From 9cb7744894bd57514408c2e77be97551caa8274f Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Sun, 12 Jul 2026 17:15:05 +0200 Subject: [PATCH] =?UTF-8?q?L=C3=B6sung=20AB3?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- content/Arbeitsmaterialien.tex | 7 +- images/AB3_GaltonVersuche_L.png | Bin 0 -> 136431 bytes images/AB3_Galton_L.png | Bin 0 -> 48526 bytes images/AB3_X_L.png | Bin 0 -> 11276 bytes material/ab3.pdf | Bin 275904 -> 274305 bytes material/ab3.tex | 22 +- material/ab3_Lös.pdf | Bin 0 -> 342493 bytes material/ab3_Lös.tex | 442 ++++++++++++++++++++++++++++++++ 8 files changed, 464 insertions(+), 7 deletions(-) create mode 100644 images/AB3_GaltonVersuche_L.png create mode 100644 images/AB3_Galton_L.png create mode 100644 images/AB3_X_L.png create mode 100644 material/ab3_Lös.pdf create mode 100644 material/ab3_Lös.tex diff --git a/content/Arbeitsmaterialien.tex b/content/Arbeitsmaterialien.tex index 00a73db..b1dba69 100644 --- a/content/Arbeitsmaterialien.tex +++ b/content/Arbeitsmaterialien.tex @@ -31,4 +31,9 @@ Bitte bedenken Sie, zumindest Lösungsskizzen zu den Aufgaben zu geben! Besser n % 2 -> Vorfaktor "abschätzen" % 3 -> potenzen Killen um E funktion zu gewinnen! % ln und umformen (Klassenstufe! -> Anschätzung und Ergebnis vorgeben, nicht rechnen) - % Rücksubstitution machen \ No newline at end of file + % Rücksubstitution machen +Das dritte Arbeitsblatt dient dem Beweis des zentralen Grenzwertsatzes von Movire-Laplace. Ziel ist es hierbei explizit nicht, dass die Schüler*innen den Beweis in jedem Detail, wie im Teil "Mathematischer Hintergrund", verstehen aber die Grundlegenden Argumentationen und mathematischen Tricks nachvollziehen. Außerdem ist vor allem das Beweisen etwas, dass im Rahmenlehrplan (\cite{lisum-2022} und \cite{lisum-2023}) nicht gefordert ist, wodurch es sich sehr gut für eine MatheAG oder Zusatzstunde handelt, in der Inhalte über den klassischen Unterricht hinaus behandelt werden.\\ +Für das Arbeitsblatt sollten die Schüler*innen bereits mit der Binomialverteilung, Zufallsvariablen, Logarithmen sowie der e-Funktion vertraut sein und mit etwas Formalismus umgehen können. Tiefgreifende Kenntnisse zur Normalverteilung sind nicht vorausgesetzt und werden auf dem Arbeitsblatt in ausreichender Form wiederholt bzw. können selbständig aufbereitet werden. Es ist empfohlen mindestens eines der Arbeitsblätter Galton x Pascall oder Galton x Plinko vorher bearbeitet zu haben, um das Konzept des Galtonbrettes und dessen Zusammenhang mit der Binomialverteilung verstanden zu haben.\\ +Auf dem Arbeitsblatt wird die formale Modellierung des Galtonbrettes vorgestellt und soll durch eine Zuordnungsaufgabe durch die Schüler*innen vertieft werden. Im Anschluss soll der Bezug zwischen Binomial- und Normalverteilung wiederholt bzw. erarbeitet werden. Das Kernstück des Arbeitsblattes stellt der Lückentext-Ähnliche Beweis dar. Hier sollen die Schüler einige Mathematische Argumentationen nachvollziehen und einige Umformungen selbst durchführen, um den Beweis nachvollziehen zu können. Sollten die Schüler leistungsschwächer sein oder an einem Punkt fest Stecken gibt es zur Differenzierung Lösungskarten für die Lücken, welche den Schülern in nicht sortierter Reihenfolge zur Verfügung gestellt