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8dba126e60
| Author | SHA1 | Date | |
|---|---|---|---|
| 8dba126e60 | |||
| 1a2987e922 | |||
| 179e9442c8 | |||
| d898c48619 | |||
| 91cf66a544 | |||
| 34769737b0 | |||
| 51718e4749 |
4
.gitignore
vendored
4
.gitignore
vendored
@@ -1,5 +1 @@
|
|||||||
build/
|
build/
|
||||||
|
|
||||||
src/*/.latexmkrc
|
|
||||||
src/*/lib
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|
||||||
src/*/src
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||||||
|
|||||||
5
.gitmodules
vendored
5
.gitmodules
vendored
@@ -1,3 +1,6 @@
|
|||||||
|
[submodule "lib/latex-common"]
|
||||||
|
path = lib/latex-common
|
||||||
|
url = ssh://git@git.mercurial-manifold.eu:2224/an.tsouchlos/latex-common.git
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||||||
[submodule "lib/cel-slides-template-2025"]
|
[submodule "lib/cel-slides-template-2025"]
|
||||||
path = lib/cel-slides-template-2025
|
path = lib/cel-slides-template-2025
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||||||
url = git@gitlab.kit.edu:kit/cel/misc/cel-slides-template-2025.git
|
url = ssh://git@git.mercurial-manifold.eu:2224/an.tsouchlos/cel-slides-template-2025.git
|
||||||
|
|||||||
@@ -1,3 +1,4 @@
|
|||||||
$pdflatex="pdflatex -shell-escape -interaction=nonstopmode -synctex=1 %O %S";
|
$pdflatex="pdflatex -shell-escape -interaction=nonstopmode -synctex=1 %O %S";
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||||||
$out_dir = 'build';
|
$out_dir = 'build';
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||||||
$pdf_mode = 1;
|
$pdf_mode = 1;
|
||||||
|
|
||||||
|
|||||||
@@ -6,7 +6,6 @@ RUN apt update -y && apt upgrade -y
|
|||||||
RUN apt install make texlive latexmk texlive-pictures -y
|
RUN apt install make texlive latexmk texlive-pictures -y
|
||||||
RUN apt install texlive-publishers texlive-science texlive-fonts-extra texlive-latex-extra -y
|
RUN apt install texlive-publishers texlive-science texlive-fonts-extra texlive-latex-extra -y
|
||||||
RUN apt install biber texlive-bibtex-extra -y
|
RUN apt install biber texlive-bibtex-extra -y
|
||||||
RUN apt install texlive-lang-german -y
|
|
||||||
|
|
||||||
RUN apt install python3 python3-pygments -y
|
RUN apt install python3 python3-pygments -y
|
||||||
|
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||||||
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|||||||
30
Makefile
30
Makefile
@@ -1,27 +1,11 @@
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PRESENTATIONS := $(patsubst src/%/presentation.tex,build/presentation_%.pdf,$(wildcard src/*/presentation.tex))
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all:
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HANDOUTS := $(patsubst build/presentation_%.pdf,build/presentation_%_handout.pdf,$(PRESENTATIONS))
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mkdir -p build/build
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|
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RC_PDFLATEX := $(shell grep '$$pdflatex' .latexmkrc \
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TEXINPUTS=./lib/cel-slides-template-2025:$$TEXINPUTS latexmk src/template/presentation.tex
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||||||
| sed -e 's/.*"\(.*\)".*/\1/' -e 's/%S//' -e 's/%O//')
|
mv build/presentation.pdf build/presentation_template.pdf
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||||||
|
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.PHONY: all
|
TEXINPUTS=./lib/cel-slides-template-2025:$$TEXINPUTS latexmk src/2025-11-07/presentation.tex
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all: $(PRESENTATIONS) $(HANDOUTS)
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mv build/presentation.pdf build/presentation_2025-11-07.pdf
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||||||
|
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build/presentation_%.pdf: src/%/presentation.tex build/prepared
|
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TEXINPUTS=./lib/cel-slides-template-2025:$(dir $<):$$TEXINPUTS \
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latexmk -outdir=build/$* $<
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cp build/$*/presentation.pdf $@
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build/presentation_%_handout.pdf: src/%/presentation.tex build/prepared
|
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TEXINPUTS=./lib/cel-slides-template-2025:$(dir $<):$$TEXINPUTS \
|
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latexmk -outdir=build/$*_handout \
|
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-pdflatex='$(RC_PDFLATEX) %O "\def\ishandout{1}\input{%S}"' $<
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cp build/$*_handout/presentation.pdf $@
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|
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build/prepared:
|
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mkdir build
|
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touch build/prepared
|
|
||||||
|
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||||||
.PHONY: clean
|
|
||||||
clean:
|
clean:
|
||||||
rm -rf build
|
rm -rf build
|
||||||
|
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|
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18
README.md
18
README.md
@@ -1,18 +0,0 @@
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# WT Tutorial Presentations
|
|
||||||
|
|
||||||
This repository contains the latex source files for the WT Tutorial slides.
