From 4d88cd33669c1c8f3569afea3f84e8b7c09bd3d4 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Tue, 16 Jun 2026 12:05:06 +0200 Subject: [PATCH] [PS] First update complete --- .../05_cont-distributions/02_cauchy.tex | 1 + .../parts/06_estimators/02_max-likelihood.tex | 32 ++++++++++------- .../parts/06_estimators/03_properties.tex | 8 +++-- .../ps/ps-jh/parts/07_tests/00_basics.tex | 33 ++++++++++++------ .../ps-jh/parts/07_tests/01_construction.tex | 13 ++++--- .../ps/ps-jh/parts/07_tests/02_examples.tex | 13 +++---- .../ps/ps-jh/parts/07_tests/03_p-value.tex | 4 +-- .../08_confidence-intervals/00_regions.tex | 4 ++- .../parts/10_tips-and-tricks/01_various.tex | 4 +++ .../probability-and-statistics-cheatsheet.pdf | Bin 1111606 -> 1115860 bytes 10 files changed, 73 insertions(+), 39 deletions(-) diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/02_cauchy.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/02_cauchy.tex index 58a2b16..a134eaf 100644 --- a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/02_cauchy.tex +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/02_cauchy.tex @@ -1,4 +1,5 @@ \subsubsection{Cauchy-Verteilung} +\label{sec:cauchy-dist} \shortdefinition $\cX \sim \text{Cauchy}(x_0, \gamma)$, falls $\displaystyle f_\cX(x) = \frac{1}{\pi} \frac{\gamma}{\gamma^2 + (x - x_0)^2}$ \shortremark[Verteilungsfunk.] $\displaystyle F_\cX(x) = \frac{1}{2} + \frac{1}{\pi} \arctan \left( \frac{x - x_0}{\gamma} \right)$ 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 e25ba41..2233db5 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 @@ -1,10 +1,12 @@ \subsection{Maximum-Likelihood-Method} +{\scriptsize Funktion $p_\cX(x_1, \ldots; \vartheta) = \P_\vartheta[\cX_1 = x_1, \ldots]$ (gleich wie normale Gew.-F.)} + In Modell $\P_\vartheta$ sind $\overrightarrow{\cX} = (\cX_1, \ldots, \cX_n)$ entweder \begin{itemize} \item \bi{diskret} (gem. Gew. $p_{\overrightarrow{\cX}} (x_1, \ldots, x_n; \vartheta)$) \item \bi{stetig} (gem. Dichte $f_{\overrightarrow{\cX}} (x_1, \ldots, x_n; \vartheta)$) \end{itemize} -Da oft $X_k$ i.i.d mit ind. Gew. $p_\cX(x; \vartheta)$ bzw. Dichte $f_\cX(x; \vartheta)$, also (mit $g$ ersetzt durch $p$ oder $f$) +Da oft $X_k$ i.i.d mit individueller Gew. $p_\cX(x; \vartheta)$ bzw. Dichte $f_\cX(x; \vartheta)$ sind, also (mit $g$ ersetzt durch $p$ oder $f$) \[ g_{\overrightarrow{\cX}}(x_1, \ldots, x_n; \vartheta) = \prod_{k = 1}^n g_\cX(x_k; \vartheta) \] @@ -13,36 +15,41 @@ Da oft $X_k$ i.i.d mit ind. Gew. $p_\cX(x; \vartheta)$ bzw. Dichte $f_\cX(x; \va \[ L(x_1, \ldots, x_n; \vartheta) = \begin{cases} p_{\overrightarrow{\cX}}(x_1, \ldots, x_n; \vartheta) & \text{im diskreten Fall} \\ - p_{\overrightarrow{\cX}}(x_1, \ldots, x_n; \vartheta) & \text{im stetigen Fall} + f_{\overrightarrow{\cX}}(x_1, \ldots, x_n; \vartheta) & \text{im