werden können. + +%Noch Checkfragen in den Beweis einbauen? \ No newline at end of file diff --git a/images/AB3_GaltonVersuche_L.png b/images/AB3_GaltonVersuche_L.png new file mode 100644 index 0000000000000000000000000000000000000000..80e71b31d0bbe7903fa074269d3d92a9dfae8a48 GIT binary patch literal 136431 zcmY&=cU+Bq_>twM2_{xAs%35A-f;zbgY18wk+9Oe-G ziO3-_M-mb&iJGFE!A;WTIx>P`)83ywX<9k7(sOIF+Prj==XW1+^A7qcg`ev>*^_jq zeO0T6)+Xb{iDizeL9E={yLVqO98n#5Cunzrr?!Ev>h(IKN%pc}?G2BFSKFD%b=#>z z>unCRZ3%O2;^&#@caEX31QIlkgcO4(`S-7Z-WadvMZY57|IeTN&;NX)K+>4Z7?DqY zPnY%o{|WpBoDKSH+bM-KKdR^}|NAWBFIlM?&W6O8J8p)OnEl^3@pBQZJANGHyoGgS zBHgxI#g+uDC51cxwt`;?3Yr1wlPmrHkIO&=tj`y?;{&y%1 z^l%z3U%rs-ugppdB^19t_qp03?RNi<_4SpzM+Bdb-I%!Do+9;!qqv)YMLnoHL--cI zxKlRE>$}YA62kU>pXm 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zlG~ffY@@I*IwiNFR~J*-((aZ2BXS*oX)t?0Z}9i_`?qFwz}_N{&YnAW@yctn{{hx) B=Jo&p diff --git a/material/ab3.tex b/material/ab3.tex index 187bce6..14adeaf 100644 --- a/material/ab3.tex +++ b/material/ab3.tex @@ -298,24 +298,29 @@ Diesen Satz wollen wir nun Semi-Formal beweisen.\\ \textbf{\underline{Aufgabe 3:}} Fülle die Lücken im Beweis aus. Wenn du dir unsicher bist kannst du dir alle fehlenden Textbausteine auf dem Zusatzblatt ansehen. \begin{enumerate} + \item Bevor wir den Satz beweisen, ist es wichtig, die Aussage zu verstehen. Formuliere die Kernaussage den Satzes kurz in eigenen Worten: + \answer{3} \item Zuerst muss uns nochmal klar werden, was unser Ausgangspunkt und und das Ziel ist:\\ Ausgangspunkt: $P(S_n = k) = $ \antwortbox[6cm] \vspace{1em} \\ Ziel: $\mathcal{N}(\mu,\sigma ^2) = \mathcal{N}(np)(npq) = $ \antwortbox[6cm] - \item Zuerst müssen wir die Binomialverteilung als diskrete Verteilung in eine stetige Verteilung überführen. Das Aspekt der Binomialverteilung welcher die Diskretheit forciert, ist die Fakultät. Die Fakultät der Reelen Zahlen ist die so genannte Gamma ($\Gamma$) Funktion. Zeichne in dem folgenden Koordinatensystem, in dem auch die Gammafunktion abgebildet ist, die Funktionswerte der Fakultät (welche nur auf $\mathbb{N}$ definiert ist) für $x\in \{0,1,2,3,4,5\}$ ein: + \item Wir wollen die Binomialverteilung als diskrete Verteilung in eine stetige Verteilung überführen. Das Aspekt der Binomialverteilung welcher die Diskretheit forciert, ist die Fakultät. Die Fakultät der Reelen Zahlen ist die so genannte Gamma ($\Gamma$) Funktion. \\ + + Zeichne in dem folgenden Koordinatensystem, in dem auch die Gammafunktion abgebildet ist, die Funktionswerte der Fakultät (welche nur auf $\mathbb{N}$ definiert ist) für $x\in \{0,1,2,3,4,5\}$ ein: \begin{figure}[H] \centering \begin{tikzpicture} \begin{axis}[ - domain=0.1:6, + domain=0.1:4.5, samples=300, axis lines=middle, xlabel={$x$}, - ylabel={$\Gamma(x)$}, + ylabel={$y$}, xmin=0, xmax=6, ymin=0, - ymax=10, - grid=both + ymax=26, + grid=both, + legend pos=north east ] \addplot[ blue, @@ -369,10 +374,16 @@ \[ P(S_n=k)= \frac{1}{\sqrt{2\pi npq}}\left(\frac{k}{np}\right)^{-k} \left(\frac{n-k}{nq}\right)^{-n+k} \Leftrightarrow e^{ln(P(S_n = k))} \sim e^{ln\left(\frac{1}{\sqrt{2\pi npq}}\left(\frac{k}{np}\right)^{-k} \left(\frac{n-k}{nq}\right)^{-n+k}\right)} \] + Beschreibe kurz, was genau die obere Gleichung aussagt und wieso wir einfach die $e$-Funktion und