From 3fef2e772bb43dca51e7f8f3f039579072c8f3ee Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Tue, 4 Aug 2026 15:38:02 +0200 Subject: [PATCH] [PS] Add tips and tricks for various concepts --- .../00_basics/03_properties-of-measures.tex | 2 ++ .../ps-jh/parts/00_basics/05_independence.tex | 2 +- .../01_random-variables/00_definition.tex | 9 ++++---- .../05_cont-distributions/03_normal.tex | 1 + .../04_joint-distribution/02_remarks.tex | 21 ++++++++++++++++-- .../ps/ps-jh/parts/07_tests/02_examples.tex | 10 ++++----- .../ps/ps-jh/parts/07_tests/03_p-value.tex | 13 +++++++++-- .../08_confidence-intervals/01_interval.tex | 16 +++++++------ .../probability-and-statistics-cheatsheet.pdf | Bin 1125880 -> 1130188 bytes .../probability-and-statistics-cheatsheet.tex | 2 +- 10 files changed, 54 insertions(+), 22 deletions(-) diff --git a/semester4/ps/ps-jh/parts/00_basics/03_properties-of-measures.tex b/semester4/ps/ps-jh/parts/00_basics/03_properties-of-measures.tex index 737ecd8..268e8a8 100644 --- a/semester4/ps/ps-jh/parts/00_basics/03_properties-of-measures.tex +++ b/semester4/ps/ps-jh/parts/00_basics/03_properties-of-measures.tex @@ -15,6 +15,8 @@ $\P\left[ \bigcup_{i = 1}^\infty A_i \right] \leq \sum_{i = 1}^{\infty} \P[A_i]$. Auch für end. n.-leere Ereignisse +Zudem: $\limni \P[\bigcup_{i = 1}^n A_i] = \P[\bigcup_{i = 1}^\8 A_i]$ + \newpage \stepLabelNumber{combined} diff --git a/semester4/ps/ps-jh/parts/00_basics/05_independence.tex b/semester4/ps/ps-jh/parts/00_basics/05_independence.tex index 8b93953..0f7f453 100644 --- a/semester4/ps/ps-jh/parts/00_basics/05_independence.tex +++ b/semester4/ps/ps-jh/parts/00_basics/05_independence.tex @@ -1,6 +1,6 @@ \subsection{Unabhängigkeit} \label{sec:basics-indep} -\bi{Siehe zudem} Abschnitt \ref{sec:rv-indep} +\bi{Siehe zudem} Abschnitt \ref{sec:rv-indep} (für Zufallsvariablen) \shortdefinition $A, B \in \cF$ \bi{unabh.} falls $\P[A \cap B] = \P[A] \cdot \P[B]$ diff --git a/semester4/ps/ps-jh/parts/01_random-variables/00_definition.tex b/semester4/ps/ps-jh/parts/01_random-variables/00_definition.tex index 2b73901..5a61a7f 100644 --- a/semester4/ps/ps-jh/parts/01_random-variables/00_definition.tex +++ b/semester4/ps/ps-jh/parts/01_random-variables/00_definition.tex @@ -1,12 +1,13 @@ % P22 \subsection{Abstrakte Definition} \shortdefinition[Zufallsvariable] kurz Z.V, ist $\cX : \Omega \rightarrow \R$, sodass $\forall a \in \R$ gilt: -$f = \{ \omega \in \Omega \divider \cX(\omega) \leq a \} \in \cF$ (notwendinge Bedingung für Wohldefiniertheit von $\P[f]$) +$f = \{ \omega \in \Omega \divider \cX(\omega) \leq a \} \in \cF$ (notwendige Bedingung für Wohldefiniertheit von $\P[f]$) \inlinenotation Ohne $\omega$: $\{ X \leq a \} = \{ \omega \in \Omega \divider X(\omega) \leq a \}$, etc \shortdefinition[Indikatorfunk.] $A \in \cF$, $\bm{1_A(\omega)} = \begin{cases} - 0 & \text{ wenn } \omega \notin A \\ - 1 & \text{ wenn } \omega \in A -\end{cases}$ + 0 & \text{ wenn } \omega \notin A \\ + 1 & \text{ wenn } \omega \in A + \end{cases}$ Das $A$ kann bspw $\cX \leq a$ sein. +Für $A \cap B$, $\E[\bm{1}_A \bm{1}_B] = 0$ weil $A$ und $B$ nie gleichzeitig eintreten diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/03_normal.