From 653a8078480982011e377af9bbf254c2bd122493 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Fri, 17 Jul 2026 12:17:14 +0200 Subject: [PATCH] [PS] Updated marginal distribution tips & tricks --- .../04_cont-dist.tex | 6 ++- .../05_cont-distributions/03_normal.tex | 1 - .../parts/03_expected-value/05_variance.tex | 3 +- .../04_joint-distribution/01_continuous.tex | 24 +++--------- .../04_joint-distribution/02_remarks.tex | 25 ++++-------- .../10_tips-and-tricks/02_diff-table.tex | 37 +++++++++--------- .../probability-and-statistics-cheatsheet.pdf | Bin 1124242 -> 1121904 bytes 7 files changed, 39 insertions(+), 57 deletions(-) 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 04fcfac..c0e85ac 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 @@ -3,7 +3,11 @@ \shortdefinition[Stetig verteilte Z.V] $\cX$ stetig, falls $\exists f_\cX : \R \rightarrow \R_+$, s.d. V.F. $F_\cX(x) = \int_{-\8}^{x} f_\cX(t) \dx t$. $f_\cX$ ist Dichte (pdf) v. $\cX$ -\shorttheorem Sei $F_\cX$ st. stückw. diff. auf Partition $-\8 = x_0 < \ldots < x_n = \8$. +Falls Z.V. Sprungstelle hat, \bi{weder stetig noch diskret}. + +\shortremark $\P[\cX = n] \approx 0$; $\P[m \leq \cX \leq n] = \int_{m}^{n} f_\cX(x) \dx x$ + +\shorttheorem $F_\cX$ stetig stückw. diff. auf Partition $-\8 = x_0 < \ldots < x_n = \8$. Dann $\cX$ stetig, mit $a_k$ beliebig und \[ f_\cX = \begin{cases} diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/03_normal.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/03_normal.tex index 4969e1c..d609f55 100644 --- a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/03_normal.tex +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/03_normal.tex @@ -11,7 +11,6 @@ $\Phi = \int_{-\8}^{x} \varphi(t) \dx t = \frac{1}{\sqrt{2\pi}} \int_{-\8}^{x} e F_\cX(x) = \P[\cX \leq x] = \P\left[ \frac{\cX - \mu}{\sigma} \leq \frac{x - \mu}{\sigma} \right] = \Phi \left( \frac{x - \mu}{\sigma} \right) \] -\newpage \shortexample für Phänomene modellierbar mit Normalverteilung: \begin{itemize} \item Streuung von Messwerten um Mittelwert diff --git a/semester4/ps/ps-jh/parts/03_expected-value/05_variance.tex b/semester4/ps/ps-jh/parts/03_expected-value/05_variance.tex index 79168e5..1de2f6e 100644 --- a/semester4/ps/ps-jh/parts/03_expected-value/05_variance.tex +++ b/semester4/ps/ps-jh/parts/03_expected-value/05_variance.tex @@ -14,7 +14,8 @@ $\V[\cX] = \E[\cX^2] - \E[\cX]^2 = a^2\E[1_\Omega] - a^2 = 0$ \shortremark $\E[\cX] < \8$, dann $\V[\cX] \geq 0$ mit $=$ g.d.w. $\cX$ konst; \hl{zudem} $\V[a \cX] = a^2 \V[\cX]$ und $\V[\cX + a] = \V[\cX]$ \shortproposition $\cX_k$ paarw. unabh. $\V\left[ \sum_{k = 1}^{n} \cX_k \right] = \sum_{k = 1}^{n} \V[\cX_k]$. -Falls $\cX_k$ abhängig, dann gilt $\neq$ +Falls $\cX_k$ abhängig, dann gilt $\neq$. +Falls \bi{mermals} selbe Z.V. (e.g. $\cX - \cY - \cY$, ist $\cX - 2\cY$, dann $\V = 2^2 \V[\cY]$) \newpage \shortexample Varianz von bekannten Verteilungen 