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Fix factorial symbol and change variable k to n

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This commit is contained in:
Andreas Tsouchlos 2026-01-17 17:55:45 +01:00
parent 9f422f859e
commit 081cad7f11
1 changed files with 10 additions and 9 deletions
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src/2026-01-16/presentation.tex
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@ -142,7 +142,7 @@
\end{gather*} \end{gather*}
\vspace*{-2mm} \vspace*{-2mm}
\begin{gather*} \begin{gather*}
P_X(k) = \frac{\lambda^k}{k!}e^{-\lambda} \\[2mm] P_X(n) = \frac{\lambda^n}{n!}e^{-\lambda} \\[2mm]
\phi_X(s) = \text{exp}\left(\lambda (e^{js} -1)\right) \phi_X(s) = \text{exp}\left(\lambda (e^{js} -1)\right)
\end{gather*} \end{gather*}
\vspace*{-2mm} \vspace*{-2mm}
@ -163,7 +163,7 @@
\vspace*{-6mm} \vspace*{-6mm}
\begin{gather*} \begin{gather*}
X \sim \text{Poisson}(\lambda) \\[3mm] X \sim \text{Poisson}(\lambda) \\[3mm]
P_X(k) = \frac{\lambda^k \cdot e^{-\lambda}}{k!} \\[4mm] P_X(n) = \frac{\lambda^n \cdot e^{-\lambda}}{n!} \\[4mm]
\phi_X(s) = \text{exp}\left(\lambda (e^{js} -1)\right) \phi_X(s) = \text{exp}\left(\lambda (e^{js} -1)\right)
\end{gather*} \end{gather*}
\end{greenblock} \end{greenblock}
@ -171,7 +171,7 @@
\vspace*{-6mm} \vspace*{-6mm}
\begin{gather*} \begin{gather*}
\nsum_{k=0}^{n} \binom{n}{k}a^k b^{n-k} = (a+b)^n, \hspace{15mm} \nsum_{k=0}^{n} \binom{n}{k}a^k b^{n-k} = (a+b)^n, \hspace{15mm}
\binom{n}{k} = \frac{n!}{(n-k!)k!} \binom{n}{k} = \frac{n!}{(n-k)!k!}
\end{gather*} \end{gather*}
\end{greenblock} \end{greenblock}
\column{\kitthreecolumns} \column{\kitthreecolumns}
@ -231,11 +231,11 @@
zweier Zufallsvariablen. zweier Zufallsvariablen.
\pause\begin{gather*} \pause\begin{gather*}
X \sim \text{Poisson}(\lambda_1) \hspace{3mm} X \sim \text{Poisson}(\lambda_1) \hspace{3mm}
\Leftrightarrow \hspace{3mm} P_X(k) \Leftrightarrow \hspace{3mm} P_X(n)
= \frac{\lambda_1^k \cdot e^{-\lambda_1}}{k!} \hspace{30mm} = \frac{\lambda_1^n \cdot e^{-\lambda_1}}{n!} \hspace{30mm}
Y \sim \text{Poisson}(\lambda_2) \hspace{3mm} Y \sim \text{Poisson}(\lambda_2) \hspace{3mm}
\Leftrightarrow \hspace{3mm} P_Y(k) \Leftrightarrow \hspace{3mm} P_Y(n)
= \frac{\lambda_2^k \cdot e^{-\lambda_2}}{k!} = \frac{\lambda_2^n \cdot e^{-\lambda_2}}{n!}
\end{gather*} \end{gather*}
\pause\begin{align*} \pause\begin{align*}
P_Z(n) &= P_{X+Y}(n) = \nsum_{k=0}^{n} P_X(k)P_Y(n-k) P_Z(n) &= P_{X+Y}(n) = \nsum_{k=0}^{n} P_X(k)P_Y(n-k)
@ -419,7 +419,7 @@
\begin{lightgrayhighlightbox} \begin{lightgrayhighlightbox}
Erinnerung: Unabhängige Ereignisse Erinnerung: Unabhängige Ereignisse
\begin{align*} \begin{align*}
X,Y \text{ \normalfont unabhängig} A,B \text{ \normalfont unabhängig}
\hspace{5mm} \Leftrightarrow \hspace{5mm} \hspace{5mm} \Leftrightarrow \hspace{5mm}
P(AB) = P(A)P(B) P(AB) = P(A)P(B)
\end{align*} \end{align*}
@ -439,7 +439,8 @@
\begin{lightgrayhighlightbox} \begin{lightgrayhighlightbox}
Erinnerung: Varianz Erinnerung: Varianz
\begin{align*} \begin{align*}
V(X) = E\big( \left(X - E(X)\right)^2 \big) = E(X^2) - E^2(X) V(X) = E\big( \left(X - E(X)\right)^2 \big) =
E(X^2) - E^2(X)
\end{align*} \end{align*}
\vspace*{-13mm} \vspace*{-13mm}
\end{lightgrayhighlightbox} \end{lightgrayhighlightbox}