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/03_normal.tex index d609f55..b663a8e 100644 --- a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/03_normal.tex +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/03_normal.tex @@ -1,4 +1,5 @@ \subsubsection{Normalverteilung} +\label{sec:dist-normal} \shortdefinition $\cX \sim \cN(\mu, \sigma^2)$ falls $f_\cX(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{\frac{1}{2} \left( \frac{x - \mu}{\sigma} \right)^2}$,\\ mit $\sigma$ Standardabweichung. Auch: Gauss'sche Verteilung 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 0883e78..3b3603f 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 @@ -11,6 +11,12 @@ Die Randdichten sind dann die Dichten von $\cX$, etc.; Für weitere Berechnungen verwendbar, wie bspw. $\E[\cX + \cY]$. +\howto{$cond$ auflösen}{ + In dem Integral sollte jeweils der äussere Integral der sein, für welchen wir volle Information haben. + Zudem, bspw., für $0 \leq x \leq y^2 \leq 1$, wissen wir zwar für $x$ und $y$ die Grenzen, wählen aber $x$ als inneren Integral, + weil $\int_0^{y^2}$ sich schöner auflöst +} + \howto{Gemeinsame Dichten finden}{ Nach Definition muss $f$ die Ableitung von $F$ sein. Oder umgekehrt, $\int f \dx \ldots$ sollte $F$ ergeben. Beispiel: @@ -20,9 +26,20 @@ Die Randdichten sind dann die Dichten von $\cX$, etc.; Für weitere Berechnungen \end{cases}$ } -\howto{Verteilungen finden}{ +\howto{Verteilungen / Dichten 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. + Schliesslich $F_Z$ aufstellen mit Bedingungen oder Indikatorvariablen. + + Zudem kann Verteilung der einzelnen Variablen (und v.a. Unabhängigkeit) nützlich werden. + + Falls Dichte, dann Verteilung ableiten. +} + +\howto{Multi-Variable Wahrscheinlichkeiten}{ + \bi{Für W. wie} $\P[\cX \leq \cY^3]$ übersetzen wir dies in Integral $\int_B f(x, y) \dx x \dx y$ mit $B = \{(x, y) \divider x \leq y^3 \}$, dann für $y \in [0, 1]$ gilt: + $\int_0^1 \int_{0}^{y^3} f(x, y) \dx x \dx y$, wobei hier die untere Grenze durch gegebenes $x \geq 0$ entsteht. + + \bi{Für W. im Exponent}, dies beschreibt typ. Repetitionen, also über die Anzahl und W. nachdenken. } 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 f53c766..3c0fa8e 100644 --- a/semester4/ps/ps-jh/parts/07_tests/02_examples.tex +++ b/semester4/ps/ps-jh/parts/07_tests/02_examples.tex @@ -1,7 +1,4 @@ \subsection{Beispiele} -\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. -} \howto{Tests}{ Was zu tun / erwähnen ist jeweils: \begin{enumerate} @@ -9,14 +6,17 @@ 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 + (bspw. mit Theoremen von \ref{sec:dist-normal}, wie Normalisierung, + und bswp. $\sum_{i = 1}^n \cX_i \sim \cN\left( \sum_{i = 1}^{n} \mu_i, \sum_{i = 1}^{n} \sigma_i^2 \right)$) + \item Verteilung von $T$: Oft $t_{n - 1}$, siehe \ref{sec:estimators-properties} unten, oder bei entsprechendem Test \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) +\bi{Verwerfungsbereich} $K_l = (c_>, \8)$ (links), $K_r = (-\8, c_<)$ (rechts), $K_b = (-\8, -c_{\neq}) \cup (c_{\neq}, \8)$ (beidseitig) +\newpage \shortexample[Gauss-Test] Voraussetz.: \bi{(1)} $\cX_i$ i.i.d. $\sim \cN(\vartheta, \sigma^2)$ unter $\P_\vartheta$ \bi{(2)} $\sigma^2$ bekannt. 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 c5a4ffe..9bb7682 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 @@ -2,8 +2,8 @@ \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 $H_A$). -Für bspw. $H_A: \vartheta > \vartheta_0$, dann ist $p\text{-Wert}(\omega) = \P_{\vartheta_0}[T > t_0] \Big|_{t_0 = t(\cX_1(\omega), \ldots)}$ +$p$-Wert ist W. unter $H_0$ einen mindestens so extremen Wert für $T$ zu bekommen wie $T(\omega) = t(x_1, \ldots, x_n)$, angenommen, dass die Nullhypothese korrekt ist. +% 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] $T$ Teststatistik, Familie von Tests $(T, K_t)_{t \geq 0}$ \bi{geordnet bezgl. $T$}, falls $\forall t$ gilt $K_t \subseteq \R$ und $\forall s \leq t$ gilt $K_s \supset K_t$. @@ -21,3 +21,12 @@ 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. + +\howto{Berechnung $p$-Wert}{ + \begin{itemize} + \item Für rechtsseitigen Test: $p_r = \P[T \geq t]$ + \item Für linksseitigen Test: $p_l = \P[T \leq t]$ + \item Für beidseitigen Test: $p = 2 \min\{ p_l, p_r \}$, falls Verteilung symmetrisch um $0$, dann $p = \P[|T| \geq |t|]$ + \end{itemize} + wobei für alle $t$ eine Realisierung der Teststatistik $T$ ist. +} diff --git a/semester4/ps/ps-jh/parts/08_confidence-intervals/01_interval.tex b/semester4/ps/ps-jh/parts/08_confidence-intervals/01_interval.tex index 9c5723b..dfb1873 100644 --- a/semester4/ps/ps-jh/parts/08_confidence-intervals/01_interval.tex +++ b/semester4/ps/ps-jh/parts/08_confidence-intervals/01_interval.tex @@ -1,33 +1,35 @@ \shortdefinition[Konfidenzintervall (K.I.)] mit Niveau $1 - \alpha$ ist Zufallsintervall $I = [A, B]$ s.d. $\forall \vartheta \in \Theta$ gilt: \[ - \P_\vartheta[[A, B] \ni \vartheta] = \P_\vartheta[A \leq \vartheta \leq B] \geq 1 - \alpha + \P_\vartheta[[A, B] \ni \vartheta] = \P_\vartheta[A \leq \vartheta \leq B] \geq 1 - \alpha \] mit $A, B$ Z.V. wie $A = a(\cX_1, \ldots, \cX_n)$, mit $a, b: \R^n \rightarrow \R$. Dabei ist $\vartheta$ deterministisch. (\hl{Berechnung}: s. \ref{sec:distribution-function}) \shortremark[Approx. K.I.] Nur wenn genaue Verteilungsaussagen zur Verfügung stehen, sind exakte K.I. möglich, -sonst mittels zentralem Grenzwertsatz approx. berechnen ($z = z_{1 - \frac{\alpha}{2}}$) +sonst mittels zentralem Grenzwertsatz approx. berechnen (\hl{$z = z_{1 - \frac{\alpha}{2}}$}) \[ - P_\vartheta[|S_n^*| \leq z] \approx 1 - \alpha + P_\vartheta[|S_n^*| \leq z] \approx 