From e194a76f9a09d3a9755bd0599758f20b590d7665 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Fri, 24 Jul 2026 10:44:40 +0200 Subject: [PATCH] [PS] Add more notes --- .../04_cont-dist.tex | 3 ++- .../parts/03_expected-value/03_properties.tex | 3 +++ .../parts/03_expected-value/05_variance.tex | 2 -- .../04_joint-distribution/02_remarks.tex | 18 +++++++++++++++++- .../parts/06_estimators/01_estimators.tex | 4 +++- .../parts/06_estimators/02_max-likelihood.tex | 7 +------ .../ps/ps-jh/parts/07_tests/02_examples.tex | 2 +- .../probability-and-statistics-cheatsheet.pdf | Bin 1123259 -> 1125880 bytes 8 files changed, 27 insertions(+), 12 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 c0e85ac..967ea16 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 +3,8 @@ \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. $\cX$ -Falls Z.V. Sprungstelle hat, \bi{weder stetig noch diskret}. +Falls Z.V. Sprungstelle hat, \bi{weder stetig noch diskret}.\\ +Zudem wenn nicht $P[a < \cX \leq b]$, dann Grenzwert an Seite mit Abweichung von diesem Schema \shortremark $\P[\cX = n] \approx 0$; $\P[m \leq \cX \leq n] = \int_{m}^{n} f_\cX(x) \dx x$ diff --git a/semester4/ps/ps-jh/parts/03_expected-value/03_properties.tex b/semester4/ps/ps-jh/parts/03_expected-value/03_properties.tex index 3134adc..4f74671 100644 --- a/semester4/ps/ps-jh/parts/03_expected-value/03_properties.tex +++ b/semester4/ps/ps-jh/parts/03_expected-value/03_properties.tex @@ -16,3 +16,6 @@ $\E\left[ \prod_{k = 1}^n \cX_k \right] = \prod_{k = 1}^n \E[\cX_k]$ \shorttheorem äquivalent: \bi{(1)} $\cX_i$ unabhängig,\\ \bi{(2)} $\forall \varphi_i$: $\E[\varphi_1(\cX_1) \cdots \varphi_n(\cX_n)] = \E[\varphi_1(\cX_1)] \cdots \E[\varphi_n(\cX_n)]$ + +\shortremark[Bedingte W.] $\E[I] = \P[K \cap R]\E[I | K, R] + \ldots$\\ +{\scriptsize Dabei beinflussen $K$ und $R$ den Wert nur durch Bedingte W.} 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 1de2f6e..a5f1d85 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 @@ -1,6 +1,4 @@ \subsection{Varianz} -$\E[\cX^2] = \sum_{x \in W} x^2 \cdot p_\cX(x)$ ($\cX$ diskret) - \shortdefinition $\cX$ mit $\E[\cX^2] < \8$, $\V[\cX] = \E[(\cX - \E[\cX])^2]$ \shortdefinition[Standardabweichung] $\sigma(\cX) = \sqrt{\V[\cX]}$ 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 1248639..0883e78 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 @@ -1,7 +1,7 @@ \subsection{Tips \& Tricks} \shortremark Die \bi{Dichten} können anstelle der Variablen in bspw. der Formel für bedingte W. verwendet werden. -\coloredbox{gray}{HOW-TO: Randdichten berechnen}{ +\howto{Randdichten berechnen}{ \begin{enumerate} \item Intervalle zur Integration für Randdichten via Indikatorvariable $\bm{1}_{cond}$ bestimmen. $\bm{1}_x$ verschwindet bei Bestimmung von $f_\cY$ in den Integral-grenzen \item $cond$ nach entsprechender Var. auflösen, wenn nötig @@ -10,3 +10,19 @@ } Die Randdichten sind dann die Dichten von $\cX$, etc.; Für weitere