|
|
||||||
|
|
||||||
## Build
|
|
||||||
|
|
||||||
### Local Environment
|
|
||||||
|
|
||||||
```bash
|
|
||||||
$ make
|
|
||||||
```
|
|
||||||
|
|
||||||
### With Docker
|
|
||||||
|
|
||||||
```bash
|
|
||||||
$ docker build . -t wt-tut
|
|
||||||
$ docker run --rm -u `id -u`:`id -g` -w $PWD -v $PWD:$PWD wt-tut make
|
|
||||||
```
|
|
||||||
1
lib/latex-common
Submodule
1
lib/latex-common
Submodule
Submodule lib/latex-common added at bded242752
1
lib/latex-common/.gitignore
vendored
1
lib/latex-common/.gitignore
vendored
@@ -1 +0,0 @@
|
|||||||
build/
|
|
||||||
@@ -1,11 +0,0 @@
|
|||||||
SRCS=$(shell find examples -type f -name "*.tex")
|
|
||||||
PDFS=$(shell echo $(SRCS:.tex=.pdf) | sed -e "s/examples\\///g")
|
|
||||||
|
|
||||||
all: $(PDFS)
|
|
||||||
|
|
||||||
%.pdf: examples/%.tex
|
|
||||||
latexmk $<
|
|
||||||
|
|
||||||
clean:
|
|
||||||
rm -rf build
|
|
||||||
|
|
||||||
@@ -1,33 +0,0 @@
|
|||||||
# latex-common
|
|
||||||
|
|
||||||
Repository containing common latex code that can be shared across multiple projects.
|
|
||||||
|
|
||||||
## Usage
|
|
||||||
|
|
||||||
Put
|
|
||||||
|
|
||||||
```latex
|
|
||||||
\input{/path/to/common.tex}
|
|
||||||
```
|
|
||||||
|
|
||||||
in your preamble. See the `examples` folder for usage examples.
|
|
||||||
|
|
||||||
## Build examples
|
|
||||||
|
|
||||||
### Build manually
|
|
||||||
|
|
||||||
```bash
|
|
||||||
$ make
|
|
||||||
```
|
|
||||||
|
|
||||||
### Build using docker
|
|
||||||
|
|
||||||
1. Build docker image
|
|
||||||
```bash
|
|
||||||
$ docker build -f dockerfiles/Dockerfile.alpine . -t latex-common
|
|
||||||
```
|
|
||||||
2. Build examples
|
|
||||||
```bash
|
|
||||||
$ docker run --rm -v $PWD:$PWD -w $PWD -u `id -u`:`id -g` latex-common make
|
|
||||||
```
|
|
||||||
|
|
||||||
@@ -1,317 +0,0 @@
|
|||||||
% Author: Andreas Tsouchlos
|
|
||||||
%
|
|
||||||
% Collection of useful commands and definitions
|
|
||||||
%
|
|
||||||
% ||====================================================================||
|
|
||||||
% || WARNING ||
|
|
||||||
% ||====================================================================||
|
|
||||||
% || The following packages have to be included before using this file: ||
|
|
||||||
% || amsmath ||
|
|
||||||
% || pgfplots ||
|
|
||||||
% ||====================================================================||
|
|
||||||
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
% Math Symbols %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
|
|
||||||
|
|
||||||
\DeclareMathOperator*{\argmin}{\arg\!\min}
|
|
||||||
\DeclareMathOperator*{\argmax}{\arg\!\max}
|
|
||||||
\DeclareMathOperator\sign{sign}
|
|
||||||
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
