diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/02_discrete-rv.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/02_discrete-rv.tex index 4ea1dab..22b3577 100644 --- a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/02_discrete-rv.tex +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/02_discrete-rv.tex @@ -7,7 +7,8 @@ \shorttheorem $(p(x))_{x \in W} = \sum_{x \in W} p(x) = 1$ -\shortremark $\forall (p(x))_{x \in W} \; \exists$ eine Z.V. mit dieser Verteilung. Können desh. schreiben: ``Sei $\cX$ disk. Z.V. mit Verteilung $(p(x))_{x \in W}$'' +\shortremark $\forall (p(x))_{x \in W} \; \exists$ Z.V. mit dieser Verteilung. Können desh. schreiben: +``Sei $\cX$ disk. Z.V. mit Verteilung $(p(x))_{x \in W}$'' \subsubsection{Zusammenhang Verteilung, Verteilungsfunktion} diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/04_cont-dist.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/04_cont-dist.tex index 7eae8a6..c55f360 100644 --- a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/04_cont-dist.tex +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/04_cont-dist.tex @@ -5,7 +5,7 @@ $f_\cX$ ist Dichte (pdf) von $\cX$ % \shortdefinition[Stückw. st. diff. F.] TODO: Consider adding this, Slides Chapter 3, P56 -\shorttheorem Sei $F_\cX$ st. stückw. diff. auf Partition $-\8 = x_0 < x_1 \ldots < x_n = \8$. +\shorttheorem Sei $F_\cX$ st. stückw. diff. auf Partition $-\8 = x_0 < x_1 < \ldots < x_n = \8$. Dann $\cX$ stetig, mit $a_k$ beliebig und \[ f_\cX = \begin{cases} diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/00_eq.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/00_eq.tex new file mode 100644 index 0000000..9e4ad3b --- /dev/null +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/00_eq.tex @@ -0,0 +1,16 @@ +\subsection{Stetige Verteilungen} + +\subsubsection{Gleichverteilung} + +\shortdefinition $\cX \sim \cU([a, b])$, f +$f_\cX = \begin{cases} + \frac{1}{b - a} & x \in [a, b] \\ + 0 & \text{sonst} + \end{cases}$ + +\shortremark $\P[\cX \in [c, c + l]] = \frac{l}{b - a}$, +$F_\cX(x) = \begin{cases} + 0 & x < a \\ + \frac{x - a}{b - a} & a \leq x \leq b \\ + 1 & x > b + \end{cases}$ diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/00_normal.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/00_normal.tex deleted file mode 100644 index 6fc1a8c..0000000 --- a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/00_normal.tex +++ /dev/null @@ -1,5 +0,0 @@ -\subsection{Stetige Verteilungen} -\shortdefinition[Gleichverteilung] $\cX \sim \cU([a, b])$, falls $f_\cX = \begin{cases} - \frac{1}{b - a} & x \in [a, b] \\ - 0 & \text{sonst} -\end{cases}$ diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/01_exp.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/01_exp.tex new file mode 100644 index 0000000..2ec82bf --- /dev/null +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/01_exp.tex @@ -0,0 +1,15 @@ +\subsubsection{Exponentialverteilung} +{\scriptsize Wie Geomemtrische Verteilung warten auf Erfolg} + +\shortdefinition $\cX \sim \text{Exp}(\lambda)$, falls +$\forall x \in \R f_\cX(x) = \begin{cases} + \lambda e^{-\lambda x} & x \geq 0 \\ + 0 & x < 0 + \end{cases}$ + +\shortremark[Gedächtnisl.] $\P[\cX > t + s | \cX > s] = \P[\cX > t]$ + +\shortremark[Verteilungsfunktion] $F_\cX(x) = \begin{cases} + 1 - e^{-\lambda x} & x \geq 0 \\ + 0 & x < 0 +\end{cases}$ diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/02_cauchy.