added 2 definitions and 2 proofs

D/exp.md

@@ -0,0 +1,42 @@
+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2020-02-08 23:48:00
+
+title: "Exponential distribution"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "Exponential distribution"
+definition: "Definition"
+
+sources:
+  - authors: "Wikipedia"
+    year: 2020
+    title: "Exponential distribution"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2020-02-08"
+    url: "https://en.wikipedia.org/wiki/Exponential_distribution#Definitions"
+
+def_id: "D8"
+shortcut: "exp"
+username: "JoramSoch"
+---
+
+
+**Definition**: Let $X$ be a random variable. Then, $X$ is said to be exponentially distributed with rate (or, inverse scale) $\lambda$
+
+$$ \label{eq:exp}
+X \sim \mathrm{Exp}(\lambda) \; ,
+$$
+
+if and only if its probability density function is given by
+
+$$ \label{eq:exp-pdf}
+\mathrm{Exp}(x; \lambda) = \lambda \exp[-\lambda x], \quad x \geq 0
+$$
+
+where $\lambda > 0$, and the density is zero, if $x < 0$.

D/gam.md

@@ -0,0 +1,43 @@
+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2020-02-08 23:29:00
+
+title: "Gamma distribution"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "Gamma distribution"
+definition: "Definition"
+
+sources:
+  - authors: "Koch, Karl-Rudolf"
+    year: 2007
+    title: "Gamma Distribution"
+    in: "Introduction to Bayesian Statistics"
+    pages: "Springer, Berlin/Heidelberg, 2007, p. 47, eq. 2.172"
+    url: "https://www.springer.com/de/book/9783540727231"
+    doi: "10.1007/978-3-540-72726-2"
+
+def_id: "D7"
+shortcut: "gam"
+username: "JoramSoch"
+---
+
+
+**Definition**: Let $X$ be a random variable. Then, $X$ is said to follow a gamma distribution with shape $a$ and rate $b$
+
+$$ \label{eq:gam}
+X \sim \mathrm{Gam}(a, b) \; ,
+$$
+
+if and only if its probability density function is given by
+
+$$ \label{eq:gam-pdf}
+\mathrm{Gam}(x; a, b) = \frac{b^a}{\Gamma(a)} x^{a-1} \exp[-b x], \quad x > 0
+$$
+
+where $a > 0$ and $b > 0$, and the density is zero, if $x \leq 0$.

I/Table_of_Contents.md

@@ -70,6 +70,14 @@ Proofs by **[Number](/I/Proof_by_Number.html)** and **[Topic](/I/Proof_by_Topic.
    &emsp;&ensp; 3.2.5. **[Mode](/P/norm-mode.html)** <br>
    &emsp;&ensp; 3.2.6. **[Variance](/P/norm-var.html)** <br>
 
+   3.3. Gamma distribution <br>
+   &emsp;&ensp; 3.3.1. *[Definition](/D/gam.html)* <br>
+   &emsp;&ensp; 3.3.2. **[Probability density function](/P/gam-pdf.html)** <br>
+
+   3.4. Exponential distribution <br>
+   &emsp;&ensp; 3.4.1. *[Definition](/D/exp.html)* <br>
+   &emsp;&ensp; 3.4.2. **[Probability density function](/P/exp-pdf.html)** <br>
+
 4. Multivariate continuous distributions
 
    4.1. Multivariate normal distribution <br>

P/exp-pdf.md

@@ -0,0 +1,37 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2020-02-08 23:53:00
+
+title: "Probability density function of the exponential distribution"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "Exponential distribution"
+theorem: "Probability density function"
+
+sources:
+
+proof_id: "P46"
+shortcut: "exp-pdf"
+username: "JoramSoch"
+---
+
+
+**Theorem:** Let $X$ be a non-negative random variable following an exponential distribution:
+
+$$ \label{eq:exp}
+X \sim \mathrm{Exp}(\lambda) \; .
+$$
+
+Then, the probability density function of $X$ is
+
+$$ \label{eq:gam-pdf}
+f_X(x) = \lambda \exp[-\lambda x] \; .
+$$
+
+
+**Proof:** This follows directly from the [definition of the exponential distribution](/D/exp.html).

P/gam-pdf.md

@@ -0,0 +1,37 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2020-02-08 23:41:00
+
+title: "Probability density function of the gamma distribution"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "Gamma distribution"
+theorem: "Probability density function"
+
+sources:
+
+proof_id: "P45"
+shortcut: "gam-pdf"
+username: "JoramSoch"
+---
+
+
+**Theorem:** Let $X$ be a positive random variable following a gamma distribution:
+
+$$ \label{eq:gam}
+X \sim \mathrm{Gam}(a, b) \; .
+$$
+
+Then, the probability density function of $X$ is
+
+$$ \label{eq:gam-pdf}
+f_X(x) = \frac{b^a}{\Gamma(a)} x^{a-1} \exp[-b x] \; .
+$$
+
+
+**Proof:** This follows directly from the [definition of the gamma distribution](/D/gam.html).