stetigen Fall} \end{cases} \] \bi{log-Likelihood-Funktion} $\log(L(x_1, \ldots, x_n; \vartheta))$. Ist im i.i.d-Fall eine Summe. -\newpage +\inlineintuition Likelihood-F. gibt W., dass in $\P_\vartheta$ die Stichprobe $\cX_1, \ldots, \cX_n$ genau Werte $x_1, \ldots x_n$ liefert. + \shortdefinition[ML-Schätzer] $T_{ML}$ für $\vartheta$ maximiert die Funktion $\vartheta \mapsto L(\cX_1, \ldots, \cX_n; \vartheta)$ für alle $\vartheta$, also \[ T_{ML} = t_{TM}(\cX_1, \ldots, \cX_n) \in \underset{\vartheta \in \Theta}{\text{argmax}}\; L(\cX_1, \ldots, \cX_n; \vartheta) \] +{\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). \shortremark -Einfacher: Statt maximieren, Nullstellen von Ableitung nach $\vartheta$. +Einfacher: Statt max., Nullstellen von Abl. nach $\vartheta$. + +\shortremark Suchen oft optimales $L(x_1, \ldots, x_n; \vartheta)$, +da es einfacher ist, ersetzen dann $x_1, \ldots$ durch $\cX_1, \ldots$ % TODO: Maybe remark from slide 356 (= p33 in 7) -\shortexample \bi{Verteilungen}\\ +\shortexample \bi{Verteilungen} (Dimension $\vartheta$ gleich dim. param der V.)\\ \highlight{Bernoulli} $\cX_i \sim \text{Ber}(p)$ i.i.d, hier $\vartheta = p$. Dabei: -$p_\cX(x; \vartheta) = \P_\vartheta[\cX = x] = \vartheta^x (1 - \vartheta)^{1 - x}$ mit $x \in \{0, 1\}$. LH-Func: +$p_\cX(x; \vartheta) = \P_\vartheta[\cX = x] = \vartheta^x (1 - \vartheta)^{1 - x}$ mit $x \in \{0, 1\}$, $S = \sum_{k = 1}^{n}x_k$ \[ - L(x_1, \ldots, x_n; \vartheta) = \vartheta^{\sum_{k = 1}^{n} x_k} (1 - \vartheta)^{n - \sum_{k = 1}^{n} x_k} + L(x_1, \ldots, x_n; \vartheta) = \vartheta^{S} (1 - \vartheta)^{n - S} \] -Log LH-Func: $\log(\vartheta) \sum_{k = 1}^{n} x_k + \log(1 - \vartheta) \left( n - \sum_{k = 1}^{n} x_k \right)$\\ +Log LH-Func: $\log(\vartheta) \cdot S + \log(1 - \vartheta) \left( n - S \right)$\\ \bi{ML-Schätzer}: $T = \frac{1}{n} \sum_{k = 1}^{n} \cX_k = \overline{\cX}_n$ \fbox{\bi{Normalverteilung}} $\cX_i \sim \cN(\mu, \sigma^2)$ i.i.d. Dabei: \[ f_\cX(x; \vartheta) = \frac{1}{\sqrt{2\pi v}} e^{-\frac{(x - \mu)^2}{2v}} \] -Weil i.i.d: +Weil i.i.d (und $\vartheta = (\mu, \sigma^2)$): \[ L(x_1, \ldots, x_n; \vartheta) = \prod_{k = 1}^n f_\cX(x_k; \vartheta) \] @@ -51,7 +58,7 @@ und somit: \log(L) = -\frac{1}{2} n(\log(2 \pi) + \log(v)) - \sum_{k = 1}^{n} \frac{(x_k - \mu)^2}{2v} \] -Der Schätzer ist $T = (T_1, T_2)$ (\bi{Momentschätzer}): +Der \bi{Schätzer} ist $T = (T_1, T_2)$ (\textbf{Momentschätzer}): \[ T_1 = \overline{\cX}_n \quad T_2 = \frac{1}{n} \sum_{k = 1}^{n} \cX_k^2 - (\overline{\cX}_n)^2 @@ -62,9 +69,10 @@ und folglich: \[ \E_\vartheta[(\overline{\cX}_n)^2] = \frac{1}{n} \E_\vartheta[\cX^2] + \frac{n^2 - n}{n^2} (\E_\vartheta[\cX])^2 \] -Erwartungstreuer Schätzer für $(\E_\vartheta[\cX], \V_\vartheta[\cX])$: +\bi{Erwartungstreuer Schätzer} für $(\E_\vartheta[\cX], \V_\vartheta[\cX])$: \[ T_1' = T_1 \quad T_2' = \frac{n}{n - 1} T_2 \] -{\scriptsize $T_2'$ ist (korrigierte) empirische (Stichproben)varianz \textit{((un)biased sample variance)}} +{\scriptsize $T_2'$ ist (korrigierte) empirische (Stichproben)varianz \textit{((un)biased sample variance)}. +Bei höherdimensionalen Parametern suchen wir den zugehörigen Nullvektor.