From e5b22cd4a3c02b31f0a3b3b5f7fb9cffb4fd88f5 Mon Sep 17 00:00:00 2001
From: Josia Pietsch <josia.pietsch@uni-bonn.de>
Date: Wed, 5 Jul 2023 17:53:33 +0200
Subject: [PATCH] some distributions

---
 inputs/a_0_distributions.tex | 86 +++++++++++++++++++++++++++++++++---
 1 file changed, 79 insertions(+), 7 deletions(-)

diff --git a/inputs/a_0_distributions.tex b/inputs/a_0_distributions.tex
index b7eabb0..338ac20 100644
--- a/inputs/a_0_distributions.tex
+++ b/inputs/a_0_distributions.tex
@@ -1,9 +1,81 @@
-\section{List of Distributions}
-Properties:
-Density, Distribution, $\Var, \bE$, $\phi$,
+\subsection{List of Distributions}
 
+{\rowcolors{2}{gray!10}{white}
+    \begin{longtable}{llllllll}
+        & Symbol & Mass (PMF) & Distribution (CDF) & $\bE$ & $\Var$ & $\phi_X(t) = \bE[e^{\i t X}]$ & $M_X(t) = \bE[e^{tX}]$ \\
+        \hline
+        Deterministic &
+            $\delta_a$ &
+            $\One_{x = a}$&
+            $\One_{[a,\infty)}$ &
+            $a$ &
+            $0$ &
+            $e^{\i t a}$ &
+            $e^{t a}$ \\
+        Bernoulli &
+            $\Bin(1,p)$ & & & & & &\\
+        Binomial &
+            $\Bin(n,p)$ &
+            $\binom{n}{k} p^{k} (1-p)^{n-k}$ &
+            $\sum_{j=0}^{\left\lfloor x \right\rfloor} \binom{n}{j} p^{j} (1-p)^{n-j}$ &
+            $n p$&
+            $n p (1-p)$ &
+            $((1-p) + pe^{\i t})^n$ &
+            $((1-p) + pe^t)^n$\\
+        Geometric &
+            $\Geo(p)$ &
+            $(1-p)^{k-1} p$ &
+            $1 - (1 - p)^{\left\lfloor x \right\rfloor}$
+            $\frac{1}{p}$ &
+            $\frac{1-p}{p^2}$&
+            $\frac{p e^{\i t}}{1 - (1 -p)e^{\i t}}$ &
+            $\frac{p e^{t}}{1 - (1 -p)e^{t}}$\\
+        Poisson &
+            $\Poi(\lambda)$ &
+            $\frac{\lambda^k e^{-\lambda}}{k!}$ &
+            $e^{-\lambda} \sum_{j=0}^{\left\lfloor x \right\rfloor} \frac{\lambda^j}{j!}$ &
+            $\lambda$ &
+            $\lambda$ &
+            $e^{\lambda (e^{\i t} -1)}$ &
+            $e^{\lambda (e^{t} -1)}$\\
+    \end{longtable}
+}
 
-Uniform
-Exponential
-Gaussian
-
+{\rowcolors{2}{gray!10}{white}
+    \begin{longtable}{llllllll}
+        & Symbol & Density (PDF) & Distribution (CDF) & $\bE$ & $\Var$ & $\phi_X(t) = \bE[e^{\i t X}]$ & $M_X(t) = \bE[e^{tX}]$ \\
+        \hline
+        Uniform &
+            $\Unif([a,b])$ &
+            $\frac{1}{b-a} \One_{[a,b]}$ &
+            $\frac{x-a}{b-a} \One_{[a,b]} + \One_{(b,\infty)}$ &
+            $\frac{a+b}{2}$ &
+            $\frac{(b-a)^2}{12}$ &
+            $\frac{e^{\i t b} - e^{\i t a}}{t (b-a)}$\footnote{$\phi_X(0) = 1$ }&
+            $\frac{e^{t b} - e^{t a}}{t (b-a)}$\footnote{$M_X(0) = 1$}\\
+        Exponential &
+            $\Exp(\lambda)$ &
+            $\One_{x \ge 0}\lambda e^{-\lambda x}$ &
+            $\One_{x \ge 0} (1 - e^{-\lambda x})$ &
+            $\frac{1}{\lambda}$ &
+            $\frac{1}{\lambda^2}$ &
+            $\frac{\lambda}{\lambda - \i t}$ &
+            $\frac{\lambda}{\lambda - t}, t < \lambda$\\
+        Cauchy &
+            $\Cauchy(x_0, \gamma)$ &
+            $\frac{1}{\pi \gamma \left(  1 + \left( \frac{x - x_0}{\gamma} \right)^2 \right) }$ &
+            $\frac{1}{\pi} \arctan \left( \frac{x - x_0}{\gamma} \right) + \frac{1}{2}$ &
+            n/a &
+            n/a &
+            $e^{x_0 \i t - \gamma |t|}$ &
+            n/a \\
+        Normal &
+            $\cN(\mu, \sigma)$ &
+            $\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(\mu - x)^2}{2 \sigma^2}}$ &
+            $\Phi\left( \frac{x - \mu}{\sigma} \right) $ &
+            $\mu$ &
+            $\sigma^2$ &
+            $e^{\i \mu t - \frac{\sigma^2 t^2}{2}}$ &
+            $e^{\mu t + \frac{\sigma^2 t^2}{2}}$\\
+  \end{longtable}
+}