\subsection{Stetige Zufallsvariablen}
\shortdefinition $\cX$ stetig $\E[\cX] = \int_{-\8}^{\8} x f_\cX(x) \dx x$

\shorttheorem $\E[\varphi(\cX)] = \int_{-\8}^{\8} \varphi(x) f_\cX(x) \dx x$, falls int. wohldef.

\shortlemma[Int über gauss. Glockenk.] $\int_{-\8}^{\8} e^{\frac{-x^2}{2\sigma^2}} \dx x = \sqrt{2 \pi \sigma^2}$

\subsubsection{Beispiele}
\begin{itemize}
    \item $\cX \sim \cU([a, b])$, $a < b$: $\E[\cX] = \frac{a + b}{2}$
    \item $\cX \sim \text{Exp}(\lambda)$, $\lambda > 0$: $\E[\cX] = \frac{1}{\lambda}$
    \item $\cX \sim \cN(\mu, \sigma^2)$: $\E[\cX] = \mu$
    \item $\cX \sim \text{Cauchy}(x_0, \gamma)$: Existiert nicht (Int. $\8$)\\
          $\E[\cX_+] = \E[\cX_-] = \8$, Median: $0$
\end{itemize}
