From 796775cd8312e6865b80ac8230966dfcc067c8e1 Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Thu, 11 Jun 2026 10:44:45 +0200 Subject: [PATCH] =?UTF-8?q?Binomialkoeffizient=20nach=20vorne=20geschoben?= =?UTF-8?q?=20und=20Referenz=20von=20Enders=20korrekt=20eingef=C3=BCgt?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- bibliography.bib | 6 ++++++ content/Mathematischer_Hintergrund.tex | 22 ++++++++++------------ galton.pdf | Bin 338406 -> 339146 bytes 3 files changed, 16 insertions(+), 12 deletions(-) diff --git a/bibliography.bib b/bibliography.bib index a9da459..210d39a 100644 --- a/bibliography.bib +++ b/bibliography.bib @@ -11,6 +11,12 @@ date = {2025}, } +@unpublished{enders_analysis_2024, + location = {Universität Potsdam}, + title = {Analysis {II}}, + author = {Enders, Dr. Jörg}, + date = {2024}, +} @book{nahrstedt_monte-carlo-methode_2015, location = {Wiesbaden}, diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index 9e56ed9..c9bb954 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -39,7 +39,13 @@ Um jeden Ausgang des Galton-Brettes durchnummeriert von links nach rechts unters Die Endposition der Kugel im Fach $k \in \{0, 1, \dots, n\}$ wird durch die Summe der Rechtsabbiegungen beschrieben. Wir definieren die Zufallsvariable: \[S_n = \sum_{i=1}^n X_i\] \end{definition} Um diese Summierte Zufallsvariable genauer zu verstehen betrachten wir zuerst den Binomialkoeffizienten. -% Def +\begin{definition}[Binomialkoeffizienten]{binomialkoeffizienten}{\cite{kosenkova_stochastik_2025}} + Für $n\in\mathbb{N}_0$ und $k\in\{0,\dots,n\}$ definieren wir den \textbf{Binomialkoeffizienten} + \[ + \binom{n}{k}=\frac{n!}{k!(n-k)!} + \] + wobei die Zahl $\binom{n}{k}$ die Anzahl der Möglichkeiten, aus einer $n$-elementigen Menge genau $k$-Elemente auszuwählen, angibt. +\end{definition} Betrachten wir nun den Binomialkoeffizienten in Bezug auf $S_n$, beschreibt dieser exakt die Anzahl der möglichen Pfade durch das Galton-Brett, bei denen die Kugel von $n$ Reihen genau $k$-mal nach rechts (und somit $(n-k)$-mal nach links) fällt. Da jeder dieser einzelnen Pfade aufgrund der Unabhängigkeit der Entscheidungen die Wahrscheinlichkeit $p^k (1-p)^{n-k}$ besitzt, ergibt sich die Gesamtwahrscheinlichkeit für das Fach $k$ durch Multiplikation. Daher rührt die Binomialverteilung: \begin{definition}[Binomialmodell]{binModell}{\cite{kosenkova_stochastik_2025}} Wiederholt man ein Bernoulli-Experiment mit Erfolgswahrscheinlichkeit $p \in [0,1]$ $n$-mal, und interessiert sich nur für die Anzahl der erfolgreichen Experimente, so wählt man @@ -242,15 +248,7 @@ Folgende Proposition ich leicht aus dem oberen Beweis ableitbar (aufgrund der Ab % Teilsortiert: \subsection{Pascalsches Dreieck} -Das Pascalsche Dreieck ist ein geometrisches Dreieck aus Zahlen, das sich unendlich nach unten fortsetzt und dabei die Binomialkoeffizienten repräsentiert. -\begin{definition}[Binomialkoeffizienten\footnote{Kosenkova, T: Stochastik für das Lehramt (Vorlesung 4), 2025}]{binomialkoeffizienten} - Für $n\in\mathbb{N}_0$ und $k\in\{0,\dots,n\}$ definieren wir den \textbf{Binomialkoeffizienten} - \[ - \binom{n}{k}=\frac{n!}{k!(n-k)!} - \] - wobei die Zahl $\binom{n}{k}$ die Anzahl der Möglichkeiten, aus einer $n$-elementigen Menge genau $k$-Elemente auszuwählen, angibt. -\end{definition} -Aus dieser Definition entwickeln wir das Pascalsche Dreieck. +Das Pascalsche Dreieck ist ein geometrisches Dreieck aus Zahlen, das sich unendlich nach unten fortsetzt und dabei die Binomialkoeffizienten repräsentiert. Aus \cref{def:binomialkoeffizienten} entwickeln wir das Pascalsche Dreieck. \begin{definition}[Pascalsches Dreieck]{pascaldreieck} Das \textbf{Pascalsche Dreieck} ist das unendliche Zahlenschema \[ @@ -329,7 +327,7 @@ Nun haben wir nachgewiesen, dass die Funktion $\ln(1-x)$ dreimal differenzierbar Mit einem Blick auf die Funktion und wie ihre Ableitungen gebildet werden, ist leicht zu erkennen, dass $\ln(x-1)$ sogar unendlich oft differenzierbar ist. \end{bemerkung} Bevor wir die Approximation nachweisen ist eine formale Definition des Satzes von Taylor und dem Taylorpolynom nötig, wobei letzteres aus dem Satz von Taylor folgt. -\begin{satz}[Satz von Taylor\footnote{Enders, J., Analysis II (2024)}]{taylor} +\begin{satz}[Satz von Taylor]{taylor}{\cite{enders_analysis_2024}} Seien $n\in\mathbb{N}_0, f:[a,b]\rightarrow\mathbb{R}$ eine $(n+1)$-mal differenzierbare Funktion und $x_0\in[a,b]$. Dann gilt für alle $x\in[a,b]$ die Darstellung @@ -349,7 +347,7 @@ Bevor wir die Approximation nachweisen ist eine formale Definition des Satzes v Die Definition des Taylorpolynoms entspringt direkt dem Satz von Taylor, nur ohne das Lagrange-Restglied. \end{bemerkung} -\begin{definition}[Taylorpolynom]{taylorpolynom} +\begin{definition}[Taylorpolynom]{taylorpolynom}{\cite{enders_analysis_2024}} \[ T_{f,x_0,n}(x)=\sum_{k=0}^{n}\frac{1}{k!}\cdot f^{(k)}(x_0)(x-x_0)^k \] diff --git a/galton.pdf b/galton.pdf index 4695fa461389935e2316dc8491b2056fc7dc089a..2bbdbeeadaead6749a2faa0d935e22d1714b4b46 100644 GIT binary patch delta 82593 zcmZVDQ*fYN)F$lM=-9Sx+qP}nd1BkPZ9C~C9osg$V|0?vyx(8*SIyL%>{7+4#)3mGU!=3*N-Il#`$%_1lW=jQHeVd4Piy>+WE=S0qhH1Maf zYG*uB^C?iCf1Gs4F$f>?&Jqbe&h3t@0C5dA<;LXuUrI`qRQlBJZ$>M8E9PZuEVm-hv9ZI#th~HD#F@y;7b;?Z@W#a0*DZbOag8ibQOJtWp~%2|)7}l|BS0R~ zsp4BfI^o;va`6*$+lX-dZISBz>GtW+&$~A?axi#ZTG!hpN%~^(fVR;C3cQ-}agbkl 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