% Data Manipulation %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
|
|
||||||
|
|
||||||
%
|
|
||||||
% Filters for Pgfplots
|
|
||||||
% Source: https://tex.stackexchange.com/a/58563 (modified)
|
|
||||||
%
|
|
||||||
|
|
||||||
\pgfplotsset{
|
|
||||||
discard if/.style 2 args={
|
|
||||||
x filter/.append code={
|
|
||||||
\edef\tempa{\thisrow{#1}}
|
|
||||||
\edef\tempb{#2}
|
|
||||||
\ifx\tempa\tempb
|
|
||||||
\def\pgfmathresult{inf}
|
|
||||||
\fi
|
|
||||||
}
|
|
||||||
},
|
|
||||||
discard if not/.style 2 args={
|
|
||||||
x filter/.append code={
|
|
||||||
\edef\tempa{\thisrow{#1}}
|
|
||||||
\edef\tempb{#2}
|
|
||||||
\ifx\tempa\tempb
|
|
||||||
\else
|
|
||||||
\def\pgfmathresult{inf}
|
|
||||||
\fi
|
|
||||||
}
|
|
||||||
},
|
|
||||||
discard if gt/.style 2 args={
|
|
||||||
x filter/.append code={
|
|
||||||
\edef\tempa{\thisrow{#1}}
|
|
||||||
\edef\tempb{#2}
|
|
||||||
\ifdim\tempa pt > \tempb pt
|
|
||||||
\def\pgfmathresult{inf}
|
|
||||||
\fi
|
|
||||||
}
|
|
||||||
},
|
|
||||||
discard if lt/.style 2 args={
|
|
||||||
x filter/.append code={
|
|
||||||
\edef\tempa{\thisrow{#1}}
|
|
||||||
\edef\tempb{#2}
|
|
||||||
\ifdim\tempa pt < \tempb pt
|
|
||||||
\def\pgfmathresult{inf}
|
|
||||||
\fi
|
|
||||||
}
|
|
||||||
}
|
|
||||||
}
|
|
||||||
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
% Graphics & Plotting %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
|
|
||||||
%
|
|
||||||
% Colors
|
|
||||||
%
|
|
||||||
|
|
||||||
% KIT Colors
|
|
||||||
|
|
||||||
\definecolor{kit-green100}{rgb}{0,.59,.51}
|
|
||||||
\definecolor{kit-green70}{rgb}{.3,.71,.65}
|
|
||||||
\definecolor{kit-green50}{rgb}{.50,.79,.75}
|
|
||||||
\definecolor{kit-green30}{rgb}{.69,.87,.85}
|
|
||||||
\definecolor{kit-green15}{rgb}{.85,.93,.93}
|
|
||||||
\definecolor{KITgreen}{rgb}{0,.59,.51}
|
|
||||||
|
|
||||||
\definecolor{KITpalegreen}{RGB}{130,190,60}
|
|
||||||
\colorlet{kit-maigreen100}{KITpalegreen}
|
|
||||||
\colorlet{kit-maigreen70}{KITpalegreen!70}
|
|
||||||
\colorlet{kit-maigreen50}{KITpalegreen!50}
|
|
||||||
\colorlet{kit-maigreen30}{KITpalegreen!30}
|
|
||||||
\colorlet{kit-maigreen15}{KITpalegreen!15}
|
|
||||||
|
|
||||||
\definecolor{KITblue}{rgb}{.27,.39,.66}
|
|
||||||
\definecolor{kit-blue100}{rgb}{.27,.39,.67}
|
|
||||||
\definecolor{kit-blue70}{rgb}{.49,.57,.76}
|
|
||||||
\definecolor{kit-blue50}{rgb}{.64,.69,.83}
|
|
||||||
\definecolor{kit-blue30}{rgb}{.78,.82,.9}
|
|
||||||
\definecolor{kit-blue15}{rgb}{.89,.91,.95}
|
|
||||||
|
|
||||||
\definecolor{KITyellow}{rgb}{.98,.89,0}
|
|
||||||
\definecolor{kit-yellow100}{cmyk}{0,.05,1,0}
|
|
||||||
\definecolor{kit-yellow70}{cmyk}{0,.035,.7,0}
|
|
||||||
\definecolor{kit-yellow50}{cmyk}{0,.025,.5,0}
|
|
||||||