den (natürlichen)Logarithmus anwenden können! + \answer{3} + Im Folgenden betrachten wir erst mal nur den Ausdruck im $ln()$ welchen wir mithilfe der Logarithmengesetze umformen können: \[ \ln\left( \left(\frac{k}{np}\right)^{-k} \left(\frac{n-k}{nq}\right)^{-n+k} \right) = (-np - z\sqrt{npq}) \ln\left(1+z\sqrt{\frac{q}{np}}\right) + (-nq + z\sqrt{npq}) \ln\left(1-z\sqrt{\frac{p}{nq}}\right) \] + Nenne die Rechengesetze und Substitutionen, welche vermutlich für diese Umformung genutzt wurden! + \answer{3} + Diese Darstellung des Thermes sieht zwar erst mal größer aus, hat aber den Vorteil, dass er aus zwei sich sehr ähnlich aussehenden Teilen besteht. Das machen wir uns zu Nutzen! Mithilfe der Tailor-Approximation für den Logarithmus lassen sich nämlich folgende Abschätzungen beweisen: \[ ln(1-\alpha) \sim -\alpha-\frac{\alpha^2}{2} @@ -397,6 +408,5 @@ \end{align*} Daher konvergiert eine Binomialverteilte Zufallsvariable (für $n\rightarrow\infty$) gegen die Dichtefunktion der Normalverteilung. \end{enumerate} - \section{Platz für Notizen und Nebenrechnungen:} \label{LastPage} \end{document} \ No newline at end of file diff --git a/material/ab3_Lös.pdf b/material/ab3_Lös.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3ecbdd7764113d7cf54127f9350ecadfbcf16762 GIT binary patch literal 342493 zcmag_1yqz>+c%7>s0hL!Dm|2xgdjb%Iw&2|jl|I1jiN9jNJH}y z!TWxn=l!qmTYuK_ysnwqaqMFsef&;3dC@n|*jc%;={g4M1{z0t{xo58Q9Y-!);Gfz z6r^HPq+%1ZRLnO!lCX$`~|BXud|9b}q^hY=JFA=gJX`9IVDXCdSNdj|jiaD|=Y|D7$G zRP3DK7X>P|H>M6$+W+hRIG+En8E;H291wP3kcFNDLKIJD5;$a`N(l zc~t-Yu<27$y^^`M(k)wE3(n50)5i{nv&$vZ*$k8J z)n4vLrtU^`S-9;Lc+Mi1f#lxJ)6M~p`TZ8XO;?bO?+KXPJgyv1W;Z)G&q*5e-o7!J 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+\definecolor{codeblue}{rgb}{0,0,0.8} +\definecolor{codered}{rgb}{0.8,0,0} +\definecolor{lightgray}{rgb}{0.95,0.95,0.95} + +\lstdefinestyle{CStyle}{ + language=C, + basicstyle=\ttfamily\small, % Monospace-Schrift + keywordstyle=\color{blue}\bfseries, % Schlüsselwörter fett und blau + stringstyle=\color{red}, % Strings rot + commentstyle=\color{codegreen}, % Kommentare grün + backgroundcolor=\color{lightgray}, % Hintergrundfarbe + numbers=left, % Zeilennummern links + numbersep=10px, % Abstand zwischen Zeilennummern und Code + numberstyle=\color{gray}\texttt, + stepnumber=1, % Zeilennummerierung Schrittweite 1 + frame=single, % Rahmen um den Code + tabsize=4, % Tabulatorgröße + breaklines=true, % Zeilenumbruch aktivieren + captionpos=below, + showstringspaces=false, + xleftmargin=15pt, % Linker Rand für den Code (verschiebt alles nach rechts) +} + + +\lstdefinestyle{ShellStyle}{ + language=Bash, + basicstyle=\ttfamily\small, % Monospace-Schrift + keywordstyle=\color{black}, % Schlüsselwörter fett und blau + stringstyle=\color{black}, % Strings rot + commentstyle=\color{codegreen}, % Kommentare grün + backgroundcolor=\color{lightgray}, % Hintergrundfarbe + numbers=left, % Zeilennummern links + numbersep=10px, % Abstand zwischen Zeilennummern und Code + numberstyle=\color{gray}\texttt, + stepnumber=1, % Zeilennummerierung Schrittweite 1 + frame=single, % Rahmen um den Code + tabsize=4, % Tabulatorgröße + breaklines=true, % Zeilenumbruch aktivieren + captionpos=below, + showstringspaces=false, + xleftmargin=15pt, % Linker Rand