From 59d0df00935ed69536a9795dac6d48e2c00f2a3b Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Wed, 11 Mar 2026 14:13:11 +0100 Subject: [PATCH] [PS] Catch up --- .../02_discrete-rv.tex | 5 ++- .../03_examples-disc-rv.tex | 34 ++++++++++++++++++ .../probability-and-statistics-cheatsheet.pdf | Bin 313994 -> 319001 bytes 3 files changed, 38 insertions(+), 1 deletion(-) diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/02_discrete-rv.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/02_discrete-rv.tex index e488281..4ea1dab 100644 --- a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/02_discrete-rv.tex +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/02_discrete-rv.tex @@ -7,6 +7,9 @@ \shorttheorem $(p(x))_{x \in W} = \sum_{x \in W} p(x) = 1$ +\shortremark $\forall (p(x))_{x \in W} \; \exists$ eine Z.V. mit dieser Verteilung. Können desh. schreiben: ``Sei $\cX$ disk. Z.V. mit Verteilung $(p(x))_{x \in W}$'' + \subsubsection{Zusammenhang Verteilung, Verteilungsfunktion} -\shorttheorem $\cX$ disk. Z.V. wie oben, dann ist Verteilungsf.: $\forall x \in \R \; F_\cX(x) = \sum_{\elementstack{y \in W}{y \leq x}}$ +\shorttheorem $\cX$ disk. Z.V. wie oben, dann ist Verteilungsfunktion: $\forall x \in \R \; F_\cX(x) = \sum_{\elementstack{y \in W}{y \leq x}} p(y)$. +\textbf{Umgekehrt:} $p(x)$ ist die ``Sprunghöhe'' im Punkt $x \in W$, $W$ pos. Sprünge in $F_\cX$ diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_examples-disc-rv.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_examples-disc-rv.tex index e69de29..863fe39 100644 --- a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_examples-disc-rv.tex +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_examples-disc-rv.tex @@ -0,0 +1,34 @@ +\newpage +\subsection{Verteilungen} +\subsubsection{Bernoulli-Verteilung} +\shortdefinition $\cX \sim \text{Ber}(p)$: $\P[\cX = 0] = 1 - p$ und $\P[\cX = 1] = p$ + + +\subsubsection{Binomialverteilung} +\shortdefinition $\cX \sim \text{Bin}(n, p)$, falls\\ +$\forall k \in \{ 0, \ldots, n \}\; \P[\cX = k] = {n \choose k} p^k (1 - p)^{n - k}$ + +\shortremark $\sum_{k = 0}^{n} p(k) = \sum_{k = 0}^{n} \P[\cX = k] = (p + 1 - p)^n = 1$ + +\shorttheorem $X_i \sim \text{Ber}(p_i)$ unab: $(S_n := \sum_{i = 0}^{n} X_i) \sim \text{Ber}(n, p)$ + +\shortremark $\text{Bin}(1, p)$ ist $\text{Ber}(p)$ verteilt. Für $X, Y \sim \text{Bin}(n_i, p)$ mit $X, Y$ unabhängig dann ist $X + Y \sim \text{Bin}(n_1 + n_2, p)$ + + +\subsubsection{Geometrische Verteilung} +\shortdefinition $\cX \sim \text{Geom}(p)$ mit $W = \N \backslash \{0\}$ falls\\ +$\forall k \in W \; \P[\cX = k] = (1 - p)^{k - 1} \cdot p$ + +\shortremark $\P[\cX = 1] = p$, da wir Konvetion $a^0 = 1$ verwenden. + +\shortremark $\sum_{k = 0}^{\8} p(k) = p \cdot \sum_{k = 0}^{\8} \P[\cX = k] = p \cdot \frac{1}{p} = 1$ + +\shorttheorem $X_i \sim \text{Ber}(p)$ für $i \in \N$.\\ +Dann $( T := \min\{ n \geq 1 \divider X_n = 1 \} )\sim \text{Geom}(p)$ + +\shortremark $T = \8$ ist möglich, $\P[T = \8] = 0$ + +\shorttheorem $T \sim \text{Geom}(p)$, dann $\forall n \geq 0 \; \forall k \geq 1 \; \P[T \geq n + k | T > n] = \P[T \geq k]$ + + +\subsubsection{Poisson-Verteilung} 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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