From 8a95d4a831767a526da9eab1d358e07398ab4a6b Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Wed, 18 Mar 2026 20:30:22 +0100 Subject: [PATCH] [PS] Start continuous distributions --- .../04_cont-dist.tex | 18 +++++++++++++++++- .../05_cont-distributions/00_normal.tex | 5 +++++ .../05_examples-cont-rv.tex | 0 .../probability-and-statistics-cheatsheet.pdf | Bin 329825 -> 332850 bytes .../probability-and-statistics-cheatsheet.tex | 3 ++- 5 files changed, 24 insertions(+), 2 deletions(-) create mode 100644 semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/00_normal.tex delete mode 100644 semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_examples-cont-rv.tex diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/04_cont-dist.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/04_cont-dist.tex index 5a5ef33..7eae8a6 100644 --- a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/04_cont-dist.tex +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/04_cont-dist.tex @@ -1,2 +1,18 @@ \newpage -\subsection{Stetige Verteilung} +\subsection{Stetige Zufallsvariablen} +\shortdefinition[Stetig verteilte Z.V] $\cX$ stetig, falls $\exists f_\cX : \R \rightarrow \R_+$, s.d. V.F. $F_\cX(x) = \int_{-\8}^{x} f_\cX(t) \dx t$. +$f_\cX$ ist Dichte (pdf) von $\cX$ + +% \shortdefinition[Stückw. st. diff. F.] TODO: Consider adding this, Slides Chapter 3, P56 + +\shorttheorem Sei $F_\cX$ st. stückw. diff. auf Partition $-\8 = x_0 < x_1 \ldots < x_n = \8$. +Dann $\cX$ stetig, mit $a_k$ beliebig und +\[ + f_\cX = \begin{cases} + F_\cX'(x) & \exists k \in \{0, 1, \ldots, n - 1\} \\ + a_k & x \in \{ x_1, \ldots, x_{n - 1} \} + \end{cases} +\] + +\shortremark $f_\cX$ fast analog zu Gew.F $p_\cX$. Also:\\ +$(\Sigma, p_\cX) \mapsto (\int, f_\cX)$ vom diskreten zu stetigen Fall diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/00_normal.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/00_normal.tex new file mode 100644 index 0000000..6fc1a8c --- /dev/null +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/00_normal.tex @@ -0,0 +1,5 @@ +\subsection{Stetige Verteilungen} +\shortdefinition[Gleichverteilung] $\cX \sim \cU([a, b])$, falls $f_\cX = \begin{cases} + \frac{1}{b - a} & x \in [a, b] \\ + 0 & \text{sonst} +\end{cases}$ diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_examples-cont-rv.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_examples-cont-rv.tex deleted file mode 100644 index e69de29..0000000 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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zkAEio+???1Xu^}(gx}sxSe(~|CN2B#ZB#4(Mi!Y?WKfYAMP?M4QDi)k@kAyQnM`Cd zk;z0R6PZk8GLfM~h7uV{WGJx>w1W=N30S7s4a%1NI~c4~d_VjT!U+8GmkAUF9Skx! NH8?Q}B_%~qMhgF8>RkW; diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex index 8bdac61..81c889e 100644 --- a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex +++ b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex @@ -47,7 +47,8 @@ \input{parts/02_discrete-continuous-rv/03_distributions/03_hyp-geom.tex} \input{parts/02_discrete-continuous-rv/03_distributions/04_poisson.tex} \input{parts/02_discrete-continuous-rv/04_cont-dist.tex} -\input{parts/02_discrete-continuous-rv/05_examples-cont-rv.tex} +\input{parts/02_discrete-continuous-rv/05_cont-distributions/00_normal.tex} +% \input{parts/02_discrete-continuous-rv/05_cont-distributions/} % \input{parts/02_discrete-continuous-rv/}