From e005c7608b595269160569db9bc6736b2bb2447a Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Wed, 15 Jul 2026 16:14:29 +0200 Subject: [PATCH] [PS] Rework parts of tests section, more changes to come --- README.md | 4 ++ .../04_cont-dist.tex | 10 ++--- .../parts/03_expected-value/00_intro.tex | 2 +- .../parts/03_expected-value/05_variance.tex | 2 +- .../parts/03_expected-value/06_covariance.tex | 3 +- .../04_joint-distribution/02_remarks.tex | 5 +++ .../parts/06_estimators/02_max-likelihood.tex | 15 +++---- .../ps/ps-jh/parts/07_tests/02_examples.tex | 38 +++++++++++++----- .../ps/ps-jh/parts/07_tests/03_p-value.tex | 1 + .../09_combinatorics/03_binomial-theorem.tex | 4 ++ .../probability-and-statistics-cheatsheet.pdf | Bin 1121817 -> 1124242 bytes .../probability-and-statistics-cheatsheet.tex | 1 + 12 files changed, 59 insertions(+), 26 deletions(-) create mode 100644 semester4/ps/ps-jh/parts/09_combinatorics/03_binomial-theorem.tex diff --git a/README.md b/README.md index e31081d..6f68237 100644 --- a/README.md +++ b/README.md @@ -84,6 +84,10 @@ git clone https://github.com/janishutz/eth-summaries You will need to also clone the helpers to `~/projects/latex`, which you can find [here](https://git.janishutz.com/janishutz/latex). If you want to use a different folder, simply duplicate the import line, change it and when opening a PR, simply change it back. +For some of the older summaries, you will need to check out the `v1` branch of the helpers, as they were written before the new helpers were created +and have not been updated to the new ones. +This likely includes all summaries and cheat sheets up to and including semester 3. + ------- ***NOTE***: It is recommended that you fork the repo to contribute back to it. To do that, click the fork button on the repo and clone it locally using ```bash 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 d49e17c..04fcfac 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 @@ -1,18 +1,18 @@ \newpage \subsection{Stetige Zufallsvariablen} \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) von $\cX$ +$f_\cX$ ist Dichte (pdf) v. $\cX$ -% \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 $-\8 = x_0 < \ldots < x_n = \8$. Dann $\cX$ stetig, mit $a_k$ beliebig und \[ f_\cX = \begin{cases} - F_\cX'(x) & \exists k \in \{0, 1, \ldots, n - 1\} : x \in (x_k, x_{k + 1}) \\ + F_\cX'(x) & \exists k \in \{0, \ldots, n - 1\} : x \in (x_k, x_{k + 1}) \\ a_k & x \in \{ x_1, \ldots, x_{n - 1} \} \end{cases} \] +\shorttheorem $\int_{-\8}^\8 f_\cX(x) \dx x = 1$ \hl{(nützl. f. berech. v. Param in $f$)} + \shortremark $f_\cX$ fast analog zu Gew.F $p_\cX$. Also:\\ $(\Sigma, p_\cX) \mapsto (\int, f_\cX)$ vom diskreten zu stetigen Fall diff --git a/semester4/ps/ps-jh/parts/03_expected-value/00_intro.tex b/semester4/ps/ps-jh/parts/03_expected-value/00_intro.tex index 34a81fc..262bdd3 100644 --- a/semester4/ps/ps-jh/parts/03_expected-value/00_intro.tex +++ b/semester4/ps/ps-jh/parts/03_expected-value/00_intro.tex @@ -10,4 +10,4 @@ \shortremark $|\cX| = \cX_+ + \cX_-$; $\cX = \cX_+ - \cX_-$. Für $\cX \geq 0$ ist $\E[\cX]$ immer definiert. Falls $\cX$ kein konst. Vorzeichen, $\E[\cX]$ undef. -\shortremark $\E[\cX] = \int_{0}^{\8} (1 - F_\cX(x)) \dx x - \int_{-\8}^{0} F_\cX(x)$ +\shortremark $\E[\cX] = \int_{0}^{\8} (1 - F_\cX(x)) \dx x - \int_{-\8}^{0} F_\cX(x) \dx x$ 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 d99d0ea..79168e5 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 @@ -5,7 +5,7 @@ $\E[\cX^2] = \sum_{x \in W} x^2 \cdot p_\cX(x)$ ($\cX$ diskret) \shortdefinition[Standardabweichung] $\sigma(\cX) = \sqrt{\V[\cX]}$ -\shortremark $\V[\cX] = \E[\cX^2] - \E[\cX]^2$ +\shortremark $\V[\cX] = \E[\cX^2] - \E[\cX]^2$ ($\V[\cX] \geq 0$ ist wahr) \shortexample $\cX$ determ. Z.V (= konst) mit Wert $a$, also $\cX = a1_\Omega$. Dann: $\E[\cX] = a \E[1_\Omega] = a \P[\Omega] = a$ und\\ $\V[\cX] = \E[\cX^2] - \E[\cX]^2 = a^2\E[1_\Omega] - a^2 = 0$ diff --git a/semester4/ps/ps-jh/parts/03_expected-value/06_covariance.tex b/semester4/ps/ps-jh/parts/03_expected-value/06_covariance.tex index 98fe343..45ee333 100644 --- a/semester4/ps/ps-jh/parts/03_expected-value/06_covariance.tex +++ b/semester4/ps/ps-jh/parts/03_expected-value/06_covariance.tex @@ -16,7 +16,8 @@ \item Wenn $\cov(\cX, \cY) < 0$, dann: $\cX$, $\cY$ \bi{negativ korreliert} \end{itemize} -\shortexample $\cX, \cY$ unkorreliert $\centernot\implies \cX, \cY$ unabhängig +\shortexample $\cX, \cY$ unkorreliert $\centernot\implies \cX, \cY$ unabhängig\\ +aber $\cX, \cY$ unabh. $\implies \cX, \cY$ unkorreliert \shortremark Eigenschaften der Kovarianz (alle $a, \ldots \in \R$): \begin{itemize} 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 5e44cd5..7c31e1b 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 @@ -1,4 +1,9 @@ \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 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 7091bcb..4b53bb6 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 @@ -36,13 +36,14 @@ Meistens sind $\cX_k$ i.i.d. unter $\P_\vartheta$, dann $L$ produkt, also besser When complete, there will be estimators for (pretty much) every covered distribution } -\shortexample[How To] 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) - \item Nullstelle davon berechnen (für Max (via e.g. $f'' < 0$)) - \item ML-Schätzer ist dann Nullstelle von Ableitung (mit $x_i$ wieder durch $\cX_i$ ersetzt) -\end{enumerate} +\coloredbox{gray}{How To: 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) + \item Nullstelle davon (für Max (via e.g. $f'' < 0$)) + \item ML-Schätzer ist dann Nullstelle von Ableitung (mit $x_i$ wieder durch $\cX_i$ ersetzt) + \end{enumerate} +} \shortexample[ML-Schätzer von Verteilungen] 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 105e701..4d7f97c 100644 --- a/semester4/ps/ps-jh/parts/07_tests/02_examples.tex +++ b/semester4/ps/ps-jh/parts/07_tests/02_examples.tex @@ -1,16 +1,31 @@ \subsection{Beispiele} -% TODO: More on Teststatistik -\shortexample[Gauss-Test] $\cX_i$ i.i.d. $\sim \cN(\vartheta, \sigma^2)$ unter $\P_\vartheta$ mit bekanntem $\sigma^2$. -Die zu testende Hypothese ist $H_0 : \vartheta = \vartheta_0$, mögliche alternative Hypothesen $H_A$ sind -$\vartheta > \vartheta_0$, $\vartheta < \vartheta_0$ (einseitig) oder $\vartheta \neq \vartheta_0$ (zweiseitig). -Teststatistik ist immer: +\coloredbox{red}{REWRITE}{ + This section is undergoing a rewrite, to be more on-point. The previous section may also see some changes as a result. +} +\coloredbox{gray}{How-To: Tests}{ + Was zu tun / erwähnen ist jeweils: + \begin{enumerate} + \item Modell: Verteilung der Stichproben $\cX_k$ unter $\P_\vartheta$, mit $\vartheta =$ gesuchte Werte. + Für $\cX_k \sim \cN(\mu, \sigma^2)$, ist $\vartheta = (\mu, \sigma^2)$, wenn beide unbekannt. + \item Hypothesen: $H_0: \vartheta = \vartheta_0 =$ Vergleichswert, $H_A: \vartheta < \vartheta_0$ (oder