Berechnungen verwendbar, wie bspw. $\E[\cX + \cY]$. + +\howto{Gemeinsame Dichten finden}{ + Nach Definition muss $f$ die Ableitung von $F$ sein. + Oder umgekehrt, $\int f \dx \ldots$ sollte $F$ ergeben. Beispiel: + $T \sim \cU[1, 2], \cX \sim \text{Exp}(\lambda), \cY \sim \text{Exp}(\gamma)$. Dann $f_{T, X, Y}(t, x, y) = \begin{cases} + \lambda \gamma \exp(-\lambda x - \gamma y) & \text{falls } 1 \leq t \leq 2 \land x, y \geq 0 \\ + 0 & \text{sonst} + \end{cases}$ +} + +\howto{Verteilungen finden}{ + Wir suchen $\P[\cZ \leq a]$ (ausser für gem. V, dann s. oben). + Dann überlegen, wie $\cZ \leq a$ erfüllt werden kann. Beispiel: + $Z = \min(\cX, \cY)$, dann gesucht: $\P[\cX \leq a \cup \cY \leq a]$ oder $1 - \P[\cZ > a]$. + Schliesslich $F_Z$ aufstellen mit Bedingungen. +} diff --git a/semester4/ps/ps-jh/parts/06_estimators/01_estimators.tex b/semester4/ps/ps-jh/parts/06_estimators/01_estimators.tex index 954a9c6..189872a 100644 --- a/semester4/ps/ps-jh/parts/06_estimators/01_estimators.tex +++ b/semester4/ps/ps-jh/parts/06_estimators/01_estimators.tex @@ -22,4 +22,6 @@ $\text{MSE}_\vartheta[T] = \E_\vartheta[(T - \vartheta)^2]$ \shortdefinition \bi{Folge von Schätzern} $T^{(n)}$, $n \in \N$ ist \bi{konsistent} für $\vartheta$, falls $T^{(n)}$ für $n \rightarrow \8$ in $\P_\vartheta$-W. gegen $\vartheta$ konvergiert, also $\forall \vartheta \in \Theta, \varepsilon > 0$: -$\limit{n}{\8} \P_\vartheta [|T^{(n)} - \vartheta| > \varepsilon] = 0$ +$\limit{n}{\8} \P_\vartheta [|T^{(n)} - \vartheta| > \varepsilon] = 0$. + +{\scriptsize Beweise oft mit GGZ (s. \ref{sec:weak-law-of-large-numbers}), also $T$ umformen, s.d. anwendbar (e.g. $T^{(n)} \div n$)} diff --git a/semester4/ps/ps-jh/parts/06_estimators/02_max-likelihood.tex b/semester4/ps/ps-jh/parts/06_estimators/02_max-likelihood.tex index 91c87d7..1c829d9 100644 --- a/semester4/ps/ps-jh/parts/06_estimators/02_max-likelihood.tex +++ b/semester4/ps/ps-jh/parts/06_estimators/02_max-likelihood.tex @@ -31,12 +31,7 @@ Meistens sind $\cX_k$ i.i.d. unter $\P_\vartheta$, dann $L$ produkt, also besser % ──────────────────────────────────────────────────────────────────── -\coloredbox{red}{REWRITE}{ - This section is undergoing a rewrite currently. - When complete, there will be estimators for (pretty much) every covered distribution -} - -\coloredbox{gray}{How To: Finden von ML-Schätzern}{ +\howto{Finden von ML-Schätzern}{ \begin{enumerate} \item log-LH-Funk. bestimmen ($x_i$ statt $\cX_i$ verwenden) \item Nach $\vartheta$ ableiten (oder wodurch $\vartheta$ ersetzt wurde) diff --git a/semester4/ps/ps-jh/parts/07_tests/02_examples.tex b/semester4/ps/ps-jh/parts/07_tests/02_examples.tex index 4d7f97c..f53c766 100644 --- a/semester4/ps/ps-jh/parts/07_tests/02_examples.tex +++ b/semester4/ps/ps-jh/parts/07_tests/02_examples.tex @@ -2,7 +2,7 @@ \coloredbox{red}{REWRITE}{ This section is undergoing a rewrite, to be more on-point. 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