diff --git a/semester4/ps/ps-jh/assets/limits.png b/semester4/ps/ps-jh/assets/limits.png new file mode 100644 index 0000000..5b71797 Binary files /dev/null and b/semester4/ps/ps-jh/assets/limits.png differ diff --git a/semester4/ps/ps-jh/parts/04_joint-distribution/01_continuous.tex b/semester4/ps/ps-jh/parts/04_joint-distribution/01_continuous.tex index ce6c239..5d4435b 100644 --- a/semester4/ps/ps-jh/parts/04_joint-distribution/01_continuous.tex +++ b/semester4/ps/ps-jh/parts/04_joint-distribution/01_continuous.tex @@ -80,4 +80,4 @@ Herleitung der Randdichte (``wegintegrieren''): \] Folglich sind die beiden Koordinaten unabhängig. -\bi{Einheitskreisscheibe} $f(x, y) \neq f_\cX(x) f_\cY(y)$, mit Dichten von oben. Also sind die beiden Koordinaten nicht unabhängig. +\bi{Einheitskreisscheibe} $f(x, y) \neq f_\cX(x) f_\cY(y)$, mit Dichten von oben. Also sind die beiden Koordinaten nicht unabh. diff --git a/semester4/ps/ps-jh/parts/04_joint-distribution/02_remarks.tex b/semester4/ps/ps-jh/parts/04_joint-distribution/02_remarks.tex new file mode 100644 index 0000000..5e44cd5 --- /dev/null +++ b/semester4/ps/ps-jh/parts/04_joint-distribution/02_remarks.tex @@ -0,0 +1,16 @@ +\subsection{Tips \& Tricks} +Zur Bestimmung von W. wie (mit $\cX, \cY \sim \cU(0, 1)$) +\[ + W(t) = \P[\cX + \cY \leq t] \ \forall t \in \R +\] +können wir dies via gemeinsamer Dichte ($f_{\cX, \cY}(x, y) = 1$ muss gegeben sein) und Menge +\[ + A_t = \{ (x, y) \in [0, 1]^2 : x + y \leq t \} +\] +bestimmen. Dann ist für $t < 0$, $W(t) = 0$. + +Für $0 \leq t \leq 1$ ist $A_t$ ein rechtwinkliges Dreieck mit Katheten der Länge $t$, also $W(t) = t^2 \div 2$. + +Für $1 < t \leq 2$ ist $A_t$ Einheitsquadrat ohne rechtw. Dreieck mit Katheten der Länge $(2 - t)$, also $W(t) = 1 - \frac{(2 - t)^2}{2}$. + +Für $t > 2$ ist $W(t) = 1$. diff --git a/semester4/ps/ps-jh/parts/10_tips-and-tricks/01_various.tex b/semester4/ps/ps-jh/parts/10_tips-and-tricks/01_various.tex new file mode 100644 index 0000000..86f1fce --- /dev/null +++ b/semester4/ps/ps-jh/parts/10_tips-and-tricks/01_various.tex @@ -0,0 +1,49 @@ +\rmvspace +\subsection{Verschiedene Funktionen} +\subsubsection{Logarithmen} +\begin{itemize} + \item \textit{(Basiswechsel)} $\log_a(x) = \frac{\ln(x)}{\ln(a)}$ + \item \textit{(Potenzen)} $\log_a(x^y) = y\log_a(x)$ + \item \textit{(Div, Mul)} $\log_a(x \cdot (\div) y) = \log_a(x) +(-) \log_a(y)$ + \item $\log_a(1) = 0 \smallhspace \forall a \in \N$ +\end{itemize} + + +\subsubsection{Trigonometrie} +$\cot(\xi) = \displaystyle\frac{\cos(\xi)}{\sin(\xi)}, \tan(\xi) = \frac{\sin(\xi)}{\cos(\xi)}$ + +$\sinh(x) := \frac{e^x - e^{-x}}{2} : \R \rightarrow \R$, +$\cosh(x) := \frac{e^x + e^{-x}}{2} : \R \rightarrow [1, \infty]$, +$\cosh(x) := \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}} : \R \rightarrow [-1, 1]$ + +\begin{enumerate} + \item $\cos(x) = \cos(-x)$ \trand $\sin(-x) = -\sin(x)$ + \item $\cos(\pi - x) = -\cos(x)$ \trand $\sin(\pi - x) \sin(x)$ + \item $\sin(x + w) = \sin(x) \cos(w) + \cos(x) \sin(w)$ + \item $\cos(x + w) = \cos(x) \cos(w) - \sin(x) \sin(w)$ + \item $\cos(x)^2 + \sin(x)^2 = 1$ + \item $\sin(2x) = 2 \sin(x) \cos(x)$ + \item $\cos(2x) = \cos(x)^2 - \sin(x)^2$ +\end{enumerate} + + +\shade{teal}{\tr{Values of trigonometric functions}{Werte der trigonometrischen Funktionen}} +\begin{tables}{ccccc}{° & rad & $\sin(\xi)$ & $\cos(\xi)$ & $\tan(\xi)$} + 0° & $0$ & $0$ & $1$ & $1$ \\ + \hline + 30° & $\frac{\pi}{6}$ & $\frac{1}{2}$ & $\frac{\sqrt{3}}{2}$ & $\frac{\sqrt{3}}{2}$ \\ + \hline + 45° & $\frac{\pi}{4}$ & $\frac{\sqrt{2}}{2}$ & $\frac{\sqrt{2}}{2}$ & $1$ \\ + \hline + 60° & $\frac{\pi}{3}$ & $\frac{\sqrt{3}}{3}$ & $\frac{1}{2}$ & $\sqrt{3}$ \\ + \hline + 90° & $\frac{\pi}{2}$ & $1$ & $0$ & $\varnothing$ \\ + \hline + 120° & $\frac{2\pi}{3}$ & $\frac{\sqrt{3}}{2}$ & $-\frac{1}{2}$ & $-\sqrt{3}$ \\ + \hline + 135° & $\frac{3\pi}{4}$ & $\frac{\sqrt{2}}{2}$ & $-\frac{\sqrt{2}}{2}$ & $-1$ \\ + \hline + 150° & $\frac{5\pi}{6}$ & $\frac{1}{2}$ & $-\frac{\sqrt{3}}{2}$ & $-\frac{\sqrt{3}}{2}$ \\ + \hline + 180° & $\pi$ & $0$ & $-1$ & $0$ \\ +\end{tables} diff --git a/semester4/ps/ps-jh/parts/10_tips-and-tricks/02_diff-table.tex b/semester4/ps/ps-jh/parts/10_tips-and-tricks/02_diff-table.tex new file mode 100644 index 0000000..ebe00e1 --- /dev/null +++ b/semester4/ps/ps-jh/parts/10_tips-and-tricks/02_diff-table.tex @@ -0,0 +1,53 @@ +\subsection{Tabelle von Auf- und Ableitungen} +\begin{scriptsize} + \begin{tables}{lll}{\tr{Antiderivative}{Stammfunktion} & \tr{Function}{Funktion} & \tr{Derivative}{Ableitung}} + $\displaystyle \frac{x^{n + 1}}{n + 1}$ & $x^n$ & $n \cdot x^{n - 1}$ \\ + $\ln|x|$ & $\displaystyle \frac{1}{x} = x^{-1}$ & $\displaystyle -x^{-2} = -\frac{1}{x^2}$ \\[0.2cm] + $\frac{2}{3} x^{\frac{3}{2}}$ & $\displaystyle \sqrt{x} = x^{\frac{1}{2}}$ & $\displaystyle \frac{1}{2 \cdot \sqrt{x}}$ \\[0.3cm] + $\frac{n}{n + 1} x^{\frac{1}{n} + 1}$ & $\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}$ & $\frac{1}{n} x^{\frac{1}{n} - 1}$ \\[0.3cm] + \hline \\[-0.2cm] + $e^x$ & $e^x$ & $e^x$ \\ + $\exp(x)$ & $\exp(x)$ & $\exp(x)$ \\ + $\frac{1}{a \cdot (n + 1)}(ax + b)^{n + 1}$ & $(ax + b)^n$ & $n\cdot (ax + b)^{n - 1} \cdot a$ \\ + $x \cdot (\ln|x| - 1)$ & $\ln(x)$ & $\frac{1}{x} = x^{-1}$ \\ + $\displaystyle \frac{1}{\ln(a)}\cdot a^x$ & $a^x$ & $a^x \cdot \ln(a)$ \\ + $\frac{x}{\ln(a)} \cdot (\ln|x| - 1)$ & $\log_a|x|$ & $\displaystyle \frac{1}{x \cdot \ln(a)}$ \\[0.3cm] + \hline \\[-0.2cm] + $-\cos(x)$ & $\sin(x)$ & $\cos(x)$ \\ + $\sin(x)$ & $\cos(x)$ & $-\sin(x)$ \\ + $-\ln|\cos(x)|$ & $\tan(x)$ & $\displaystyle \frac{1}{\cos^2(x)}$ \\[0.3cm] + $x \cdot \arcsin(x) + \sqrt{1 - x^2}$ & $\arcsin(x)$ & $\displaystyle\frac{1}{\sqrt{1 - x^2}}$ \\ + $x \cdot \arccos(x) - \sqrt{1 - x^2}$ & $\arccos(x)$ & $\displaystyle -\frac{1}{\sqrt{1 - x^2}}$ \\ + $\displaystyle x \cdot \arctan(x) - \frac{\ln(x^2 + 1)}{2}$ & $\arctan(x)$ & $\displaystyle \frac{1}{x^2 + 1}$ \\[0.2cm] + $\ln|\sin(x)|$ & $\cot(x)$ & $\displaystyle -\frac{1}{\sin^2(x)}$ \\ + $\cosh(x)$ & $\sinh(x)$ & $\cosh(x)$ \\ + $\sinh(x)$ & $\cosh(x)$ & $\sinh(x)$ \\ + $\ln|\cosh(x)|$ & $\tanh(x)$ & $\displaystyle \frac{1}{\cosh^2(x)}$ \\ + & $\arcsinh(x)$ & $\frac{1}{\sqrt{1 + x^2}}$ \\ + & $\arccosh(x)$ & $\frac{1}{\sqrt{x^2 - 1}}$ \\ + & $\arctanh(x)$ & $\frac{1}{1 - x^2}$ \\ + \end{tables} +\end{scriptsize} + +\shade{teal}{Weitere Ableitungen} +\begin{tables}{cc}{$F(x)$ & $f(x)$} + $\frac{1}{a} \ln|ax + b|$ & $\frac{1}{ax + b}$ \\ + $\frac{ax}{c} - \frac{ad - bc}{c^2} \ln|cx + d|$ & $\frac{a (cx + d) - c(ax + b)}{(cx + d)^2}$ \\ + $\frac{x}{2} f(x) + \frac{a^2}{2} \ln|x + f(x)|$ & $\sqrt{a^2 + x^2}$ \\ + $\frac{x}{2} f(x) - \frac{a^2}{2} \arcsin\left( \frac{x}{|a|} \right)$ & $\sqrt{a^2 - x^2}$ \\ + $\frac{x}{2} f(x) - \frac{a^2}{2} \ln|x + f(x)|$ & $\sqrt{x^2 - a^2}$ \\ + $\ln(x + \sqrt{x^2 \pm a^2})$ & $\frac{1}{\sqrt{x^2 \pm a^2}}$ \\ + $\arcsin \left( \frac{x}{|a|} \right)$ & $\frac{1}{\sqrt{x^2 - a^2}}$ \\ + $\frac{1}{a}\arctan \left( \frac{x}{|a|} \right)$ & $\frac{1}{a^2 - x^2}$ \\ +\end{tables} +\begin{tables}{cc}{$F(x)$ & $f(x)$} + $-\frac{1}{a} \cos(ax + b)$ & $\sin(ax + b)$ \\ + $\frac{1}{a} \sin(ax + b)$ & $\cos(ax + b)$ \\[1mm] + \hline + $x^x$ & $x^x \cdot (1 + \ln|x|)$ \\ + $(x^x)^x$ & $(x^x)^x \cdot (x + 2x\ln|x|)$ \\ + $x^{(x^x)}$ & $x^{(x^x)} \cdot (x^{x - 1} + \ln|x| \cdot x^x (1 + \ln|x|))$ \\ + \hline \\[-3mm] + $\frac{1}{2}(x - \frac{1}{2} \sin(2x))$ & $\sin(x)^2$ \\[1mm] + $\frac{1}{2}(x + \frac{1}{2} \sin(2x))$ & $\cos(x)^2$ \\ +\end{tables} diff --git a/semester4/ps/ps-jh/parts/10_tips-and-tricks/03_limits.tex b/semester4/ps/ps-jh/parts/10_tips-and-tricks/03_limits.tex new file mode 100644 index 0000000..3d1370c --- /dev/null +++ b/semester4/ps/ps-jh/parts/10_tips-and-tricks/03_limits.tex @@ -0,0 +1,2 @@ +\subsection{Limits} +\includegraphics[width=1\columnwidth]{./assets/limits.png} diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index ee521bf..388bed9 100644 Binary files a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf and b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf differ diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex index 38326d1..9de09d6 100644 --- a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex +++ b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex @@ -78,6 +78,7 @@ \section{Gemeinsame Verteilungen} \input{parts/04_joint-distribution/00_discrete.tex} \input{parts/04_joint-distribution/01_continuous.tex} +\input{parts/04_joint-distribution/02_remarks.tex} % \input{parts/04_joint-distribution/} @@ -122,8 +123,11 @@ \newsectionNoPB -\section{Kombinatorik} +\section{Tabellen} \input{parts/10_tips-and-tricks/00_phi-func-table.tex} +\input{parts/10_tips-and-tricks/01_various.tex} +\input{parts/10_tips-and-tricks/02_diff-table.tex} +\input{parts/10_tips-and-tricks/03_limits.tex} % \input{parts/10_tips-and-tricks/} \end{document}