} diff --git a/semester4/ps/ps-jh/parts/06_estimators/03_properties.tex b/semester4/ps/ps-jh/parts/06_estimators/03_properties.tex index 6da4fa1..0b98955 100644 --- a/semester4/ps/ps-jh/parts/06_estimators/03_properties.tex +++ b/semester4/ps/ps-jh/parts/06_estimators/03_properties.tex @@ -1,6 +1,8 @@ +\newpage \subsection{Verteilungsaussagen} -Approx. von Verteilung von Schätzer unter $\P_\vartheta$ (wenn $T$ Summe mit $\cY_k$ im $\P_\vartheta$). -Solche sind approx. Normalverteilt unter $\P_\vartheta$, mit Parametern $\mu = n \E_\vartheta[\cY_k]$ und $\sigma^2 = n \V_\vartheta[\cY_k]$. +\label{sec:estimators-properties} +Approx. V. von Schätzer unter $\P_\vartheta$ (wenn $T = \sum_{k = 1}^{n} \cY_k$, mit $\cY_k$ i.i.d in $\P_\vartheta$). +Dabei $T \sim \cN(n \E_\vartheta[\cY_k], n \V_\vartheta[\cY_k])$ unter $\P_\vartheta$ \shortdefinition[$\chi^2$-Verteilung] $\cX$ ist $\chi^2$-verteilt mit $m$ Freiheitsgraden, falls Dichte für $x \geq 0$ (wir schreiben $\cX \sim \chi^2_m$): \[ @@ -21,7 +23,7 @@ $\chi^2_2 = \text{Exp}(\frac{1}{2})$ \] \shortremark $\cX \sim \cN(0, 1)$, $\cY \sim \chi^2_m$ unabh., dann: $\frac{\cX}{\sqrt{\frac{1}{m} \cY}} \sim t_m$ -Mit $m = 1$ Cauchy-V. mit $m \rightarrow \8$ asympt. $\cN(0, 1)$. $t_m$ symm. um $0$ wie $\cN(0, 1)$, aber \bi{langschwänziger} +Mit $m = 1$ Cauchy-V. mit $m \rightarrow \8$ asympt. $\cN(0, 1)$. $t_m$ symm. um $0$ wie $\cN(0, 1)$, aber \bi{langschwänziger} (see \ref{sec:cauchy-dist}) \shorttheorem Für $\cX_k \sim \cN(\mu, \sigma^2)$ i.i.d. und \[ diff --git a/semester4/ps/ps-jh/parts/07_tests/00_basics.tex b/semester4/ps/ps-jh/parts/07_tests/00_basics.tex index a909667..87655f0 100644 --- a/semester4/ps/ps-jh/parts/07_tests/00_basics.tex +++ b/semester4/ps/ps-jh/parts/07_tests/00_basics.tex @@ -1,6 +1,6 @@ \subsection{Grundbegriffe} Wiederum Stichprobe $\cX_1, \ldots, \cX_n$, mögliche Modelle durch Familie $(\P_\vartheta)_{\theta \in \Theta}$ beschrieben und $\vartheta$ ein-/mehrdimensional. -Wir haben \textit{Vermutung} wo in $\Theta$ richtiges $\vartheta$ liegt. +Wir haben \bi{Vermutung} wo in $\Theta$ richtiges $\vartheta$ liegt. Wir entscheiden zwischen folgenden, mit $\Theta_0 \cap \Theta_A = \emptyset$.: \begin{itemize} @@ -8,25 +8,38 @@ Wir entscheiden zwischen folgenden, mit $\Theta_0 \cap \Theta_A = \emptyset$.: \item \bi{Alternative} $\Theta_A \subseteq \Theta$ (oft: $H_A: \vartheta \in \Theta_A$) \end{itemize} Falls keine Alternative spezifiziert, so gilt $\Theta_A = \Theta_0^C = \Theta \backslash \Theta_0$. -Sie heissen \bi{Einfach}, falls $\Theta_0 = \{ \vartheta_0 \}$, resp. $\Theta_A = \{ \vartheta_A \}$, sonst \bi{Zusammengesetzt} +Sie heissen \bi{Einfach}, falls $\Theta_0 = \{ \vartheta_0 \}$, resp. $\Theta_A = \{ \vartheta_A \}$, sonst \bi{Zusammengesetzt} (e.g. $\Theta_A = (0.5, 1]$) \shortdefinition[Test] ist ein Paar $(T, K)$ mit \begin{itemize} \item \bi{Teststatistik} $T = t(\cX_1, \ldots, \cX_n)$ eine Zufallsvariable mit messbarer Funktion $t: \R^n \rightarrow \R$ - \item \bi{Verwerfungsbereich} $K \subseteq \R$ + \item \bi{Verwerfungsbereich} $K \subseteq \R$ (auch \textit{kritischer Bereich}) \end{itemize} \shade{gray}{Entscheidungsregel} $H_0$ wird \textit{verworfen}, falls $T(\omega) \in K$, sonst \textit{angenommen}, bzw. nicht verworfen -\shade{gray}{Fehlerarten} -\bi{Fehler 1. Art} (Hypothese abgelehnt, ist aber richtig), falls $\vartheta \in \Theta_0$, aber $T \in K$, also -$\P_\vartheta[T \in K]$ für $\vartheta \in \Theta_0$ ist W. für F. 1. Art\\ -\bi{Fehler 2. Art} (Hypothese angenommen, ist aber falsch), falls $\vartheta \in \Theta_A$, aber $T \notin K$, also -$\P_\vartheta[T \notin K]$ für $\vartheta \in \Theta_A$ ist W. für F. 2. Art +\shortdefinition[Fehler 1. Art] (Hypothese abgelehnt, ist aber richtig), falls $\vartheta \in \Theta_0$, aber $T \in K$, also +$\P_\vartheta[T \in K]$ für $\vartheta \in \Theta_0$ ist W. für F. 1. Art + +\shortdefinition[Fehler 2. Art] (Hypothese angenommen, aber falsch), falls $\vartheta \in \Theta_A$, aber $T \notin K$, also +$\P_\vartheta[T \notin K] = 1 - \P_\vartheta[T \in K]$ für $\vartheta \in \Theta_A$ ist W. für F. 2. Art \shortremark Entscheidung hängt dabei von \bi{Realisierung} $\omega$ ab. Da $T$ eine Z.V. ist, können $\P_\vartheta[T \in K]$ in jedem $\P_\vartheta$ betrachten -\bi{Signifikanzniveau} $\alpha \in (0, 1)$. Ziel: $\underset{\vartheta \in \Theta_0}{\sup} \P_\vartheta[T \in K] \leq \alpha$ +\shortdefinition[Signifikanzniveau] $\alpha \in (0, 1)$. Ziel: $\underset{\vartheta \in \Theta_0}{\sup} \P_\vartheta[T \in K] \leq \alpha$ (Fehler 1. Art so klein wie möglich). Typisch: $\alpha = 0.05$ -\bi{Macht} $\beta: \Theta_A \rightarrow [0, 1]$, mit $\beta(\vartheta) = \P_\vartheta[T \in K]$ soll möglichst gross werden (Fehler 2. Art so klein wie möglich) +\shortdefinition[Macht] $\beta: \Theta_A \rightarrow [0, 1]$, mit $\beta(\vartheta) = \P_\vartheta[T \in K]$ soll möglichst gross werden (Fehler 2. Art so klein wie möglich) + +\shortremark Seriöser Test nutzt immer \highlight{negation der Aussage} als Hypothese (da Hypothese eher angenommen als abgelehnt wird). +Entscheidung ist nicht \textit{Beweis}, sondern \textit{Interpr.