From 93dffbedea37e80ce37bb999f8d559f3938848fc Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Tue, 3 Mar 2026 16:04:04 +0100 Subject: [PATCH] [PS] Random variables --- .../00_basics/03_properties-of-measures.tex | 2 +- .../01_random-variables/00_definition.tex | 4 +--- .../01_distribution-function.tex | 12 ++++++++++++ .../01_random-variables/02_independence.tex | 16 ++++++++++++++++ .../03_transformations.tex | 4 ++++ .../01_random-variables/04_construction.tex | 6 ++++++ .../00_continuity-of-pdf.tex | 1 + .../probability-and-statistics-cheatsheet.pdf | Bin 283035 -> 294966 bytes .../probability-and-statistics-cheatsheet.tex | 10 +++++++++- 9 files changed, 50 insertions(+), 5 deletions(-) create mode 100644 semester4/ps/ps-jh/parts/01_random-variables/01_distribution-function.tex create mode 100644 semester4/ps/ps-jh/parts/01_random-variables/02_independence.tex create mode 100644 semester4/ps/ps-jh/parts/01_random-variables/03_transformations.tex create mode 100644 semester4/ps/ps-jh/parts/01_random-variables/04_construction.tex create mode 100644 semester4/ps/ps-jh/parts/02_discrete-continuous-rv/00_continuity-of-pdf.tex 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 f8795eb..6f0d2e6 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 @@ -8,7 +8,6 @@ \item $B$ Ereignis, dann $\P[A \cup B] = \P[A] + \P[B] - \P[A \cap B]$ \end{enumerate} -\newpage \subsubsection{Nützliche Ungleichungen} \shorttheorem[Monot.] $A, B \in \cF$, dann $A \subseteq B \Rightarrow \P[A] \leq \P[B]$ @@ -18,6 +17,7 @@ Auch für endl. n.-leere Ereignisse \stepLabelNumber{combined} +\newpage \subsubsection{Anwendungen der Ungleichungen} Sie sind nützlich für schwer zu berechnende W. 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 2b27d5f..47f3654 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 @@ -1,8 +1,6 @@ +% 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 $\P[f]$) \inlinenotation Ohne $\omega$: $\{ X \leq a \} = \{ \omega \in \Omega \divider X(\omega) \leq a \}$, etc - -\subsection{Verteilungsfunktion} -\shortdefinition $F_\cX : \R \rightarrow [0, 1]$, def: $\forall a \in \R \quad F_\cX(a) = \P[\cX \leq a]$ diff --git a/semester4/ps/ps-jh/parts/01_random-variables/01_distribution-function.tex b/semester4/ps/ps-jh/parts/01_random-variables/01_distribution-function.tex new file mode 100644 index 0000000..20d47be --- /dev/null +++ b/semester4/ps/ps-jh/parts/01_random-variables/01_distribution-function.tex @@ -0,0 +1,12 @@ +% P23 +\subsection{Verteilungsfunktion} +\shortdefinition $F_\cX : \R \rightarrow [0, 1]$, def: $\forall a \in \R \quad F_\cX(a) = \P[\cX \leq a]$ + +\shorttheorem $a < b \in \R$. Dann: $\P[a < X \leq b] = F(b) - F(a)$ + +\shorttheorem $\cX$ Z.V. auf $(\Omega, \cF, \P)$ und V.F. $F = F_\cX$. Eig.: +\begin{enumerate} + \item $F$ ist monoton wachsend + \item $F$ ist rechtsseitig ($F(a) = \lim_{h \downarrow 0} F(a + h) \; \forall a \in \R$) + \item $\limit{a}{-\8} F(a) = 0$ und $\limit{a}{\8} F(a) = 1$ +\end{enumerate} diff --git a/semester4/ps/ps-jh/parts/01_random-variables/02_independence.tex b/semester4/ps/ps-jh/parts/01_random-variables/02_independence.tex new file mode 100644 index 0000000..dcee58d --- /dev/null +++ b/semester4/ps/ps-jh/parts/01_random-variables/02_independence.tex @@ -0,0 +1,16 @@ +% P26 +\subsection{Unabhängigkeit} +\shortdefinition Z.V. $\cX_1, \ldots, \cX_n$ sind \bi{unabh.} falls +$\forall x_1, \ldots, x_n \in \R$ $\P[\cX_1 \leq x_1, \ldots, \cX_n \leq x_n] = \P[\cX_1 \leq x_1] \cdots \P[\cX_n \leq x_n]$ + +\shortremark Alternativ: ZVs unabhängig, falls $\forall I_1 \subseteq \R, \ldots, I_n \subseteq \R$ Intervalle +$\{ \cX_1 \in I_1 \}, \ldots, \{ \cX_n \in I_n \}$ unabhängig + + +\subsubsection{Gruppierung} +\shorttheorem $n$ ZV $\cX_i$ und $1 \leq i_1 < \ldots < i_k \leq n$ sind indizes und $\phi_1, \ldots, \phi_k$ Abbildungen. Dann sind unabhängig:\\ +$Y_1 = \phi_1(\cX_1, \ldots, \cX_{i_1}), \ldots, Y_k = \phi_k(X_{i_{k - 1} + 1}, \ldots, X_{i_k})$ + +\subsubsection{Unabhängig identisch verteilte ZV} +\shortdefinition Eine Folge von ZV ist \bi{(1)} unabh. falls $X_i$ unabh. sind und \bi{(2)} uiv, falls unabh. und die ZV dieselbe Verteilungsf. haben, also: +$\forall i, j \quad F_{\cX_i} = F_{\cX_j}$ diff --git a/semester4/ps/ps-jh/parts/01_random-variables/03_transformations.tex b/semester4/ps/ps-jh/parts/01_random-variables/03_transformations.tex new file mode 100644 index 0000000..380df0d --- /dev/null +++ b/semester4/ps/ps-jh/parts/01_random-variables/03_transformations.tex @@ -0,0 +1,4 @@ +% P27 +\subsection{Transformation von Zufallsvariablen} +Da ZV Funk. $\Omega \rightarrow \R$ sind, mit komposition neue ZV aus and. ZV, bspw: +$Z_1 = \exp(\cX_1)$ oder $Z_2 = \cX_1 + \cX_2$ diff --git a/semester4/ps/ps-jh/parts/01_random-variables/04_construction.tex b/semester4/ps/ps-jh/parts/01_random-variables/04_construction.tex new file mode 100644 index 0000000..3ceef16 --- /dev/null +++ b/semester4/ps/ps-jh/parts/01_random-variables/04_construction.tex @@ -0,0 +1,6 @@ +% P28 +\subsection{Konstruktion von Zufallsvariablen} +\shortdefinition[Bernoulli ZV] mit param. $p \in [0, 1]$ falls $\P[\cX = 0] = 1 - p$ und $\P[\cX = 1] = p$. +Wir schreiben: $\cX \sim \text{Ber}(p)$ + +\shorttheorem[$\exists$-T v. 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