From 19743a0a9f4818c09251d25c0528f7b73ae9b23c Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Thu, 16 Jul 2026 11:28:59 +0200 Subject: [PATCH] =?UTF-8?q?fachliches=20Ged=C3=B6ns=20in=20AB's=20gefixt?= =?UTF-8?q?=20und=20pdf(s)=20aktualisiert?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- galton.pdf | Bin 820870 -> 699725 bytes material/ab3.pdf | Bin 146831 -> 147605 bytes material/ab3.tex | 20 ++++++++++---------- material/ab3_Lös.pdf | Bin 210707 -> 211193 bytes material/ab3_Lös.tex | 16 ++++++++-------- 5 files changed, 18 insertions(+), 18 deletions(-) diff --git a/galton.pdf b/galton.pdf index 1712d647be88af1b747375b0a21d8bb09529284a..d0f704ac62e62cf0485ef84dc6c3f36ce69b79fb 100644 GIT binary patch delta 497287 zcmc$_1z45ax;8vb6a*6`B&3n<4(aYjO1is|R}eu!Kw4V5LApVtTe`cuyZ!;V;++5N z{e9owXYIwc7857DbByuybKlQ!_n~eZdMSl3!NW&JMa}?^KeMrV08c}OO^I###sHp` zl|tJ3jg`HbHie+EhORaRx2ch-g|wN5rZxqSwvE1~wiG`Xg}lC&l^!+~1098cw!W^O 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zuO52el-cQ0(YKd;d3V*yGST)fUq1Y@vRt&Iz?aTuD=S1la}Qs-tQ3oW4cT5N`Yqe` zO3^>JZI_Dv4ccBM+8wsNDZ1Njua-b;@7>l%ckg$VNFect?V}P1<=S2o{WsOx-h$gA za0l+h-M9zCxDRWv7VGd3p1@Oh2G8LI3FKrt_a8xfbOM*qI0GFFFgZCkF$yImMNdWw D>r^ea diff --git a/material/ab3_Lös.tex b/material/ab3_Lös.tex index b1675d0..953556b 100644 --- a/material/ab3_Lös.tex +++ b/material/ab3_Lös.tex @@ -291,13 +291,13 @@ \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] + \begin{satzbox}[Lokaler 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 Zähldichte von $S_n$ durch die Dichtefunktion der Normalverteilung annähern: + Für große $n$ lässt sich die Zähldichte von $S_n$ durch die Glockenkurve (Dichtefunktion der Normalverteilung) annähern: \[ - S_n \sim \mathcal{N}(\mu,\sigma ^2) + P(S_n=k) \sim f_{\mathcal{N}(\mu,\sigma ^2)}(k) \] - wobei $\mathbb{E}(X)=\mu$, $Var(X)=\sigma ^2$ und $\frac{k-np}{\sqrt{npq}}$ begrenzt ist. + wobei $\mathbb{E}(S_n)=\mu$, $Var(S_n)=\sigma ^2$ und $ z = \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. @@ -308,8 +308,8 @@ \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: $\frac{1}{\sqrt{2\pi \sigma ^2}}\cdot e^{-\frac{\left( x-\mu \right)^2}{2\sigma ^2}} = \frac{1}{\sqrt{2\pi npq}}e^{\left(-\frac{(k-np)^2}{2npq}\right)}$ \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. \\ + Ziel: $f_{\mathcal{N}(\mu,\sigma ^2)}(k) = \frac{1}{\sqrt{2\pi \sigma ^2}}\cdot e^{-\frac{\left( x-\mu \right)^2}{2\sigma ^2}} = \frac{1}{\sqrt{2\pi npq}}e^{\left(-\frac{(k-np)^2}{2npq}\right)}$ \vspace{3em} + \item Wir wollen die Zähldichte der Binomialverteilung (diskrete Verteilung) mithilfe der Dichte einer einer stetige Verteilung approximieren. 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] @@ -361,7 +361,7 @@ \[ P(S_n = k) \sim \frac{1}{\sqrt{2\pi}}\sqrt{\frac{n}{k(n-k)}} \left(\frac{np}{k}\right)^k \left(\frac{nq}{n-k}\right)^{n-k} \] - \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: + \item Aus dieser Formel lässt sich nun leicht schon einmal der Vorfaktor der Glockenkurve erschließen. Hierzu und zu Vereinfachung weiterer Rechnungen standardisieren wir $S_n$. Dies entspricht auch einer einfachen Substitution und daher vollkommen legitim. Hierzu definieren wir: \[ Z_n=\frac{S_n-E(S_n)}{\sigma(S_n)} \] @@ -437,7 +437,7 @@ &= \frac{1}{\sqrt{2\pi npq}}e^{\left(-\frac{(k-np)^2}{2npq}\right)} \vspace{1em} \\ &= \frac{1}{\sqrt{2\pi \sigma ^2}}\cdot e^{-\frac{\left( x-\mu \right)^2}{2\sigma ^2}} \end{align*} - Daher konvergiert eine Binomialverteilte Zufallsvariable (für $n\rightarrow\infty$) gegen die Dichtefunktion der Normalverteilung. + Daher nährt sich die Zähldichte einer binomialverteilten Zufallsvariable (für $n\rightarrow\infty$) der Dichtefunktion der Normalverteilung an. \end{enumerate} \label{LastPage} \end{document} \ No newline at end of file

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