From 1f2bdb65f53b926b40cf83a54d0207ba15051b29 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Mon, 6 Jul 2026 18:12:31 +0200 Subject: [PATCH] [PS] Improve Geometric distribution definition --- .../03_distributions/01_geom.tex | 6 +++--- .../probability-and-statistics-cheatsheet.pdf | Bin 1120932 -> 1121086 bytes 2 files changed, 3 insertions(+), 3 deletions(-) diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/01_geom.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/01_geom.tex index b301c4b..69e2294 100644 --- a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/01_geom.tex +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/01_geom.tex @@ -1,8 +1,8 @@ \subsubsection{Geometrische Verteilung} {\scriptsize Warten auf den ersten Erfolg (in $\8$ Folge von Bernoulli-Experimenten)} -\shortdefinition $\cX \sim \text{Geom}(p)$ mit $W = \N \backslash \{0\}$ falls\\ -$\forall k \in W \; \P[\cX = k] = (1 - p)^{k - 1} \cdot p$ +\shortdefinition $\cX \sim \text{Geom}(p)$ mit $W = \N \backslash \{0\}$ falls $\forall k \in W$\\ +$\P[\cX = k] = \P[\cX > (k - 1)] \cdot p$ mit $\P[\cX > k] = (1 - p)^{k}$ \shortremark $\P[\cX = 1] = p$, da wir Konvetion $a^0 = 1$ verwenden. @@ -13,5 +13,5 @@ Dann $( T := \min\{ n \geq 1 \divider X_n = 1 \} )\sim \text{Geom}(p)$ \shortremark $T = \8$ ist möglich, $\P[T = \8] = 0$ -\shorttheorem $T \sim \text{Geom}(p)$, dann\\ +\shorttheorem[Gedächtnislosigkeit] $T \sim \text{Geom}(p)$, dann\\ $\forall n \geq 0 \; \forall k \geq 1 \; \P[T \geq n + k \divider T > n] = \P[T \geq k]$ diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index 729ddddfa9212f78c49420015d7a4bb425c46c67..791c752f1a7df1df935846dc8f2b05a0603bbc88 100644 GIT binary patch delta 39789 zcmV(#K;*xqwM4$QM6jO_e|Ot9mhbr$>YIEfLh%hSJ>A)4C+W7cnbU64&F-dsP;A9k zBFhjMUz&*0)#Sy4pB_*TNcFSR=Zq2q 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