added proof "cuni-pdf"

I/Table_of_Contents.md

@@ -58,6 +58,7 @@ 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>
 
    3.2. Normal distribution <br>
    &emsp;&ensp; 3.2.1. *[Definition](/D/norm.html)* <br>

P/cuni-pdf.md

@@ -0,0 +1,70 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2020-01-31 15:41:00
+
+title: "Probability density function of the continuous uniform distribution"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "Continuous uniform distribution"
+theorem: "Probability density function"
+
+sources:
+
+proof_id: "P37"
+shortcut: "cuni-pdf"
+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 probability density function of $X$ is
+
+$$ \label{eq:cuni-pdf}
+f_X(x) =
+\begin{cases}
+\frac{1}{b-a} & , \text{if} \; a \leq x \leq b \\
+\;\; 0 & , \text{otherwise} \; .
+\end{cases}
+$$
+
+
+**Proof:** A [continuous uniform variable is defined as](/D/cuni.html) having a constant probability density between minimum $a$ and maximum $b$. Therefore,
+
+$$ \label{eq:cuni-pdf-s1}
+\begin{split}
+f_X(x) &\propto 1 \quad \text{for all} \quad x \in [a,b] \quad \text{and} \\
+f_X(x) &= 0, \quad\!\! \text{if} \quad x < a \quad \text{or} \quad x > b \; .
+\end{split}
+$$
+
+To ensure that $f_X(x)$ [is a proper probability density function](/D/pdf.html), the integral over all non-zero probabilities has to sum to $1$. Therefore,
+
+$$ \label{eq:cuni-pdf-s2}
+f_X(x) = \frac{1}{c(a,b)} \quad \text{for all} \quad x \in [a,b]
+$$
+
+where the normalization factor $c(a,b)$ is specified, such that
+
+$$ \label{eq:cuni-pdf-s3}
+\frac{1}{c(a,b)} \int_{a}^{b} 1 \, \mathrm{d}x = 1 \; .
+$$
+
+Solving this for $c(a,b)$, we obtain:
+
+$$ \label{eq:cuni-pdf-s4}
+\begin{split}
+\int_{a}^{b} 1 \, \mathrm{d}x &= c(a,b) \\
+[x]_a^b &= c(a,b) \\
+c(a,b) &= b-a \; .
+\end{split}
+$$