\definecolor{kit-yellow30}{cmyk}{0,.015,.3,0}
|
|
||||||
\definecolor{kit-yellow15}{cmyk}{0,.0075,.15,0}
|
|
||||||
|
|
||||||
\definecolor{KITorange}{rgb}{.87,.60,.10}
|
|
||||||
\definecolor{kit-orange100}{cmyk}{0,.45,1,0}
|
|
||||||
\definecolor{kit-orange70}{cmyk}{0,.315,.7,0}
|
|
||||||
\definecolor{kit-orange50}{cmyk}{0,.225,.5,0}
|
|
||||||
\definecolor{kit-orange30}{cmyk}{0,.135,.3,0}
|
|
||||||
\definecolor{kit-orange15}{cmyk}{0,.0675,.15,0}
|
|
||||||
|
|
||||||
\definecolor{KITred}{rgb}{.63,.13,.13}
|
|
||||||
\definecolor{kit-red100}{cmyk}{.25,1,1,0}
|
|
||||||
\definecolor{kit-red70}{cmyk}{.175,.7,.7,0}
|
|
||||||
\definecolor{kit-red50}{cmyk}{.125,.5,.5,0}
|
|
||||||
\definecolor{kit-red30}{cmyk}{.075,.3,.3,0}
|
|
||||||
\definecolor{kit-red15}{cmyk}{.0375,.15,.15,0}
|
|
||||||
|
|
||||||
\definecolor{KITpurple}{RGB}{160,0,120}
|
|
||||||
\colorlet{kit-purple100}{KITpurple}
|
|
||||||
\colorlet{kit-purple70}{KITpurple!70}
|
|
||||||
\colorlet{kit-purple50}{KITpurple!50}
|
|
||||||
\colorlet{kit-purple30}{KITpurple!30}
|
|
||||||
\colorlet{kit-purple15}{KITpurple!15}
|
|
||||||
|
|
||||||
\definecolor{KITcyanblue}{RGB}{80,170,230}
|
|
||||||
\colorlet{kit-cyanblue100}{KITcyanblue}
|
|
||||||
\colorlet{kit-cyanblue70}{KITcyanblue!70}
|
|
||||||
\colorlet{kit-cyanblue50}{KITcyanblue!50}
|
|
||||||
\colorlet{kit-cyanblue30}{KITcyanblue!30}
|
|
||||||
\colorlet{kit-cyanblue15}{KITcyanblue!15}
|
|
||||||
|
|
||||||
% Matplotlib Colors
|
|
||||||
|
|
||||||
\definecolor{Mpl1}{HTML}{1f77b4}
|
|
||||||
\definecolor{Mpl2}{HTML}{ff7f0e}
|
|
||||||
\definecolor{Mpl3}{HTML}{2ca02c}
|
|
||||||
\definecolor{Mpl4}{HTML}{d62728}
|
|
||||||
\definecolor{Mpl5}{HTML}{9467bd}
|
|
||||||
\definecolor{Mpl6}{HTML}{8c564b}
|
|
||||||
\definecolor{Mpl7}{HTML}{e377c2}
|
|
||||||
\definecolor{Mpl8}{HTML}{7f7f7f}
|
|
||||||
\definecolor{Mpl9}{HTML}{bcbd22}
|
|
||||||
\definecolor{Mpl10}{HTML}{17becf}
|
|
||||||
|
|
||||||
%
|
|
||||||
% Color Schemes
|
|
||||||
%
|
|
||||||
|
|
||||||
% Define colormaps
|
|
||||||
|
|
||||||
\pgfplotsset{
|
|
||||||
colormap={mako}{
|
|
||||||
rgb=(0.18195582, 0.11955283, 0.23136943)
|
|
||||||
rgb=(0.25307401, 0.23772973, 0.48316271)
|
|
||||||
rgb=(0.21607792, 0.39736958, 0.61948028)
|
|
||||||
rgb=(0.20344718, 0.56074869, 0.65649508)
|
|
||||||
rgb=(0.25187832, 0.71827158, 0.67872193)
|
|
||||||
rgb=(0.54578602, 0.8544913, 0.69848331)
|
|
||||||
},
|
|
||||||
colormap={rocket}{
|
|
||||||
rgb=(0.20973515, 0.09747934, 0.24238489)
|
|
||||||
rgb=(0.43860848, 0.12177004, 0.34119475)
|
|
||||||
rgb=(0.67824099, 0.09192342, 0.3504148)
|
|
||||||
rgb=(0.8833417, 0.19830556, 0.26014181)
|
|
||||||
rgb=(0.95381595, 0.46373781, 0.31769923)
|
|
||||||
rgb=(0.96516917, 0.70776351, 0.5606593)
|
|
