From b10514d7fa808d52fb68b576c24c66a63793f99e Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Wed, 15 Apr 2026 14:15:58 +0200 Subject: [PATCH] [PS] Joint discrete distribution mostly complete --- .../04_joint-distribution/00_discrete.tex | 52 +++++++++++++++++- .../probability-and-statistics-cheatsheet.pdf | Bin 366867 -> 373673 bytes .../probability-and-statistics-cheatsheet.tex | 2 +- 3 files changed, 52 insertions(+), 2 deletions(-) diff --git a/semester4/ps/ps-jh/parts/04_joint-distribution/00_discrete.tex b/semester4/ps/ps-jh/parts/04_joint-distribution/00_discrete.tex index af1610a..8715dec 100644 --- a/semester4/ps/ps-jh/parts/04_joint-distribution/00_discrete.tex +++ b/semester4/ps/ps-jh/parts/04_joint-distribution/00_discrete.tex @@ -1,2 +1,52 @@ \subsection{Gemeinsame Diskrete Verteilung} -\shortdefinition $\cX_i$ haben gemeinsame +\shortdefinition $\cX_i$ mit Mengen $W_k \subseteq \R$ endlich oder abzählbar mit $\cX_k \in W_k$ fast sicher. +Die gemeinsame Verteilung (\textit{Joint probability distribution}) von $(\cX_1, \ldots, \cX_n)$ ist Familie von W.: +\[ + \{ p(x_1, \ldots, x_n) \}_{x_1 \in W_1, \ldots, x_n \in W_n} +\] +mit $p : \R^n \rightarrow [0, 1]$ die gemeinsame Gewichtsfunktion (joint probability mass function) mit +\[ + p(x_1, \ldots, x_n) = \P[\cX_1 = x_1, \ldots, \cX_n = x_n] +\] + +\shorttheorem Gemeinsame Verteilung von $\cX_i$ erfüllt stets +\[ + \sum_{x_1 \in W_1, \ldots, x_n \in W_n} p(x_1, \ldots, x_n) = 1 +\] + +\shortremark Umgekehrt: Für endliche oder abzählbare $W_i$ und Funktion $p : W_1 \times \ldots \times W_n \rightarrow [0, 1]$, die obiges erfüllen, gibt es W-Raum mit Verteilung $p$ + +\shortproposition Aus $p$ Verteilungsfunktion (analog zu ZH $F_\cX \; \& \; p_\cX$): +\begin{align*} + F(x_1, \ldots, x_n) & = \P[\cX_1 \leq x_1, \ldots, \cX_n \leq x_n] \\ + & = \sum_{y_1 \leq x_1, \ldots, y_n \leq x_n} p(y_1, \ldots, y_n) +\end{align*} + +\shorttheorem[Verteilung Bild] Sei $\phi : \R^n \rightarrow \R$, $\cX_i$ disk. Z.V. mit Werten jeweils f.s. in $W_1, \ldots, W_n$. +Dann $\cZ = \varphi(\cX_1, \ldots, \cX_n)$ disk. Z.V. mit Werten f.s. in $W = \varphi(W_1 \times \ldots \times W_n)$. Verteilung von $\cZ$ dann gegeben durch ($\forall z \in W$): +\[ + \P[\cZ = z] = \sum_{\elementstack{x_1 \in W_1, \ldots, x_n \in W_n}{\varphi(x_1, \ldots, x_n) = z}} \P[\cX_1 = x_1, \ldots, X_n = x_n] +\] + +\shorttheorem[Randverteilung] $\cX_i$ disk. Z.V. mit gem. $p$.\\ +$\forall k \in \{ 1, \ldots, n \}$ und $\forall x \in W_k$ gilt: +\[ + \P[\cX_k = x] = \sum_{\elementstack{x_l \in W_l}{l \in \{1, \ldots, n\} \backslash \{k\}}} p(x_1, \ldots, x_{k - 1}, x, x_{k + 1}, \ldots, x_n) +\] +{\scriptsize Also: Elimination der anderen Variable(n) in $p$} + +\shortremark Verteilungsf. d. $k$-ten Randv. $F_{\cX_k}(x) = \P[\cX_k \leq x]$ +\[ + F_{\cX_k}(x) = \lim_{\elementstack{x_l \rightarrow \8}{l \in \{ 1, \ldots, n \} \backslash \{ k \}}} F(x_1, \ldots, x_{k - 1}, x, x_{k + 1}, \ldots, x_n) +\] + +\shorttheorem[Erwartungswert Bild] (Solange Summe wohldefiniert ist, summieren über $x_1 \in W_1, \ldots, x_n \in W_n$) +\[ + \E[\varphi(\cX_1, \ldots, \cX_n)] = \sum_{x_1, \ldots, x_n} \varphi(x_1, \ldots, x_n) p(x_1, \ldots, x_n) +\] + +\shorttheorem Folgende Aussagen sind äquivalent (für $\cX_i$ mit Verteilung $\{p(x_1, \ldots, x_n)\}_{x_1 \in W_1, \ldots, x_n \in W_n}$): +\begin{itemize} + \item $\cX_1, \ldots, \cX_n$ sind unabhängig + 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@@ \input{parts/03_expected-value/06_covariance.tex} % \input{parts/03_expected-value/} -\newsectionNoPB +\newsection \section{Gemeinsame Verteilungen} \input{parts/04_joint-distribution/00_discrete.tex} \input{parts/04_joint-distribution/01_continuous.tex}