für den Code (verschiebt alles nach rechts) +} + + +% Umgebung für Listings mit Titel und Zähler +\newenvironment{codeexample}[1][]{ + \refstepcounter{commoncounter} % Zähler erhöhen + \lstset{ + style=CSharpStyle, + caption={Listing \thecommoncounter: #1}, % Titel mit Zähler + label={listing:\thecommoncounter} + } +}{} + +\makeatletter +% Internen Zähler von subsubsection auf commoncounter umbiegen +\let\c@subsubsection\c@commoncounter + +% Anzeige der Nummer angleichen +\renewcommand{\thesubsubsection}{\thecommoncounter} +\makeatother + +\setlength{\parindent}{0pt} + +\sethlcolor{black!10} % Farbe des "Textmarkers" + +\newcommand{\curdate}{09.07.2026} +\newcommand{\curtitle}{ZSG Movire-Laplace} + +\title{\curtitle} +\author{Fierke, Janik, Weidlich} +\pagestyle{scrheadings} +\date{\curdate} +\begin{document} + \ihead{\normalfont\normalsize Mathematik} + \chead{\normalfont\normalsize \curtitle} + \ohead{\normalfont\normalsize \curdate} + \ofoot{\normalfont\normalsize Seite {\pagemark} von \pageref{LastPage}} + \ifoot{\normalfont\normalsize Fierke, Janik, Weidlich} + + \section{\curtitle} + Das Galtonbrett lässt sich wir folgt formal mathematisch formulieren: + \begin{definitionbox}[Modell des Galtonbretts] + Sei $(\Omega, \mathcal{F}, P)$ ein Wahrscheinlichkeitsraum. \\ + Der Fall einer Murmel durch ein Galtonbrett mit \hl{$\bm{n \in \mathbb{N}}$} Reihen wird modelliert durch eine Folge von stochastisch unabhängigen und identisch verteilten Zufallsvariablen $X_1, X_2, \dots, X_n$, wobei \hl{$\bm{X_i \in \{0, 1\}}$}. + Dabei beschreibt \hl{$\bm{X_i = 1}$} den Fall nach rechts in der $i$-ten Reihe und \hl{$\bm{X_i = 0}$} den Fall nach links. Die Wahrscheinlichkeit sei $P(X_i = 1) = p$ und $P(X_i = 0) = 1-p = q$. Bei einem symmetrischen Brett gilt $p = q = 0.5$.\\ + Die Endposition der Murmel im Fach $k \in \{0, 1, \dots, n\}$ wird durch die Summe der Rechtsabbiegungen beschrieben. Wir definieren die Zufallsvariable: \[\colorbox{black!10}{$\displaystyle \bm{S_n=\sum_{i=1}^n X_i}$}\] Hierbei ist $S_n$ Binomialverteilt. + \end{definitionbox} + + \textbf{\underline{Aufgabe 1:}} Um die Definition genauer zu verstehten, ordne die in der Definition hervorgehobenen Teile den Folgenden Abbildungen des Galtonbrettes zu: + \begin{figure}[H] + \centering + % Erstes Bild + \begin{minipage}[t]{0.45\textwidth} + \centering + \includegraphics[width=\linewidth]{../images/AB3_Galton_L.png} + \caption{Darstellung des Galtonbretts nach Brünner (2025)} + \label{fig:galton} + \end{minipage} + \hfill + % Zweites Bild + \begin{minipage}[t]{0.45\textwidth} + \centering + \includegraphics[width=\linewidth]{../images/AB3_X_L.png} + \caption{Darstellung des Falls einer Murmel im Galtonbrett.} + \label{fig:galtonXi} + \end{minipage} + \end{figure} + \textbf{\underline{Aufgabe 2:}} Sobald sehr viele Murmeln durch das Galtonbrett gewandert sind, entsteht eine Verteilung, wie in Grafik 3 für ein symmetrisches Brett dargestellt wird. + \begin{enumerate} + \item Probiere das ganze selbst einmal auf folgender Website aus: \url{https://www.arndt-bruenner.de/mathe/scripts/galtonbrett.htm}.\\ Was fällt dir auf? \\ \textit{Achtung: stelle p = 0,5 für ein symmetrisches Brett ein!