diff --git a/semester4/ps/ps-jh/parts/00_basics/05_independence.tex b/semester4/ps/ps-jh/parts/00_basics/05_independence.tex new file mode 100644 index 0000000..098fd99 --- /dev/null +++ b/semester4/ps/ps-jh/parts/00_basics/05_independence.tex @@ -0,0 +1,18 @@ +\subsection{Unabhängigkeit} +\shortdefinition $A, B \in \cF$ \bi{unabh.} falls $\P[A \cap B] = \P[A] \cdot \P[B]$ + +\shortremark Falls $\P[A] \in \{0, 1\}$: $\forall B \in \cF \; \P[A \cap B] = \P[A] \P[B]$. +Falls $A$ unabh. von sich selbst ($\P[A \cap A] = \P[A]^2$), dann $\P[A] \in \{0, 1\}$. +\bi{Implik.}: $A$ unabh. v. $B$ $\Leftrightarrow$ $A$ unabh. v. $B^C$ + +\shorttheorem Sei $\P[A], \P[B] > 0$. Dann ist equivalent: +\begin{enumerate}[label=\textit{(\roman*)}] + \item $\P[A \cap B] = \P[A] \cdot \P[B]$\quad \textit{($A$ und $B$ unabhängig)} + \item $\P[A | B] = \P[A]$\quad \textit{(Eintreten von $B$ beinflusst $A$ nicht)} + \item $\P[B | A] = \P[B]$\quad \textit{(Eintreten von $A$ beinflusst $B$ nicht)} +\end{enumerate} + +\shortdefinition $I$ eine beliebige Menge. $(A_i)_{i \in I}$ \bi{unabhängig} falls: +\[ + \forall j \subseteq I \text{ endlich} \quad \P\left[ \bigcap_{j \in J} A_j \right] = \prod_{j \in J} \P[A_j] +\] diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index ea233c4..4ab8a66 100644 Binary files a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf and b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf differ diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex index 132fc1e..8116fb2 100644 --- a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex +++ b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex @@ -21,8 +21,10 @@ \input{parts/00_basics/02_properties-of-events.tex} \input{parts/00_basics/03_properties-of-measures.tex} \input{parts/00_basics/04_conditional-probability.tex} -% \input{parts/00_basics/} +\input{parts/00_basics/05_independence.tex} + +\newsection \section{Zufallsvar., Verteilungsfunktionen} \input{parts/01_random-variables/00_definition.tex} % \input{parts/01_random-variables/}