From 383d26adc6aea2c0a58a815c991218e7c171dcc6 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Mon, 23 Feb 2026 07:55:56 +0100 Subject: [PATCH] [PS] Intro: Independence and finish --- .../ps-jh/parts/00_basics/05_independence.tex | 18 ++++++++++++++++++ .../probability-and-statistics-cheatsheet.pdf | Bin 274666 -> 279920 bytes .../probability-and-statistics-cheatsheet.tex | 4 +++- 3 files changed, 21 insertions(+), 1 deletion(-) create mode 100644 semester4/ps/ps-jh/parts/00_basics/05_independence.tex 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 ea233c481392ee0201c9b86234c9967d78952855..4ab8a6695cbb39da4ad2aa1e4c3507df63be93af 100644 GIT binary patch delta 153725 zcmZs?V|1o%&~_Qy9ox2T8yzPd+jg$lzGK@R+fK(xM;)tUcI?dae)G-DnlW>e0XFGIKvfUiYDyrb{WyOl@yp{l&THc7y2=$XjxIgmZfG)X7yW zTWDiWo6qbUyZ(Hq;Q(e|?q}dR7GDlukM5iFYSUvEr%<0u_3V`ceLpTGr&8d^j44+Y 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