}\\ + Die Verteilung entspricht einer Binomialverteilung. + \item Recherchiere selbstständig und zeichne den Graphen der Normalverteilung über die Verteilung des Galtonbrettes ein: + \begin{figure}[H] + \centering + \includegraphics[width=0.8\linewidth]{../images/AB3_GaltonVersuche_L.png} + \caption{Galtonbrett nach c.a. 7000 durchgefallenen Murmeln nach Brünner (2025)} + \end{figure} + \item Wiederhole oder Recherchiere selbstständig die Eigenschaften der Normalverteilung und Vergleiche diese mit der Binomialverteilung (die Verteilung des Galtonbrettes): + \begin{table}[H] + \centering + \begin{tabularx}{\textwidth}{|X|X|} + \hline + \centering\textbf{Normalverteilung} & + \centering\arraybackslash\textbf{Binomialverteilung} \\ + \hline + Stetig & Diskret \\ + Parameter $\mu, \sigma$ & Parameter n, p\\ + Glockenkurve & Histogramm \\ + $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$ & $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$\\ + \hline + \end{tabularx} + \end{table} + \end{enumerate} + \vspace{4em} + Wir wollen nun das schöne mathematische Phänomen, welches sich in Aufgabe 2.2 offenbart, genauer betrachten. Hierzu formulieren wir dies zuerst mathematisch formal sauber: + \begin{satzbox}[Zentraler Grenzwertsatz von de Movire-Laplace] + Sei $S_n \sim \mathcal{B}in_{n,p}$ die Anzahl der Erfolge bei $n$ unabhängigen Bernoulli-Versuchen mit Erfolgswahrscheinlichkeit $p \in (0, 1)$ und sei $q = 1-p$ die Gegenwahrscheinlichkeit. (Beispielsweise bei einem Galtonbrett)\\ + Für große $n$ lässt sich die Wahrscheinlichkeit, dass $S_n$ den Wert $k \in \{x| x \in\mathbb{N}_0 \land x \leq n\}$ annimmt, durch die Dichtefunktion der Normalverteilung annähern: + \[ + P(S_n = k) \sim \mathcal{N}(\mu,\sigma ^2) + \] + wobei $\mathbb{E}(X)=\mu$, $Var(X)=\sigma ^2$ und $\frac{k-np}{\sqrt{npq}}$ begrenzt ist. + \end{satzbox} + Diesen Satz wollen wir nun Semi-Formal beweisen.\\ + \textbf{\underline{Aufgabe 3:}} Fülle die Lücken im Beweis aus. Wenn du dir unsicher bist kannst du dir alle fehlenden Textbausteine auf dem Zusatzblatt ansehen. + \begin{enumerate} + \item Bevor wir den Satz beweisen, ist es wichtig, die Aussage zu verstehen. Formuliere die Kernaussage den Satzes kurz in eigenen Worten:\\ + Die Binomialverteilung lässt sich mithilfe der Normalverteilung für eine hohe Anzahl an Durchführungen approximieren. + \vspace{3em} + \item Zuerst muss uns nochmal klar werden, was unser Ausgangspunkt und und das Ziel ist: + \vspace{3em} + Ausgangspunkt: $P(S_n = k) = \frac{n!}{k!(n-k)!} p^k q^{n-k}$ \vspace{2em} \\ + Ziel: $\mathcal{N}(\mu,\sigma ^2) = \mathcal{N}(np)(npq) = \frac{1}{\sqrt{2\pi npq}}e^{\left(-\frac{(k-np)^2}{2npq}\right)}$ \vspace{3em} + \item Wir wollen die Binomialverteilung als diskrete Verteilung in eine stetige Verteilung überführen. Das Aspekt der Binomialverteilung welcher die Diskretheit forciert, ist die Fakultät. Die Fakultät der Reelen Zahlen ist die so genannte Gamma ($\Gamma$) Funktion. \\ + + Zeichne in dem folgenden Koordinatensystem, in dem auch die Gammafunktion abgebildet ist, die Funktionswerte der Fakultät (welche nur auf $\mathbb{N}$ definiert ist) für $x\in \{0,1,2,3,4\}$ ein: + \begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + domain=0.1:4.5, + samples=300, + axis lines=middle, + xlabel={$x$}, + ylabel={$y$}, + xmin=0, + xmax=5, + ymin=0, + ymax=26, + grid=both, + legend pos=north west + ] + + \addplot[ + blue, + thick + ] + {sqrt(2*pi)*((x+1/2)^(x+1/2))*exp(-(x+1/2))}; + \addlegendentry{Fortsetzung} + + \addplot[ + only marks, + mark=*, + mark options={fill=red, draw=red, scale=1.2} + ] + coordinates { + (0, 1) + (1, 1) + (2, 2) + (3, 6) + (4, 24) + }; + \addlegendentry{Fakultät $x!