From 3b8d85523bdaa2fabcd0875756c2ef192cacd748 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Sun, 22 Feb 2026 16:14:49 +0100 Subject: [PATCH] [PS] Intro: Probably (lol) catch up --- .../00_basics/03_properties-of-measures.tex | 13 +++++++++++++ .../00_basics/04_conditional-probability.tex | 18 ++++++++++++++++++ .../probability-and-statistics-cheatsheet.pdf | Bin 270022 -> 274666 bytes .../probability-and-statistics-cheatsheet.tex | 1 + 4 files changed, 32 insertions(+) create mode 100644 semester4/ps/ps-jh/parts/00_basics/04_conditional-probability.tex diff --git a/semester4/ps/ps-jh/parts/00_basics/03_properties-of-measures.tex b/semester4/ps/ps-jh/parts/00_basics/03_properties-of-measures.tex index 28549f8..67dd950 100644 --- a/semester4/ps/ps-jh/parts/00_basics/03_properties-of-measures.tex +++ b/semester4/ps/ps-jh/parts/00_basics/03_properties-of-measures.tex @@ -15,5 +15,18 @@ \shorttheorem[Union Bound] Für $A_1, A_2, \ldots$ (mögl. disj.) gilt: $\P\left[ \bigcup_{i = 1}^\infty A_i \right] \leq \sum_{i = 1}^{\infty} \P[A_i]$. Auch für endl. n.-leere Ereignisse +\stepLabelNumber{combined} + \subsubsection{Anwendungen der Ungleichungen} +Sie sind nützlich für schwer zu berechnende W. + +\shorttheorem $(A_n)$ mit $A_n \subseteq A_{n + 1}$ (mon. wachsend). Dann: + +{\centering $\limni P[A_n] = \P\left[ \bigcup_{n = 1}^\8 A_n \right]$\\} + +und für $(B_n)$ mit $B_n \supseteq B_{n + 1}$ (mon. fallend) gilt:\\ +$\limni P[B_n] = \P\left[ \bigcap_{n = 1}^\8 \right]$ + +\shortremark Mit Monotonie: $\P[A_n] \leq \P[A_{n + 1}]$ und\\ +$\P[B_n] \geq \P[B_{n + 1}]$. Grenzwerte oben wohldefiniert. diff --git a/semester4/ps/ps-jh/parts/00_basics/04_conditional-probability.tex b/semester4/ps/ps-jh/parts/00_basics/04_conditional-probability.tex new file mode 100644 index 0000000..9ec96e1 --- /dev/null +++ b/semester4/ps/ps-jh/parts/00_basics/04_conditional-probability.tex @@ -0,0 +1,18 @@ +\subsection{Bedingte Wahrscheinlichkeit} +\shortdefinition Für $(\Omega, \cF, \P)$ mit $A, B \in \cP(\Omega)$ mit $\P[B] > 0$: +\[ + \P[A | B] = \frac{\P[A \cap B]}{\P[B]} +\] +\shortremark $\P[B | B] = 1$ + +\shorttheorem $B \in \cP(\Omega)$, dann ist $\P[ \cdot | B]$ ein W-Mass auf $\Omega$ + +\shorttheorem[Totale W.] $\Omega = B_1 \cup \dots \cup B_n$ mit $B_i$s eine Partition von $\Omega$, mit $B_i$ paarw. disj. und $\P[B_i] > 0\; \forall 1 \leq i \leq n$. Dann: +\[ + \forall A \in \cF \quad \P[A] = \sum_{i = 1}^{n} \P[A | B_i] \cdot \P[B_i] +\] + +\shorttheorem[Bayes] $B_i$ wie oben, damm $\forall A$ mit $\P[A] > 0$: +\[ + \forall i = 1, \ldots, n \quad \P[B_i | A] = \frac{\P[A | B_i] \P[B_i]}{\sum_{j = 1}^{n} \P[A | B_j] \P[B_j]} +\] diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index 045fdfbba8585b00277ea6d94a89438bd48ccce4..ea233c481392ee0201c9b86234c9967d78952855 100644 GIT binary patch delta 129244 zcmV)5K*_(xy%6f*5U@rGm*CR_6a+UiGc=Q71So&)JzbLULZP`k6cl(<}l%e8WP$zQ*205by&f#D2EIkL4?mdPDrfCkX$?`{;6 zi%Bv0>fWR0_r7>S3o|K6shFBPKc7%d3V5K3f(R`q&(9_=rvQcytjWXuO_C@#ApHEmy>7rez;fodDov?ea86{=9|UMX7LtV!iZ`@DJi97b~1m8 zXc%wC3^65jZ~k?@THL_9Oi%y&cJsl#2FMb;27nSxrCrcc_z^ERn|C)4zWCw|zJCdC zlGE#})S!$LW)uv{i4cC0j1e4CO`n{=xYcX?|N3@rKim(jV9NIfun5A1aAy}ShqG~eRn_?0!8Fbl zfpGxQ{xN~={%T_P{GP=%3CM2$r2G3TV5Go*Bv*pHN~HjE`%?qyeO@J7^x9PVR9@lq{)iA3|3EG%J5fZrX?N 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