From 5240f4f75ce25384de63a2ac4691e220db229600 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Tue, 18 Aug 2026 16:59:42 +0200 Subject: [PATCH] [PS] More detailed descriptions --- .../parts/03_expected-value/05_variance.tex | 3 ++- .../04_joint-distribution/02_remarks.tex | 4 ++++ .../03_central-limit-theorem.tex | 4 ++++ .../parts/06_estimators/02_max-likelihood.tex | 3 +++ .../parts/10_tips-and-tricks/01_various.tex | 10 +++++++++- .../probability-and-statistics-cheatsheet.pdf | Bin 1132192 -> 1134841 bytes 6 files changed, 22 insertions(+), 2 deletions(-) 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 0b3f5c3..050aba5 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 @@ -15,7 +15,8 @@ $\V[\cX] = \E[\cX^2] - \E[\cX]^2 = a^2\E[1_\Omega] - a^2 = 0$ Falls $\cX_k$ abhängig, dann gilt $\neq$. Falls \bi{mermals} selbe Z.V. (e.g. $\cX - \cY - \cY$, ist $\cX - 2\cY$, dann $\V = \V[\cX] + 2^2 \V[\cY]$) -\newpage +\shortremark Für Varianz von bspw. $Y = \sum_{i = 1}^{n} \cX_i$, $\cX_i$ i.i.d: Unabh. nutzen (s. oben), dann ist Berechnung einfach + \shortexample Varianz von bekannten Verteilungen \begin{itemize} \item $\cX \sim \text{Ber}(p)$, $\V[\cX] = p (1 - p)$ 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 4d67aa7..b2b0059 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 @@ -7,6 +7,7 @@ Dies ist besonders nützlich für die Berechnung von bedingten Dichten, etc. und \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!) + \item Übrig bleibende Indikatorvariablen wieder anfügen \end{enumerate} } @@ -48,4 +49,7 @@ Die Randdichten sind dann die Dichten von $\cX$, etc.; Für weitere Berechnungen Für unbekannte gemeinsame Dichte, die Eigenschaften ausfächern, Unabhängigkeit, etc ausnutzen. \bi{Für W. im Exponent}, dies beschreibt typ. Repetitionen, also über die Anzahl und W. nachdenken. + Beispielsweise für $\E[\left( \frac{5}{6} \right)^{\cX - 2} = k]$ berechnen wir $\P[\cX = k] = \left( \frac{1}{6} \right)^{k - 1} \cdot \frac{5}{6}$, + dann ist $\E[\ldots] = \sum_{k = 1}^{\8} \left( \frac{5}{6} \right)^{k - 2} \cdot \left( \frac{1}{6} \right)^{k - 1} \cdot \frac{5}{6} + = \sum_{k = 1}^{n} \left( \frac{5}{36} \right)^{k - 1}$ } diff --git a/semester4/ps/ps-jh/parts/05_limit-theorems/03_central-limit-theorem.tex b/semester4/ps/ps-jh/parts/05_limit-theorems/03_central-limit-theorem.tex index 89d735a..b426cfb 100644 --- a/semester4/ps/ps-jh/parts/05_limit-theorems/03_central-limit-theorem.tex +++ b/semester4/ps/ps-jh/parts/05_limit-theorems/03_central-limit-theorem.tex @@ -28,3 +28,7 @@ $\P[a < S_n \leq b] \approx \Phi \left( \frac{b + \frac{1}{2} - np}{\sqrt{np(1 - \shortremark Für Bestimmung von $\P[S_n \leq y]$, ZGS direkt verwenden, dann die Ungleichung in gewünschte Form umformen und $\E[\cX_i]$ und $\V[\cX_i]$ berechnen und einsetzen. \shortremark Im Allgemeinen für ZGS, gegebenen Ausdruck umformen, und allfällig Werte bestimmen. +\hl{WICHTIG}: Muss genau in dieser Form sein. + +\shortexample $\limni \P\left[ \frac{S_n}{\sqrt{n}} > 1 \right]$ mit $S_n$ wie oben, $\E[S_n] = 0$, dann berechnen $1 - \P[ \frac{S_n}{\sqrt{n} \sigma} \leq \frac{1}{\sigma} ]$, +durch Umformung der Ungl. 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 c7c4d9f..0157429 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 @@ -29,6 +29,9 @@ Da oft $X_k$ i.i.d mit individueller Gew. $p_\cX(x; \vartheta)$ bzw. Dichte $f_\ {\scriptsize Wir suchen besten $t_{TM}$, durch geschickte Wahl von $\vartheta$}\\ Meistens sind $\cX_k$ i.i.d. unter $\P_\vartheta$, dann $L$ produkt, also besser $\log(L)$ maximieren (da Summe). +\shortdefinition[Momentschätzer (MM)] $T_{MM}^k = \frac{1}{n} \sum_{i = 1}^{n} X_i^k$, +und die Lösung ist definiert als $T_{MM}^k = \E[\cX^k]$. Für die Normalverteilung wird das Erste und Zweite Moment geschätzt. + % ──────────────────────────────────────────────────────────────────── \howto{Finden von ML-Schätzern}{ 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 index 6f34f3f..d4f2357 100644 --- 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 @@ -53,5 +53,13 @@ $\cosh(x) := \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}} : \R \subsubsection{Reihen} -\label{sec:series} $\prod_{k = 1}^{n} \vartheta (1 - \vartheta)^{x_k - 1} = \vartheta^n (1 - \vartheta)^{S - n}$ mit $S = \sum_{k = 1}^{n} x_k$ + +$\sum_{k = 1}^n k = \frac{n \cdot (n + 1)}{2}$; $\sum_{k = 1}^{n} k^2 = \frac{n \cdot (n + 1) \cdot (2n + 1)}{6}$; + +$\sum_{k = 1}^{n} k^3 = \left( \frac{n \cdot (n + 1)}{2} \right)^2$; +$\sum_{k = 1}^{n} \frac{1}{k^2} = \frac{\pi^2}{6}$ + +Arithmetic series: 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