||||||
},
|
|
||||||
colormap={cividis}{
|
|
||||||
rgb=(0.130669, 0.231458, 0.43284)
|
|
||||||
rgb=(0.298421, 0.332247, 0.423973)
|
|
||||||
rgb=(0.42512, 0.431334, 0.447692)
|
|
||||||
rgb=(0.555393, 0.537807, 0.471147)
|
|
||||||
rgb=(0.695985, 0.648334, 0.440072)
|
|
||||||
rgb=(0.849223, 0.771947, 0.359729)
|
|
||||||
},
|
|
||||||
colormap={cel}{
|
|
||||||
color=(KITred!90!black);
|
|
||||||
color=(kit-blue100);
|
|
||||||
color=(kit-green70);
|
|
||||||
color=(kit-yellow70!80!kit-orange70);
|
|
||||||
},
|
|
||||||
colormap={matplotlib}{
|
|
||||||
% Source: https://github.com/matplotlib/matplotlib/blob/e5a85f960b2d47eac371cff709b830d52c36d267/lib/matplotlib/_cm.py#L1114
|
|
||||||
rgb=(0.2298057, 0.298717966, 0.753683153)
|
|
||||||
rgb=(0.26623388, 0.353094838, 0.801466763)
|
|
||||||
rgb=(0.30386891, 0.406535296, 0.84495867 )
|
|
||||||
rgb=(0.342804478, 0.458757618, 0.883725899)
|
|
||||||
rgb=(0.38301334, 0.50941904, 0.917387822)
|
|
||||||
rgb=(0.424369608, 0.558148092, 0.945619588)
|
|
||||||
rgb=(0.46666708, 0.604562568, 0.968154911)
|
|
||||||
rgb=(0.509635204, 0.648280772, 0.98478814 )
|
|
||||||
rgb=(0.552953156, 0.688929332, 0.995375608)
|
|
||||||
rgb=(0.596262162, 0.726149107, 0.999836203)
|
|
||||||
rgb=(0.639176211, 0.759599947, 0.998151185)
|
|
||||||
rgb=(0.681291281, 0.788964712, 0.990363227)
|
|
||||||
rgb=(0.722193294, 0.813952739, 0.976574709)
|
|
||||||
rgb=(0.761464949, 0.834302879, 0.956945269)
|
|
||||||
rgb=(0.798691636, 0.849786142, 0.931688648)
|
|
||||||
rgb=(0.833466556, 0.860207984, 0.901068838)
|
|
||||||
rgb=(0.865395197, 0.86541021, 0.865395561)
|
|
||||||
rgb=(0.897787179, 0.848937047, 0.820880546)
|
|
||||||
rgb=(0.924127593, 0.827384882, 0.774508472)
|
|
||||||
rgb=(0.944468518, 0.800927443, 0.726736146)
|
|
||||||
rgb=(0.958852946, 0.769767752, 0.678007945)
|
|
||||||
rgb=(0.96732803, 0.734132809, 0.628751763)
|
|
||||||
rgb=(0.969954137, 0.694266682, 0.579375448)
|
|
||||||
rgb=(0.966811177, 0.650421156, 0.530263762)
|
|
||||||
rgb=(0.958003065, 0.602842431, 0.481775914)
|
|
||||||
rgb=(0.943660866, 0.551750968, 0.434243684)
|
|
||||||
rgb=(0.923944917, 0.49730856, 0.387970225)
|
|
||||||
rgb=(0.89904617, 0.439559467, 0.343229596)
|
|
||||||
rgb=(0.869186849, 0.378313092, 0.300267182)
|
|
||||||
rgb=(0.834620542, 0.312874446, 0.259301199)
|
|
||||||
rgb=(0.795631745, 0.24128379, 0.220525627)
|
|
||||||
rgb=(0.752534934, 0.157246067, 0.184115123)
|
|
||||||
rgb=(0.705673158, 0.01555616, 0.150232812)
|
|
||||||
}
|
|
||||||
}
|
|
||||||
|
|
||||||
% Define cycle lists
|
|
||||||
|
|
||||||
\pgfplotscreateplotcyclelist{mako}{%
|
|
||||||
[samples of colormap={4} of mako]%
|
|
||||||
}
|
|
||||||
\pgfplotscreateplotcyclelist{rocket}{%
|
|
||||||