grösser, $\neq$, etc). \hl{Umgekehrt, s. links} + \item Teststatistik: Basierend auf Verteilung von $\cX_k$ und Bekanntheit von Parametern + \item Verteilung von $T$: Oft $t_{n - 1}$, siehe \ref{sec:estimators-properties} unten + \item Verwerfungsbereich: Nach Anleitung unten + \item Wert der $T$: $T(\omega)$ berechnen, $\omega$ ist beobachteter Wert (bspw. Durchschnitt $\overline{\cX}_n$) + \item Testentscheid: Basierend auf Resultat von vorherigem und $K$ entscheiden. + \end{enumerate} +} +\bi{Verwerfungsbereich} $K_l = (c_>, \8)$ (links), $K_r = (-\8, c_<)$ (rechts), $K_b = (-\8, c_{\neq}) \cup (c_{\neq}, \8)$ (beidseitig) + +\shortexample[Gauss-Test] Voraussetz.: +\bi{(1)} $\cX_i$ i.i.d. $\sim \cN(\vartheta, \sigma^2)$ unter $\P_\vartheta$ +\bi{(2)} $\sigma^2$ bekannt. +\textbf{Teststatistik}: \[ T = \frac{\overline{\cX}_n - \vartheta_0}{\frac{\sigma}{\sqrt{n}}} \sim \cN(0, 1) \qquad \text{unter } \P_{\vartheta_0} \] -$K$: $(c_>, \8)$, bzw. $(-\8, c_<)$ für einseitig, $(-\8, -c_{\neq}) \cup$\\ -$(c_{\neq}, \8)$ für zweiseitig (verwerfen $H_0$ falls $|T| > c_{\neq}$). - -\bi{Bestimmung $c$s}: Mit Verteilung von $T$. Bsp: $\alpha = \P_{\vartheta_0}[T > c_>] = 1 - \Phi(c_>)$, folglich $c_> = \Phi^{-1}(1 - \alpha) =: z_{1 - \alpha}$ +\bi{Bestimmung $c$s}: $z_{1 - \alpha} = \Phi^{-1}(1 - \alpha)$.\\ +$c_> = z_{1 - \alpha}; c_< = -z_{1 - \alpha}; c_{\neq} = z_{1 - 0.5 \alpha}$ \shortexample[t-Test] $\cX_i$ i.i.d. $\sim \cN(\mu, \sigma^2)$ unter $\P_{\overrightarrow{\vartheta}}$ @@ -19,8 +34,9 @@ Testen wieder $H_0 : \mu = \mu_0$. Teststatistik ist hier (see \ref{sec:estimato \[ T = \frac{\overline{X}_n - \mu_0}{\frac{S}{\sqrt{n}}} \qquad S^2 = \frac{1}{n - 1} \sum_{k = 1}^{n} (\cX_k - \overline{\cX}_n)^2 \] -$K$: $c_> = t_{n - 1, 1 - \alpha}$; $c_< = t_{n - 1, \alpha} = -c_>$; $c_{\neq} = t_{n - 1, 1 - 0.5 \alpha}$, -mit $t_{m, \gamma}$ das $\gamma$-Quantil, mit $\P[\cX \leq t_{m, \gamma}] = \gamma$ für $\cX \sim t_m$ +$K$: $c_> = t_{n - 1, 1 - \alpha}$; $c_< = t_{n - 1, \alpha} = -c_>$; $c_{\neq} = t_{n - 1, 1 - 0.5 \alpha}$,\\ +mit $t_{m, \gamma}$ das $\gamma$-Quantil von $t_m$-Verteilung, welches Wert ist, so dass $\P[\cX \leq t_{m, \gamma}] = \gamma$ für $\cX \sim t_m$. +Wert ist tabelliert! \shortremark[Zweistichproben-Tests] $\cX_k$ (i.i.d.) und jeweils unabhängig von allen $\cY_l$ diff --git a/semester4/ps/ps-jh/parts/07_tests/03_p-value.tex b/semester4/ps/ps-jh/parts/07_tests/03_p-value.tex index b7dccc5..c5a4ffe 100644 --- a/semester4/ps/ps-jh/parts/07_tests/03_p-value.tex +++ b/semester4/ps/ps-jh/parts/07_tests/03_p-value.tex @@ -1,3 +1,4 @@ +\newpage \subsection{p-Wert} Für Stichprobe $\cX_i$ ($n$ el.) testen $H_0 : \vartheta = \vartheta_0$ gegen eine $H_A : \vartheta \in \Theta_A$. diff --git a/semester4/ps/ps-jh/parts/09_combinatorics/03_binomial-theorem.tex b/semester4/ps/ps-jh/parts/09_combinatorics/03_binomial-theorem.tex new file mode 100644 index 0000000..5849522 --- /dev/null +++ b/semester4/ps/ps-jh/parts/09_combinatorics/03_binomial-theorem.tex @@ -0,0 +1,4 @@ +\subsection{Binomialsatz} +\[ + (x + y)^n = \sum_{i = 0}^{n} {n \choose i} x^i y^{n - i} = \sum_{i = 0}^{n} {n \choose i} x^{n - i} y^i +\] diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index fbe238bd0f984fa76e8aaf9a7d5f6d83b835407a..78cf66eb4116565ee4cc07675dc1b49fbdd3c4c9 100644 GIT binary patch delta 107640 zcmV(~K+nIKyhM`EM6g{D0XMT=5V`?>lmv%V^B->3t8#FbF%>MUT`npJ7pjCJ=R{>N 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