} + +\shortexample[Tea Tasting Lady] Lady behauptet, sie kann schmecken, ob zuerst Tee oder Milch eingegossen wurde. +Hypothese: ist Raten ($H_0: \vartheta = 0.5$, $H_A: \vartheta > 0.5$). Unter $\P_\vartheta$, $\cX_i \sim \text{Ber}(\vartheta)$ i.i.d und +$S_n = \sum_{k = 1}^{n} \cX_k \sim \text{Bin}(n, \vartheta)$. Grosser Wert von $S_n$ eher $H_A$ als $H_0$. + +\bi{Test}: $T = S_n$, $K = (c, \8)$. Für Bestimmung von $\alpha$ brauchen wir $\P_\vartheta[S_n > c]$ für $\vartheta = 0.5$. +Berechnen Verteilung von $T$ (nicht immer möglich, wenn nicht, nutzen Approximation). Für $\alpha$ muss $\P_\vartheta[S_n > c] \leq \alpha$ sein. +Wählen $c$ so, dass $\alpha$ ungefähr 5\% ist. + +\bi{Macht} ersichtlich durch Berechnung von $\beta(0.6)$ hier. Fehler 2. Art ist gross ($1 - \beta(\vartheta)$ gross) diff --git a/semester4/ps/ps-jh/parts/07_tests/01_construction.tex b/semester4/ps/ps-jh/parts/07_tests/01_construction.tex index 3fe1c13..b82d416 100644 --- a/semester4/ps/ps-jh/parts/07_tests/01_construction.tex +++ b/semester4/ps/ps-jh/parts/07_tests/01_construction.tex @@ -1,3 +1,4 @@ +\newpage \subsection{Konstruktion von Tests} % Slide 401 (P24) $\cX_i$ diskret oder gemeinsam stetig unter $\P_{\vartheta_0}$ und $\P_{\vartheta_A}$. Sei $L(x_1, \ldots, x_n; \vartheta)$ die Likelihood-Funktion @@ -6,20 +7,22 @@ $\cX_i$ diskret oder gemeinsam stetig unter $\P_{\vartheta_0}$ und $\P_{\varthet $R(x_1, \ldots, x_n; \vartheta_0, \vartheta_A) = \frac{L(x_1, \ldots, x_n; \vartheta_A)}{L(x_1, \ldots, x_n; \vartheta_0)}$.\\ $R(x_1, \ldots; \vartheta_0, \vartheta_A) = +\8$ wenn $L(x_1, \ldots; \vartheta_0) = 0$. Wenn $R$ gross ist, sind Beobachtungen in $\P_{\vartheta_A}$ deutlich wahrscheinlicher als in $\P_{\vartheta_0}$. +\hl{Oft verwendeter Test} -% TODO: More places with highlighting -\shortdefinition[Likelihood-Quotienten-Test] Sei $c \geq 0$. Der LQT mit param $c$ ist Test $(T, K)$ mit $T = R(\cX_1, \ldots, \cX_n; \vartheta_0, \vartheta_A)$ +\shortdefinition[Likelihood-Quotienten-Test] Sei $c \geq 0$. Der \bi{LQT} mit param $c$ ist Test $(T, K)$ mit $T = R(\cX_1, \ldots, \cX_n; \vartheta_0, \vartheta_A)$ und $K = (c, \8)$. $H_0$ wird verworfen wenn $R$ gross wird. + \shorttheorem[Neyman-Pearson] Seien $\Theta_0, \Theta_A$ \textit{einfach} und $(T, K)$ ein LQT mit $c$ und sig.-niv. $\alpha^* := \P_{\vartheta_0}[T \in K]$. Anderer Test $(T', K')$ mit $\P_{\vartheta_0}[T' \in K'] =: \alpha \leq \alpha^*$, so gilt auch $\P_{\vartheta_A}[T' \in K'] \leq \P_{\vartheta_A}[T \in K]$. \inlineintuition Jeder Test mit kleinerem Signifikanzniveau hat auch kleinere Macht \shortdefinition[Verallgemeinerte LQ] ist gegeben durch -\[ - R(x_1, \ldots, x_n) = \frac{\sup_{\vartheta \in \Theta_A} L(x_1, \ldots, x_n; \vartheta)}{\sup_{\vartheta \in \Theta_0} L(x_1, \ldots, x_n; \vartheta)} -\] +\begin{align*} + R(x_1, \ldots, x_n) = \frac{\sup_{\vartheta \in \Theta_A} L(x_1, \ldots, x_n; \vartheta)}{\sup_{\vartheta \in \Theta_0} L(x_1, \ldots, x_n; \vartheta)} \\ + \tilde{R}(x_1, \ldots, x_n) = \frac{\sup_{\vartheta \in \Theta_A \cup \Theta_0} L(x_1, \ldots, x_n; \vartheta)}{\sup_{\vartheta \in \Theta_0} L(x_1, \ldots, x_n; \vartheta)} +\end{align*} % 407 % TODO: Likelihood-Functions and quotients of common distributions 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 d013dd2..dba82e1 100644 --- a/semester4/ps/ps-jh/parts/07_tests/02_examples.tex +++ b/semester4/ps/ps-jh/parts/07_tests/02_examples.tex @@ -6,18 +6,19 @@ Teststatistik ist immer: \[ T = \frac{\overline{\cX}_n - \vartheta_0}{\frac{\sigma}{\sqrt{n}}} \sim \cN(0, 1) \qquad \text{unter } \P_{\vartheta_0} \] -Kritischer Bereich $K$: $(c_>, \8)$, bzw. $(-\8, c_<)$ für einseitig, $(-\8, -c_{\neq}) \cup (c_{\neq}, \8)$ für zweiseitig (verw. falls $|T| > c_{\neq}$). +$K$: $(c_>, \8)$, 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 $c_> = \Phi^{-1}(1 - \alpha) =: z_{1 - \alpha}$ \shortexample[t-Test] $\cX_i$ i.i.d. $\sim \cN(\mu, \sigma^2)$ unter $\P_{\overrightarrow{\vartheta}}$ mit $\overrightarrow{\vartheta} := (\mu, \sigma^2)$ (und insbesondere $\sigma^2)$ unbekannt. -Testen wieder $H_0 : \mu = \mu_0$. Teststatistik ist hier: +Testen wieder $H_0 : \mu = \mu_0$. Teststatistik ist hier (see \ref{sec:estimators-properties}, $t_m$-V.): \[ T = \frac{\overline{X}_n - \mu_0}{\frac{S}{\sqrt{n}}} \qquad S^2 = \frac{1}{n - 1} \sum_{k = 1}^{n} (\cX_k - \overline{\cX}_n)^2 \] -Kritischer Bereich $K$: $c_> = t_{n - 1, 1 - \alpha}$, $c_< = t_{n - 1, \alpha} = -c_>$ und $c_{\neq} = t_{n - 1, 1 - 0.5 \alpha}$, +$K$: $c_> = t_{n - 1, 1 - \alpha}$; $c_< = t_{n - 1, \alpha} = -c_>$; $c_{\neq} = t_{n - 1, 1 - 0.5 \alpha}$, mit $t_{m, \gamma}$ das $\gamma$-Quantil, mit $\P[\cX \leq t_{m, \gamma}] = \gamma$ für $\cX \sim t_m$ @@ -30,15 +31,15 @@ Differenzen $\cZ_k = \cX_k - \cY_k$ unter $\P_\vartheta$ i.i.d. $\sim \cN(\mu_\c Wenn nicht unabhängig: $\sim \cN(\mu_\cX - \mu_\cY, 2(1 - \rho) \sigma^2)$ mit $\rho \in (-1, 1)$ Korrelation und $\rho = 0$ Unabhängigkeit. -\shortexample[Ungepaart] $\cX_i$, $\cY_j$ wie zuvor ($n$ und $m$ je). +\shortexample[Ungepaart] $\cX_i$, $\cY_j$ wie zuvor ($n$ und $m$ je). -$z$-Test (falls $\sigma^2$ bekannt): +\bi{$z$-Test} (falls $\sigma^2$ bekannt): \[ T = \frac{(\overline{\cX}_n - \overline{\cY}_m) - (\mu_\cX - \mu_\cY)}{\sigma \sqrt{\frac{1}{n} + \frac{1}{m}}} \sim \cN(0, 1) \] $\mu_\cX - \mu_\cY$ muss aus $H_0$ bekannt sein. Der kritische Bereich ist wieder Quantile der $\cN(0, 1)$-Verteilung. -$t$-Test (falls $\sigma^2$ unbekannt): +\bi{$t$-Test} (falls $\sigma^2$ unbekannt): \[ T = \frac{(\overline{\cX}_n - \overline{\cY}_m) - (\mu_\cX - \mu_\cY)}{S \sqrt{\frac{1}{n} + \frac{1}{m}}} \sim