From 45075d2952f78e5e3fe485155c638bc5292e8671 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Wed, 22 Jul 2026 16:11:11 +0200 Subject: [PATCH] [PS] Analysis stuff, fixes --- .../04_joint-distribution/02_remarks.tex | 4 ++-- .../parts/06_estimators/02_max-likelihood.tex | 6 ++++++ .../10_tips-and-tricks/02_diff-table.tex | 19 ++++++++++++++++++ .../probability-and-statistics-cheatsheet.pdf | Bin 1121904 -> 1123085 bytes 4 files changed, 27 insertions(+), 2 deletions(-) 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 1b2447e..1248639 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 @@ -3,10 +3,10 @@ \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 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 \item Bei Integration Indikatorvariable als konstante betrachten (muss auch integriert werden!) \end{enumerate} } -Die Randdiochten sind dann die Dichten von $\cX$, etc; Für weitere Berechnungen verwendbar, wie bspw. $\E[\cX + \cY]$. +Die Randdichten sind dann die Dichten von $\cX$, etc.; Für weitere Berechnungen verwendbar, wie bspw. $\E[\cX + \cY]$. 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 4b53bb6..91c87d7 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 @@ -78,6 +78,12 @@ $\cX_i \sim \cN(\mu, \sigma^2)$ i.i.d. und $\vartheta = (\mu, \sigma^2) = (\mu, \[ l(\vartheta) = -\frac{1}{2} n(\log(2 \pi) + \log(v)) - \sum_{k = 1}^{n} \frac{(x_k - \mu)^2}{2v} \] + +\[ + \partial_{\mu} l = 2 \sum_{k = 1}^{n} \frac{x_k - \mu}{2 v} + \quad + \partial_v l = -\frac{n}{2v} + 2v^{-2}\sum_{k = 1}^{n} (x_k - \mu)^2 +\] Der \bi{Momentschätzer} ist $T = (T_1, T_2) = (E_\vartheta[\cX], \V_\vartheta[\cX])$: \[ T_1 = \overline{\cX}_n \qquad 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 cf4a4c6..9bc5061 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 @@ -2,6 +2,25 @@ \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. + +\shortremark[Integration by parts] +\[ + \int_{a}^{b} f(x) g'(x) \dx x = \left[f(x) g(x)\right]^b_a - \int_{a}^{b} f'(x) g(x) \dx x +\] +$f(x)$ einfach ableitbar, $g'(x)$ einfach aufleitbar wählen. + +\shortremark[Integration by substitution] +\[ + \int_{a}^{b} f(\varphi(t)) \varphi'(t) \dx t = \int_{\varphi(a)}^{\varphi((b))} f(x) \dx x +\] +dabei die innere Funktion gut wählen, ableiten und zurück substituieren. Grenzen des Integrals anpassen! + +\shortremark[Multiplication rule] $f'(x)g(x) + f(x)g'(x)$ + +\shortremark[Quotient rule] $\displaystyle \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$ + + + \begin{scriptsize} \begin{tables}{lll}{\tr{Antiderivative}{Stammfunktion} & \tr{Function}{Funktion} & \tr{Derivative}{Ableitung}} $\displaystyle \frac{x^{n + 1}}{n + 1}$ & $x^n$ & $n \cdot x^{n - 1}$ \\ diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index eb4a1e8f3db406373c75da177d70420163881ba4..b0f81f402acf069253fde137657f742637d6da9f 100644 GIT binary patch delta 89948 zcmV(xKsW z4KInHn$sDaE6QaKs9QgRU15p>Qm=z&hBC%ylo-Xi!i8kg07_RK)DKQ)54a>s=CH%{ zL+FYt!m^y3<{>;$Qo^~ed-tD|Fr3;C%ZJraYwuYFPt&`DxPievfSzht-6K0eH=k;v z_0RBm^(pjnF?;ZK_TVSG#;sK?SKV%Z7GNixNx(LZ8-)!r_s{?R^m0~iV*H3#oh?`f z-2nmOG9!{Q?9!4Jz`};KxJFeQi-}K)^X=-FjPhO*;~`nA6By{HW>O0~t3^GnpJwjA z%`9kE(Mu0iyOdA@AdyTq7J^gz 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