1 - \alpha \] Auflösen von untenstehendem nach $\vartheta$ ($S_n$ wie in \ref{sec:weak-law-of-large-numbers}) \[ - |S_n - n \vartheta| \leq z \sqrt{n \vartheta(1 - \vartheta)} \text{ bzw. } (S_n - n \vartheta)^2 \leq z^2 n \vartheta(1 - \vartheta) + |S_n - n \vartheta| \leq z \sqrt{n \vartheta(1 - \vartheta)} \text{ bzw. } (S_n - n \vartheta)^2 \leq z^2 n \vartheta(1 - \vartheta) \] Ist kompliziert, ersetzen der Ugl. durch Gl. (quad. Gl.) \[ - \hat{\vartheta}_\pm = \frac{2 S_n + z^2 \pm \sqrt{(2 S_n + z^2)^2 - 4S_n^2(1 + \frac{z^2}{n})}}{2n(1 + \frac{z^2}{n})} + \hat{\vartheta}_\pm = \frac{2 S_n + z^2 \pm \sqrt{(2 S_n + z^2)^2 - 4S_n^2(1 + \frac{z^2}{n})}}{2n(1 + \frac{z^2}{n})} \] Konfidenzintervall ist $[\hat{\vartheta}_-, \hat{\vartheta}_+]$. \shade{gray}{Alternative Methoden} (Annahme $\vartheta(1 - \vartheta) \approx \frac{1}{4}$): \[ - \left[\overline{S}_n - \frac{z}{2 \sqrt{n}}, \overline{S}_n + \frac{z}{2 \sqrt{n}}\right] + \left[\overline{S}_n - \frac{z}{2 \sqrt{n}}, \overline{S}_n + \frac{z}{2 \sqrt{n}}\right] \] (Ersetzen von $\vartheta$ durch $\overline{S}_n$): \[ - \left[ \overline{S}_n - \frac{z}{2 \sqrt{n}} \sqrt{\overline{S}_n (1 - \overline{S}_n)}, \overline{S}_n + \frac{z}{2 \sqrt{n}} \sqrt{\overline{S}_n (1 - \overline{S}_n)} \right] + \left[ \overline{S}_n - \frac{z}{2 \sqrt{n}} \sqrt{\overline{S}_n (1 - \overline{S}_n)}, \overline{S}_n + \frac{z}{2 \sqrt{n}} \sqrt{\overline{S}_n (1 - \overline{S}_n)} \right] \] Alle Methoden liefern unterschiedliche Resultate \bi{Normalverteilung}: $\hat{\vartheta} = \overline{\cX}_n \pm z \frac{\sigma}{\sqrt{n}}$, $I = [\hat{\vartheta}_-, \hat{\vartheta}_+]$ + +\shortdefinition $z_{1 - \alpha} = \Phi^{-1}(1 - \alpha)$. $\alpha$ ist Verwerfungsbereich, nicht ``Akzeptanzbereich'' diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index 25dfb467bb491dfbccae691e566caa43a465fed0..2394acd7e02bde24bd4f6b7c9ce82d8b202c85c8 100644 GIT binary patch delta 146078 zcmV(xKlOVY&e_dO19JiHz*RL4A4J#VBaR=n`W~D^(+F7OS zmAu-G;)fC0B1aky70%Gg>Lq`D&q3dSZlHU*XDBNnt1Of3M&o*LaIPP++4(G+efQwW z%LiXSqnynwv5KkL%eOPF6%mrn6lFy4*~_!p>p4AH&@9X5PaePc{tqusetG%Be}ktl zAN=hBg-Ws+ol#A)%*<3~h!SFUdimh>FWKx2-u*Dk2)AbT(UrWMnM@O-1^m9Ay?F4` zgRJaG^|MpwA|o0)Hlj*5CM&!l32ID0>||Gn6qtqr@oA6;33R z2I#t~pnh;Vd%z`8(!w&=51}fq2+J%t%|m#iq=d~~*X}jl%Mw>M(`be&)3o81z3?{$8i 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zkut5DRbYMIpz4Viw)y0j7a0|SGCsEX^Sob)f@~^fd<^nU*}h7r?UarAs91n-`)4ci zQ7`LEX3tzYSCVq?i1GtwJpPZGV)*}8A%6b$5dT1klQXWM;D757aa>kbRed$w{|}M^ B{mlRX diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex index dd7e312..329c1dc 100644 --- a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex +++ b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex @@ -88,7 +88,7 @@ % \input{parts/04_joint-distribution/} -\newsection +\newsectionNoPB \section{Das Gesetz der grossen Zahlen} \input{parts/05_limit-theorems/00_intro.tex} \input{parts/05_limit-theorems/01_weak-law-of-large-numbers.tex}