From d1db481fb86e865fe9a3cb677ac10bd12ecbfd0a Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Thu, 16 Apr 2026 12:00:45 +0200 Subject: [PATCH] [PS] Joint distributions done, start limit theorems / law of large numbers --- .../04_joint-distribution/01_continuous.tex | 53 +++++++++++++++++- .../parts/05_limit-theorems/00_intro.tex | 12 ++++ .../probability-and-statistics-cheatsheet.pdf | Bin 375516 -> 384798 bytes .../probability-and-statistics-cheatsheet.tex | 6 ++ 4 files changed, 68 insertions(+), 3 deletions(-) create mode 100644 semester4/ps/ps-jh/parts/05_limit-theorems/00_intro.tex diff --git a/semester4/ps/ps-jh/parts/04_joint-distribution/01_continuous.tex b/semester4/ps/ps-jh/parts/04_joint-distribution/01_continuous.tex index 371e209..ce6c239 100644 --- a/semester4/ps/ps-jh/parts/04_joint-distribution/01_continuous.tex +++ b/semester4/ps/ps-jh/parts/04_joint-distribution/01_continuous.tex @@ -14,11 +14,11 @@ Jeder Funk. $f$ die obiges erfüllt ein W-Raum $(\Omega, \cF, \P)$ und $n$-Z.V. \shortexample Gleichverteilungen: \begin{itemize} - \item \bi{Einheistquadrat} $f(x, y) = \begin{cases} + \item \bi{Einheitsquadrat} $f(x, y) = \begin{cases} 1 & (x, y) \in [0, 1]^2 \\ 0 & \text{sonst} \end{cases}$ - \item \bi{Einheistkreisscheibe} $D = \{ (x, y) \in \R^2 \divider x^2 + y^2 \leq 1 \}$ + \item \bi{Einheitskreisscheibe} $D = \{ (x, y) \in \R^2 \divider x^2 + y^2 \leq 1 \}$ $f(x, y) = \begin{cases} \frac{1}{\pi} & (x, y) \in D \\ 0 & \text{sonst} \\ @@ -30,7 +30,54 @@ Jeder Funk. $f$ die obiges erfüllt ein W-Raum $(\Omega, \cF, \P)$ und $n$-Z.V. \E[\varphi(\cX_1, \ldots)] \! = \! \int_{-\8}^{\8} \ldots \int_{-\8}^{\8} \varphi(x_1, \ldots) f(x_1, \ldots) \dx x_n \ldots \] -\shortdefinition[Randverteilung] Falls $\cX, \cY$ gemeinsame Verteilung $F$ haben, so ist $F_\cX : \R \rightarrow [0, 1]$ gegeben durch: +\shortdefinition[Randverteilung] Falls $\cX, \cY$ gemeinsame Verteilung $F$ haben, so ist die Verteilungsfunktion der \bi{Randverteilung} von $\cX$, +$F_\cX : \R \rightarrow [0, 1]$ gegeben durch: \[ x \mapsto F_\cX(x) \! = \! \P[\cX \leq x] \! = \! \P[\cX \leq x, \cY < \8] \! = \! \limit{y}{\8} F(x, y) \] +Analog für $\cY$ ist sie $F_y : \R \rightarrow [0, 1]$: +\[ + y \mapsto F_\cY(y) \! = \! \P[\cY \leq y] \! = \! \P[\cY < \8, \cY \leq y] \! = \! \limit{x}{\8} F(x, y) +\] +Die Dichten der Randverteilungen sind: +\[ + f_\cX(x) = \int_{-\8}^{\8} f(x, y) \dx y + \qquad + f_\cY(y) = \int_{-\8}^{\8} f(x, y) \dx x +\] +Herleitung der Randdichte (``wegintegrieren''): +\begin{align*} + f_\cX(x) & = \diff{x} F_\cX(x) = \diff{x}\left( \int_{-\8}^{x} \int_{-\8}^{\8} f(s, t) \dx t \dx s \right) \\ + & =\int_{-\8}^{\8} f(x, t) \dx t +\end{align*} +{\scriptsize (Hier wieder Umwandlung von Summe zu Integral von Diskret zu Stetig)} + +\shortexample Beispiele von gemeinsamen stetigen Verteilungen\\ +\bi{Einheitsquadrat} {\scriptsize ($f(x, y) = 1_{(x, y) \in [0, 1]^2} = 1_{x \in [0, 1]} 1_{y \in [0, 1]}$)} +\begin{align*} + f_\cX(x) & = \int_{0}^{1} 1_{x \in [0, 1]} 1_{y \in [0, 1]} \dx y = 1_{x \in [0, 1]} \\ + f_\cY(y) & = \int_{0}^{1} 1_{x \in [0, 1]} 1_{y \in [0, 1]} \dx x = 1_{y \in [0, 1]} +\end{align*} + +\bi{Einheitskreisscheibe} {\scriptsize ($f(x, y) = \frac{1}{\pi} \cdot 1_{x^2 + y^2 \leq 1}$)} +\begin{align*} + f_\cX(x) \! = \! \int_{-\8}^{\8} \frac{1}{\pi} 1_{y^2 \leq 1 - x^2} \dx y \! = \! \int_{-\sqrt{1 - x^2}}^{\sqrt{1 - x^2}} \frac{1}{\pi} \dx y \! + = \! \frac{2}{\pi} \sqrt{1 - x^2} \\ + f_\cY(y) \! = \! \int_{-\8}^{\8} \frac{1}{\pi} 1_{x^2 \leq 1 - y^2} \dx x \! = \! \int_{-\sqrt{1 - y^2}}^{\sqrt{1 - y^2}} \frac{1}{\pi} \dx x \! = \! \frac{2}{\pi} \sqrt{1 - y^2} +\end{align*} + +\shorttheorem[Unabhängigkeit] $\cX_i$ mit Dichten $f_{\cX_i}$, dann ist äquiv.: +\begin{itemize} + \item $\cX_1, \ldots, \cX_n$ sind unabhängig + \item $\cX_1, \ldots, \cX_n$ sind gem. stetig mit gem. Dichte (Prod. Randdichten): + $f(x_1, \ldots, x_n) = f_{\cX_1}(x_1) \cdot \ldots \cdot f_{\cX_n}(x_n)$ +\end{itemize} + +\shortexample[Gleichverteilungen]\\ +\bi{Einheitsquadrat} Wieder $f(x, y) = 1_{(x, y) \in [0, 1]^2}$, dann: +\[ + f(x, y) = 1_{(x, y) \in [0, 1]^2} = 1_{x \in [0, 1]} 1_{y \in [0, 1]} = f_\cX(x) f_\cY(y) +\] +Folglich sind die beiden Koordinaten unabhängig. + +\bi{Einheitskreisscheibe} $f(x, y) \neq f_\cX(x) f_\cY(y)$, mit Dichten von oben. Also sind die beiden Koordinaten nicht unabhängig. diff --git a/semester4/ps/ps-jh/parts/05_limit-theorems/00_intro.tex b/semester4/ps/ps-jh/parts/05_limit-theorems/00_intro.tex new file mode 100644 index 0000000..d2ced9d --- /dev/null +++ b/semester4/ps/ps-jh/parts/05_limit-theorems/00_intro.tex @@ -0,0 +1,12 @@ +\shortdefinition $\displaystyle \overline{\cX}_n = \frac{1}{n} \sum_{k = 1}^{n} \cX_k$ ist das \bi{arithmetische Mittel} der $\cX_k$. + +{\scriptsize \shortterm In Zusammenhang mit Zufallsvariablen auch \bi{Sichprobenmittel} genannt. Realisierung wird \bi{empirischer Mittelwert} genannt} + +% TODO: Possibly add the remark from P288 (P6 in Slide Deck 6) on expected value + +\shorttheorem[Schwaches Ges. der grossen Zahlen] Sei $K = \{ 1, 2, \ldots \}$ und $\forall k \in K : \cX_k$ unabh. Z.V. mit $\E[\cX_k] = \mu$; $\V[\cX_k] = \sigma^2$: +\[ + \overline{\cX}_n = \frac{1}{n} S_n = \frac{1}{n} \sum_{k = 1}^{n} \cX_k +\] +Dann konvergiert $\overline{\cX}_n$ für $n \rightarrow \8$ in Wahrscheinlichkeit gegen $\mu = \E[\cX_k]$, +also $\forall \varepsilon > 0$ gilt $\P[|\overline{\cX}_n - \mu| > \varepsilon] \overset{n \rightarrow \8}{\longrightarrow} 0$ diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index d48ee4860b83796b301fee9cd3724bfb0128fdfb..c8f17ac3956c0ca61b9c366d8c28df56d94cccf9 100644 GIT binary patch delta 179870 zcmV(!K;^&O^%kDK7qCwd0W_0P5ETJ2lVJoXvyKpX0Rd{W2ohZbf0!avBgZO0UvV!d z(cb|XuG=Ctdrhd7lW29b*bf+D2%{!Om0hQyy_ulX?2RAE?QG#a*J69zuFK4hYtchc z;Kvl~7Q_H6Qm{u50Hd-I>{qa&c6wH>M~w!`st_r>A3B)pnN$l#ss-~OpwFGu%X9*2 z9j?@tN^}K6t|dq;e`R)tMG17Fx~VXKG|w-+lMV- zuxc0xgH`X1H0WXYBWZ6O*eBniz13|h?d|&gNO>?A;sCuZz|Klgsu<9LB3HSp zM6TEFK$VN?FZRz9$8ujfio}v+TP-S=5w{q}1%^v~Z@x^m37L0x6Mmq#+pQv&_Wd 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