label{eq:cuni-cdf-s2}
+\begin{split}
+p &= \frac{x-a}{b-a} \\
+x &= p(b-a) + a \\
+x &= bp + a(1-p) = Q_X(p) \; .
+\end{split}
+$$