t_{n + m - 2} \] 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 7ee859a..b7dccc5 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 @@ -1,7 +1,7 @@ \subsection{p-Wert} Für Stichprobe $\cX_i$ ($n$ el.) testen $H_0 : \vartheta = \vartheta_0$ gegen eine $H_A : \vartheta \in \Theta_A$. -$p$-Wert ist W unter $H_0$ einen mind. so extremen Wert für $T$ zu bekommen wie $T(\omega) = t(x_1, \ldots, x_n)$ (extrem bezieht sich auf $H_A$). +$p$-Wert ist W. unter $H_0$ einen mind. so extremen Wert für $T$ zu bekommen wie $T(\omega) = t(x_1, \ldots, x_n)$ (extrem bezieht sich auf $H_A$). Für bspw. $H_A: \vartheta > \vartheta_0$, dann ist $p\text{-Wert}(\omega) = \P_{\vartheta_0}[T > t_0] \Big|_{t_0 = t(\cX_1(\omega), \ldots)}$ \shortdefinition[Geordnete Testsammlung] @@ -19,4 +19,4 @@ Er ist eine Zufallsvariable, hängt direkt von $x_i$ ab. Wiederholen des Tests g Wenn $T$ stetig und $K_t = (t, \8)$, dann gleichverteilt auf $[0, 1]$. \inlineintuition Besagt, welche Tests $H_0$ ablehnen würden (alle Tests mit $\alpha > p$ lehnen $H_0$ ab). $H_A$ spielt \textit{keine} Rolle. -Es heisst \textit{NICHT}, dass der $p$-Wert die W \textit{IST}, dass die Hypothese richtig ist. +Es heisst \textit{NICHT}, dass der $p$-Wert die W. \textit{IST}, dass die Hypothese richtig ist. diff --git a/semester4/ps/ps-jh/parts/08_confidence-intervals/00_regions.tex b/semester4/ps/ps-jh/parts/08_confidence-intervals/00_regions.tex index f501005..14b6cc1 100644 --- a/semester4/ps/ps-jh/parts/08_confidence-intervals/00_regions.tex +++ b/semester4/ps/ps-jh/parts/08_confidence-intervals/00_regions.tex @@ -1,5 +1,7 @@ \shortdefinition[Konfidenzbereich] ist für $\vartheta$ zu Daten $x_i$ eine Menge $C(x_1, \ldots, x_n) \subseteq \Theta$. -Mit gen. Z.V. ist $\tilde{C}(\cX_1, \ldots \cX_n)$ eine \bi{zuf. Teilm. v.} $\Theta$, Real. $\tilde{C}(\omega) = C(\cX_1(\omega), \ldots, \cX_n(\omega))$. +Mit genrierenden Z.V. ist $\tilde{C}(\cX_1, \ldots \cX_n)$ eine \bi{zufällige Teilmenge von} $\Theta$. + +\bi{Realisierung} $\tilde{C}(\omega) = C(\cX_1(\omega), \ldots, \cX_n(\omega))$. Für $\alpha \in [0, 1]$ ist $C$ \bi{Konfidenzbereich zum Niveau} $1 - \alpha$, falls \[ \P_\vartheta[C(\cX_1, \ldots, \cX_n) \ni \vartheta] \geq 1 - \alpha \quad \forall \vartheta \in \Theta 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 86f1fce..8eda31d 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 @@ -47,3 +47,7 @@ $\cosh(x) := \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}} : \R \hline 180° & $\pi$ & $0$ & $-1$ & $0$ \\ \end{tables} + + +\subsubsection{Reihen} +$\prod_{k = 1}^{n} \vartheta (1 - \vartheta)^{x_k - 1} = \vartheta^n (1 - \vartheta)^{S - n}$ mit $S = \sum_{k = 1}^{n} x_k$ 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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