[PS] Start adding more expected values / variance defs

semester4/ps/ps-jh/parts/03_expected-value/01_disc.tex

@@ -10,6 +10,9 @@
 \begin{itemize}
     \item $\cX \sim \text{Ber}(p)$: $\E[\cX] = p$ ($\E[1_A] = \P[A]$)
     \item $\cX \sim \text{Bin}(n, p)$: $\E[\cX] = np$
+    \item $\cX \sim \text{NBin}(r, p)$: $\E[\cX] = \frac{r(1 - p)}{p}$
+    \item $\cX \sim \text{Geom}(p)$: $\E[\cX] = \frac{1}{p}$
+    \item $\cX \sim \text{H}(n, r, m)$: $\E[\cX] = m\frac{n}{r}$
     \item $\cX \sim \text{Poisson}(\lambda)$: $\E[\cX] = \lambda$
 \end{itemize}
 

semester4/ps/ps-jh/parts/03_expected-value/05_variance.tex

@@ -16,10 +16,14 @@ $\V[\cX] = \E[\cX^2] - \E[\cX]^2 = a^2\E[1_\Omega] - a^2 = 0$
 \shortproposition $\cX_k$ paarw. unabh. $\V\left[ \sum_{k = 1}^{n} \cX_k \right] = \sum_{k = 1}^{n} \V[\cX_k]$.
 Falls $\cX_k$ abhängig, dann gilt $\neq$
 
+\newpage
 \shortexample Varianz von bekannten Verteilungen
 \begin{itemize}
     \item $\cX \sim \text{Ber}(p)$, $\V[\cX] = p (1 - p)$
     \item $\cX \sim \text{Bin}(n, p)$, $\V[\cX] = n p (1 - p)$
+    \item $\cX \sim \text{NBin}(r, p)$, $\V[\cX] = \frac{r(1 - p)}{p^2}$
+    \item $\cX \sim \text{Geom}(p)$, $\V[\cX] = \frac{1 - p}{p^2}$
+    \item $\cX \sim \text{H}(n, r, m)$, $\V[\cX] = m\frac{rn - r^2}{n^2} \frac{n - m}{n - 1}$
     \item $\cX \sim \text{Poisson}(\lambda)$, $\V[\cX] = \lambda = \E[\cX]$
     \item $\cX \sim \cU([a, b])$, $\V[\cX] = \frac{(b - a)^2}{12}$
     \item $\cX \sim \cN(\mu, \sigma^2)$, $\V[\cX] = \sigma^2$

semester4/ps/ps-jh/parts/03_expected-value/06_covariance.tex

@@ -1,4 +1,3 @@
-\newpage
 \subsection{Kovarianz}
 \shortdefinition $\cov(\cX, \cY) = \E[(\cX - \E[\cX])(\cY - \E[\cY])]$