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 3562d26..eebd554 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 @@ -46,12 +46,6 @@ Die Dichten der Randverteilungen sind: \qquad f_\cY(y) = \int_{-\8}^{\8} f(x, y) \dx x \] -Herleitung der Randdichte (``wegintegrieren''): -\begin{align*} - f_\cX(x) & = \diff{x} F_\cX(x) = \diff{x}\left( \int_{-\8}^{x} \int_{-\8}^{\8} f(s, t) \dx t \dx s \right) \\ - & =\int_{-\8}^{\8} f(x, t) \dx t -\end{align*} -{\scriptsize (Hier wieder Umwandlung von Summe zu Integral von Diskret zu Stetig)} \shortexample Randdichten mit Gleichverteilung\\ \bi{Einheitsquadrat} {\scriptsize ($f(x, y) = 1_{(x, y) \in [0, 1]^2} = 1_{x \in [0, 1]} 1_{y \in [0, 1]}$)} @@ -67,18 +61,12 @@ Herleitung der Randdichte (``wegintegrieren''): f_\cY(y) \! = \! \int_{-\8}^{\8} \frac{1}{\pi} 1_{x^2 \leq 1 - y^2} \dx x \! = \! \int_{-\sqrt{1 - y^2}}^{\sqrt{1 - y^2}} \frac{1}{\pi} \dx x \! = \! \frac{2}{\pi} \sqrt{1 - y^2} \end{align*} -\shorttheorem[Unabhängigkeit] $\cX_i$ mit Dichten $f_{\cX_i}$, dann ist äquiv.: -\begin{itemize} - \item $\cX_1, \ldots, \cX_n$ sind unabhängig - \item $\cX_1, \ldots, \cX_n$ sind gem. stetig mit gem. Dichte (Prod. Randdichten): - $f(x_1, \ldots, x_n) = f_{\cX_1}(x_1) \cdot \ldots \cdot f_{\cX_n}(x_n)$ -\end{itemize} +\shorttheorem[Unabhängigkeit] $\cX_i$ unabhängig g.d.w. +\[ + f_{\cX_1, \ldots, \cX_n}(x_1, \ldots, x_n) = f_{\cX_1}(x_1) \cdot \ldots \cdot f_{\cX_n}(x_n) +\] \shortexample Unabhängigkeit bei Gleichverteilungen\\ -\bi{Einheitsquadrat} Wieder $f(x, y) = 1_{(x, y) \in [0, 1]^2}$, dann: -\[ - f(x, y) = 1_{(x, y) \in [0, 1]^2} = 1_{x \in [0, 1]} 1_{y \in [0, 1]} = f_\cX(x) f_\cY(y) -\] -Folglich sind die beiden Koordinaten unabhängig. +\bi{Einheitsquadrat} Koordinaten sind unabhängig -\bi{Einheitskreisscheibe} $f(x, y) \neq f_\cX(x) f_\cY(y)$, mit Dichten von oben. Also sind die beiden Koordinaten nicht unabh. +\bi{Einheitskreisscheibe} Koordinaten nicht unabhängig 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 index 7c31e1b..1b2447e 100644 --- a/semester4/ps/ps-jh/parts/04_joint-distribution/02_remarks.tex +++ b/semester4/ps/ps-jh/parts/04_joint-distribution/02_remarks.tex @@ -1,21 +1,12 @@ \subsection{Tips \& Tricks} -\coloredbox{red}{REWRITE}{ - This section is going to be completely rewritten -} \shortremark Die \bi{Dichten} können anstelle der Variablen in bspw. der Formel für bedingte W. verwendet werden. -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 $t < 0$, $W(t) = 0$. +\coloredbox{gray}{HOW-TO: Randdichten berechnen}{ + \begin{enumerate} + \item Intervalle zur Integration für Randdichten via Indikatorvariable $\bm{1}_{cond}$ bestimmen. $\bm{1}_x$ verschwindet bei Bestimmung von $f_\cY$ + \item $cond$ nach entsprechender Var. auflösen, wenn nötig + \item Bei Integration Indikatorvariable als konstante betrachten (muss auch integriert werden!) + \end{enumerate} +} -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 $W(t) = 1$. +Die Randdiochten sind dann die Dichten von $\cX$, etc; Für weitere Berechnungen verwendbar, wie bspw. $\E[\cX + \cY]$. 