$} + + \end{axis} + \end{tikzpicture} + \end{figure} + Mithilfe dieser Funktion lässt sich die Fakultät leicht approximieren. Diese Approximation nennt sich Sterling Approximation und besagt: + \[ + n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n \sim \Gamma(n) + \] + Setzen wir das in die Gleichung für $P(S_n = k)$ ein, erhalten wir nach umstellen: + \[ + P(S_n = k) \sim \frac{1}{\sqrt{2\pi}}\sqrt{\frac{n}{k(n-k)}} \left(\frac{np}{k}\right)^k \left(\frac{nq}{n-k}\right)^{n-k} + \] + \item Aus dieser Formel lässt sich nun leicht schon einmal der Vorfaktor der Normalverteilung erschließen. Hierzu und zu Vereinfachung weiterer Rechnungen standardisieren wir $S_n$. Dies entspricht auch einer einfachen Substitution und daher vollkommen legitim. Hierzu definieren wir: + \[ + Z_n = Z_n=\frac{S_n-E(S_n)}{\sigma(S_n)} + \] + nimmt $Z_n$ die folgenden Werte an: + \[ + z= \frac{k-np}{\sqrt{npq}} + \] + \vspace{3em} + Nach dem Umstellen nach $k$ und $n-k$ folgt dann: + \[ + k= np + z\sqrt{npq} + \] + \vspace{3em} + \[ + n-k= nq - z\sqrt{npq} + \] + \vspace{3em} + Multiplizieren wir nun diese beiden Ausdrücke miteinander erhalten wir: + \[ + k(n-k)= n^2pq + (nz\sqrt{nqp}(q-p)-z^2npq) + \] + Da wir uns nur für eine große Anzahl an Durchführungen also große $n$ ($n\rightarrow \infty$) interessieren, ist beobachtbar, dass für große n der lineare Teil kaum noch einen Einfluss auf den Term hat. Daher kann dieser in diesem Fall weggelassen werden. Daher gilt: + \[ + k(n-k) \sim n^2pq + \] + Setzen wir dies nun in unsere diskrete Variante von $P(S_n = k)$ ein, erhalten wir den passenden Vorfaktor: + \begin{align*} + P(S_n = k) &\sim \frac{1}{\sqrt{2\pi}}\sqrt{\frac{n}{k(n-k)}} \left(\frac{np}{k}\right)^k \left(\frac{nq}{n-k}\right)^{n-k} \\ + &\sim \frac{1}{\sqrt{2\pi}}\sqrt{\frac{n}{n^2pq}} \left(\frac{np}{k}\right)^k \left(\frac{nq}{n-k}\right)^{n-k} \vspace{1em} \\ + &= \frac{1}{\sqrt{2\pi}}\frac{1}{\sqrt{npq}} \left(\frac{np}{k}\right)^k \left(\frac{nq}{n-k}\right)^{n-k} \\ + &= \frac{1}{\sqrt{2\pi npq}}\left(\frac{k}{np}\right)^{-k} \left(\frac{n-k}{nq}\right)^{-n+k} + \end{align*} + \item Zum Schluss müssen wir noch die $e$-Funktion mit in die Gleichung bekommen. Das können wir machen, indem wir die Identität ausnutzen: $x = e^{ln(x)}$. Also gilt insbesondere, dass: + \[ + P(S_n=k)= \frac{1}{\sqrt{2\pi npq}}\left(\frac{k}{np}\right)^{-k} \left(\frac{n-k}{nq}\right)^{-n+k} \Leftrightarrow e^{ln(P(S_n = k))} \sim e^{ln\left(\frac{1}{\sqrt{2\pi npq}}\left(\frac{k}{np}\right)^{-k} \left(\frac{n-k}{nq}\right)^{-n+k}\right)} + \] + Beschreibe kurz, was genau die obere Gleichung aussagt und wieso wir einfach die $e$-Funktion und den (natürlichen)Logarithmus anwenden können!\\ + Da $e(ln(x))=x$ können wir diese Identität auf beide Seiten der Gleichung anwenden, ohne das Ergebnis zu verändern. Der Logarithmus ist nur für $x>0$ definiert, das bereitet hier aber keine Probleme, da $0