From 788a2b26c3b03988da8f8fe9a186051b75ddb800 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Wed, 15 Apr 2026 21:08:31 +0200 Subject: [PATCH] [PS] Joint continuous distribution start --- .../04_joint-distribution/01_continuous.tex | 31 ++++++++++++++++++ .../probability-and-statistics-cheatsheet.pdf | Bin 373673 -> 375516 bytes 2 files changed, 31 insertions(+) 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 c79b52b..371e209 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 @@ -3,3 +3,34 @@ \[ \P[\cX_1 \! \leq \! a_1, \ldots, \cX_n \! \leq \! a_n]\! = \! \int_{-\8}^{a_1}\! \ldots \! \int_{-\8}^{a_n} \! f(x_1, \ldots) \dx x_n \ldots \mbox{d} x_1 \] + +\shorttheorem $f$ gem. Dichte +\[ + \int_{-\8}^{\8}\! \ldots \! \int_{-\8}^{\8} \! f(x_1, \ldots) \dx x_n \ldots \mbox{d} x_1 = 1 +\] +Jeder Funk. $f$ die obiges erfüllt ein W-Raum $(\Omega, \cF, \P)$ und $n$-Z.V. $\cX_i$ zugeo. werden, s.d. $f$ die gem. Dichte von $\cX_i$ ist. + +\inlineintuition $f(x_1, \ldots) \dx x_1, \ldots$ beschreibt die W., dass ein zu-fälliger Punkt $(\cX_1, \ldots)$ in $[x_1, x_1 + \dx x_1] \times \ldots$ liegt + +\shortexample Gleichverteilungen: +\begin{itemize} + \item \bi{Einheistquadrat} $f(x, y) = \begin{cases} + 1 & (x, y) \in [0, 1]^2 \\ + 0 & \text{sonst} + \end{cases}$ + \item \bi{Einheistkreisscheibe} $D = \{ (x, y) \in \R^2 \divider x^2 + y^2 \leq 1 \}$ + $f(x, y) = \begin{cases} + \frac{1}{\pi} & (x, y) \in D \\ + 0 & \text{sonst} \\ + \end{cases}$ +\end{itemize} + +\shorttheorem[Erwartungswert Bild] +\[ + \E[\varphi(\cX_1, \ldots)] \! = \! \int_{-\8}^{\8} \ldots \int_{-\8}^{\8} \varphi(x_1, \ldots) f(x_1, \ldots) \dx x_n \ldots +\] + +\shortdefinition[Randverteilung] Falls $\cX, \cY$ gemeinsame Verteilung $F$ haben, so ist $F_\cX : \R \rightarrow [0, 1]$ gegeben durch: +\[ + x \mapsto F_\cX(x) \! = \! \P[\cX \leq x] \! = \! \P[\cX \leq x, \cY < \8] \! = \! \limit{y}{\8} F(x, y) +\] diff --git 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