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 index b90bfca..cf4a4c6 100644 --- 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 @@ -1,6 +1,7 @@ \newpage \subsection{Tabelle von Auf- und Ableitungen} Für Integrale von stückweisen Funktionen, stückweise Integrieren und aufsummieren. +Für ungerade Funktionen (mit $f(-x) = -f(x)$, bspw $f(x) = x$ und $f(x) = \cos(x)$) ist Integral $\int_{-C}^C f(x) \dx x = 0$, auch bei Multiplikation von mehreren Funktionen. \begin{scriptsize} \begin{tables}{lll}{\tr{Antiderivative}{Stammfunktion} & \tr{Function}{Funktion} & \tr{Derivative}{Ableitung}} $\displaystyle \frac{x^{n + 1}}{n + 1}$ & $x^n$ & $n \cdot x^{n - 1}$ \\ @@ -21,28 +22,26 @@ Für Integrale von stückweisen Funktionen, stückweise Integrieren und aufsummi $x \cdot \arcsin(x) + \sqrt{1 - x^2}$ & $\arcsin(x)$ & $\displaystyle\frac{1}{\sqrt{1 - x^2}}$ \\ $x \cdot \arccos(x) - \sqrt{1 - x^2}$ & $\arccos(x)$ & $\displaystyle -\frac{1}{\sqrt{1 - x^2}}$ \\ $\displaystyle x \cdot \arctan(x) - \frac{\ln(x^2 + 1)}{2}$ & $\arctan(x)$ & $\displaystyle \frac{1}{x^2 + 1}$ \\[0.2cm] - $\ln|\sin(x)|$ & $\cot(x)$ & $\displaystyle -\frac{1}{\sin^2(x)}$ \\ - $\cosh(x)$ & $\sinh(x)$ & $\cosh(x)$ \\ - $\sinh(x)$ & $\cosh(x)$ & $\sinh(x)$ \\ - $\ln|\cosh(x)|$ & $\tanh(x)$ & $\displaystyle \frac{1}{\cosh^2(x)}$ \\ - & $\arcsinh(x)$ & $\frac{1}{\sqrt{1 + x^2}}$ \\ - & $\arccosh(x)$ & $\frac{1}{\sqrt{x^2 - 1}}$ \\ - & $\arctanh(x)$ & $\frac{1}{1 - x^2}$ \\ + % $\ln|\sin(x)|$ & $\cot(x)$ & $\displaystyle -\frac{1}{\sin^2(x)}$ \\ + % $\cosh(x)$ & $\sinh(x)$ & $\cosh(x)$ \\ + % $\sinh(x)$ & $\cosh(x)$ & $\sinh(x)$ \\ + % $\ln|\cosh(x)|$ & $\tanh(x)$ & $\displaystyle \frac{1}{\cosh^2(x)}$ \\ + % & $\arcsinh(x)$ & $\frac{1}{\sqrt{1 + x^2}}$ \\ + % & $\arccosh(x)$ & $\frac{1}{\sqrt{x^2 - 1}}$ \\ + % & $\arctanh(x)$ & $\frac{1}{1 - x^2}$ \\ \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}{$F(x)$ & $f(x)$} +% \shade{teal}{Weitere Ableitungen} +% \begin{tables}{cc}{$F(x)$ & $f(x)$} +% $\frac{1}{a} \ln|ax + b|$ & $\frac{1}{ax + b}$ \\ +% $\frac{ax}{c} - \frac{ad - bc}{c^2} \ln|cx + d|$ & $\frac{a (cx + d) - c(ax + b)}{(cx + d)^2}$ \\ +% $\frac{x}{2} f(x) + \frac{a^2}{2} \ln|x + f(x)|$ & $\sqrt{a^2 + x^2}$ \\ +% $\frac{x}{2} f(x) - \frac{a^2}{2} \arcsin\left( \frac{x}{|a|} \right)$ & $\sqrt{a^2 - x^2}$ \\ +% $\frac{x}{2} f(x) - \frac{a^2}{2} \ln|x + f(x)|$ & $\sqrt{x^2 - a^2}$ \\ +% $\ln(x + \sqrt{x^2 \pm a^2})$ & $\frac{1}{\sqrt{x^2 \pm a^2}}$ \\ +% $\arcsin \left( \frac{x}{|a|} \right)$ & $\frac{1}{\sqrt{x^2 - a^2}}$ \\ +% $\frac{1}{a}\arctan \left( \frac{x}{|a|} \right)$ & $\frac{1}{a^2 - x^2}$ \\ % $-\frac{1}{a} \cos(ax + b)$ & $\sin(ax + b)$ \\ % $\frac{1}{a} \sin(ax + b)$ & $\cos(ax + b)$ \\[1mm] % \hline diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index 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