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/02_cauchy.tex new file mode 100644 index 0000000..58a2b16 --- /dev/null +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/02_cauchy.tex @@ -0,0 +1,6 @@ +\subsubsection{Cauchy-Verteilung} +\shortdefinition $\cX \sim \text{Cauchy}(x_0, \gamma)$, falls $\displaystyle f_\cX(x) = \frac{1}{\pi} \frac{\gamma}{\gamma^2 + (x - x_0)^2}$ + +\shortremark[Verteilungsfunk.] $\displaystyle F_\cX(x) = \frac{1}{2} + \frac{1}{\pi} \arctan \left( \frac{x - x_0}{\gamma} \right)$ + +\shortdefinition[Langschwänzige Verteilung] für $|x| \rightarrow \8$ nur sehr langsam gegen $0$ (quadratisch vs. exponentiell bei Norm. V) diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/03_normal.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/03_normal.tex new file mode 100644 index 0000000..f9c4dc8 --- /dev/null +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/03_normal.tex @@ -0,0 +1,28 @@ +\subsubsection{Normalverteilung} +\shortdefinition $\cX \sim \cN(\mu, \sigma^2)$ falls $f_\cX(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{\frac{1}{2} \left( \frac{x - \mu}{\sigma} \right)^2}$,\\ +mit $\sigma$ Standardabweichung. Auch: Gauss'sche Verteilung + +\shortdefinition[Standardnormalverteilung] $\cX \sim \cN(0, 1)$:\\ +$f_\cX = \varphi$ und $\F_\cX = \Phi = \int_{-\8}^{x} \varphi(t) \dx t = \frac{1}{\sqrt{2\phi}} \int_{-\8}^{x} e^\frac{-t^2}{2} \dx t$ + +\shorttheorem $cX \sim \cN(\mu, \sigma^2)$, dann $\frac{\cX - \mu}{\sigma} \sim \cN(0, 1)$, also: +\[ + F_\cX(x) = \P[\cX \leq x] = \P\left[ \frac{\cX - \mu}{\sigma} \leq \frac{x - \mu}{\sigma} \right] = \Phi \left( \frac{x - \mu}{\sigma} \right) +\] + +\shortexample für Phänomene modellierbar mit Normalverteilung: +\begin{itemize} + \item Streuung von Messwerten um Mittelwert + \item Grösse und Gewicht der Bevölkerung + \item Renditen von Aktien +\end{itemize} + +\shortremark Für $\cX_i \sim \cN(\mu_i, \sigma_i^2)$ unabhängig gilt: +\[ + \cY := \mu_0 + \sum_{k = 1}^{n} a_k \cX_k \sim \cN\left( \mu_0 + \sum_{k = 1}^{n} a_k \mu_k, \sum_{k = 1}^{n} a_k^2 \sigma_k^2 \right) +\] + +\shortremark Für $\mu \in \R, \sigma^2 > 0$ und $\cZ \sim \cN(0, 1)$ gilt\\ +$\mu + \sigma \cZ \sim \cN(\mu, \sigma^2)$ (nützlich für Simulation) + +\shortremark $\P[|\cX - \mu| \geq 3\sigma] \leq 0.0027$ diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index 6800e8e..775ea50 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 81c889e..9bb7e0f 100644 --- a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex +++ b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex @@ -9,6 +9,9 @@ \setupCheatSheet{Wahrscheinlichkeit und Statistik} \loadGerman +\setnumberingpreset{off} +\renewcommand{\definitionShortNamingDE}{Def} +\renewcommand{\remarkShortNamingDE}{Bem} \setsubsectionnumbering{section} \renewcommand{\examplenumbering}{off} @@ -47,8 +50,10 @@ \input{parts/02_discrete-continuous-rv/03_distributions/03_hyp-geom.tex} \input{parts/02_discrete-continuous-rv/03_distributions/04_poisson.tex} \input{parts/02_discrete-continuous-rv/04_cont-dist.tex} -\input{parts/02_discrete-continuous-rv/05_cont-distributions/00_normal.tex} -% \input{parts/02_discrete-continuous-rv/05_cont-distributions/} +\input{parts/02_discrete-continuous-rv/05_cont-distributions/00_eq.tex} +\input{parts/02_discrete-continuous-rv/05_cont-distributions/01_exp.tex} +\input{parts/02_discrete-continuous-rv/05_cont-distributions/02_cauchy.tex} +\input{parts/02_discrete-continuous-rv/05_cont-distributions/03_normal.tex} % \input{parts/02_discrete-continuous-rv/}