added proof "cuni-cdf"

I/Table_of_Contents.md

@@ -59,6 +59,7 @@ Proofs by **[Number](/I/Proof_by_Number.html)** and **[Topic](/I/Proof_by_Topic.
    3.1. Continuous uniform distribution <br>
    &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>
 
    3.2. Normal distribution <br>
    &emsp;&ensp; 3.2.1. *[Definition](/D/norm.html)* <br>

P/cuni-cdf.md

@@ -0,0 +1,86 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2020-01-02 18:05:00
+
+title: "Cumulative distribution function of the continuous uniform distribution"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "Continuous uniform distribution"
+theorem: "Cumulative distribution function"
+
+sources:
+
+proof_id: "P38"
+shortcut: "cuni-cdf"
+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 cumulative distribution function of $X$ is
+
+$$ \label{eq:cuni-cdf}
+F_X(x) =
+\begin{cases}
+\;\; 0 & , \text{if} \; x < a \\
+\frac{x-a}{b-a} & , \text{if} \; a \leq x \leq b \\
+\;\; 1 & , \text{if} \; x > b \; .
+\end{cases}
+$$
+
+
+**Proof:** The [probability density function of the continuous uniform distribution](/P/cuni-pdf.html) is:
+
+$$ \label{eq:cuni-pdf}
+\mathcal{U}(x; a, b) =
+\begin{cases}
+\frac{1}{b-a} & , \text{if} \; a \leq x \leq b \\
+\;\; 0 & , \text{otherwise} \; .
+\end{cases}
+$$
+
+Thus, the [cumulative distribution function](/D/cdf.html) is:
+
+$$ \label{eq:cuni-cdf-s1}
+F_X(x) = \int_{-\infty}^{x} \mathcal{U}(z; a, b) \, \mathrm{d}z
+$$
+
+If $x < a$, we have
+
+$$ \label{eq:cuni-cdf-s2a}
+F_X(x) = \int_{-\infty}^{x} 0 \, \mathrm{d}z = 0 \; .
+$$
+
+If $a \leq x \leq b$, we have
+
+$$ \label{eq:cuni-cdf-s2b}
+\begin{split}
+F_X(x) &= \int_{-\infty}^{a} \mathcal{U}(z; a, b) \, \mathrm{d}z + \int_{a}^{x} \mathcal{U}(z; a, b) \, \mathrm{d}z \\
+&= \int_{-\infty}^{a} 0 \, \mathrm{d}z + \int_{a}^{x} \frac{1}{b-a} \, \mathrm{d}z \\
+&= 0 + \frac{1}{b-a} [z]_a^x \\
+&= \frac{x-a}{b-a} \; .
+\end{split}
+$$
+
+If $x > b$, we have
+
+$$ \label{eq:cuni-cdf-s2c}
+\begin{split}
+F_X(x) &= \int_{-\infty}^{b} \mathcal{U}(z; a, b) \, \mathrm{d}z + \int_{b}^{x} \mathcal{U}(z; a, b) \, \mathrm{d}z \\
+&= F_X(b) + \int_{b}^{x} 0 \, \mathrm{d}z \\
+&= \frac{b-a}{b-a} + 0 \\
+&= 1 \; .
+\end{split}
+$$
+
+This completes the proof.