[samples of colormap={4} of rocket]%
|
|
||||||
}
|
|
||||||
\pgfplotscreateplotcyclelist{cividis}{%
|
|
||||||
[samples of colormap={4} of cividis]%
|
|
||||||
}
|
|
||||||
\pgfplotscreateplotcyclelist{viridis}{%
|
|
||||||
[samples of colormap={4} of viridis]%
|
|
||||||
}
|
|
||||||
\pgfplotscreateplotcyclelist{cel}{%
|
|
||||||
[samples of colormap={4} of cel]%
|
|
||||||
}
|
|
||||||
\pgfplotscreateplotcyclelist{matplotlib}{%
|
|
||||||
{Mpl1},{Mpl2},{Mpl3},{Mpl4}
|
|
||||||
}
|
|
||||||
|
|
||||||
% Define 'scolX' colors
|
|
||||||
|
|
||||||
\makeatletter
|
|
||||||
|
|
||||||
\def\extractcolormapcolor#1#2{%
|
|
||||||
\expandafter\pgfplotscolormapaccess\expandafter[\pgfplotspointmetatransformedrange]%
|
|
||||||
[1.0]%
|
|
||||||
{#2}%
|
|
||||||
{\pgfkeysvalueof{/pgfplots/colormap name}}%
|
|
||||||
\def\pgfplots@loc@TMPb{\pgfutil@definecolor{#1}{\csname pgfpl@cm@\pgfkeysvalueof{/pgfplots/colormap name}@colspace\endcsname}}%
|
|
||||||
\expandafter\pgfplots@loc@TMPb\expandafter{\pgfmathresult}%
|
|
||||||
}%
|
|
||||||
|
|
||||||
\def\getcolorbyvalue#1{
|
|
||||||
\csname pgfpl@cm@\pgfkeysvalueof{/pgfplots/colormap name}@colspace\endcsname
|
|
||||||
}
|
|
||||||
|
|
||||||
\makeatother
|
|
||||||
|
|
||||||
\def\setschemecolorsfrommap{
|
|
||||||
\extractcolormapcolor{scol0}{0}
|
|
||||||
\extractcolormapcolor{scol1}{333}
|
|
||||||
\extractcolormapcolor{scol2}{666}
|
|
||||||
\extractcolormapcolor{scol3}{1000}
|
|
||||||
}
|
|
||||||
|
|
||||||
\newcommand{\setschemecolorsmanually}[4]{
|
|
||||||
\colorlet{scol0}{#1}
|
|
||||||
\colorlet{scol1}{#2}
|
|
||||||
\colorlet{scol2}{#3}
|
|
||||||
\colorlet{scol3}{#4}
|
|
||||||
}
|
|
||||||
|
|
||||||
% Define color schemes
|
|
||||||
|
|
||||||
\pgfplotsset{
|
|
||||||
/pgfplots/colorscheme/cel/.style={
|
|
||||||
colormap name={cel},
|
|
||||||
cycle list name={cel},
|
|
||||||
/utils/exec={\setschemecolorsfrommap},
|
|
||||||
},
|
|
||||||
/pgfplots/colorscheme/rocket/.style={
|
|
||||||
colormap name={rocket},
|
|
||||||
cycle list name={rocket},
|
|
||||||
/utils/exec={\setschemecolorsfrommap},
|
|
||||||
},
|
|
||||||
/pgfplots/colorscheme/viridis/.style={
|
|
||||||
colormap name={viridis},
|
|
||||||
cycle list name={viridis},
|
|
||||||
/utils/exec={\setschemecolorsfrommap},
|
|
||||||
},
|
|
||||||
/pgfplots/colorscheme/mako/.style={
|
|
||||||
colormap name={mako},
|
|
||||||
cycle list name={mako},
|
|
||||||
/utils/exec={\setschemecolorsfrommap},
|
|
||||||
},
|
|
||||||
/pgfplots/colorscheme/cividis/.style={
|
|
||||||
colormap name={cividis},
|
|
||||||
cycle list name={cividis},
|
|
||||||
/utils/exec={\setschemecolorsfrommap},
|
|
||||||
},
|
|
||||||
/pgfplots/colorscheme/matplotlib/.style={
|
|
||||||
colormap name={matplotlib},
|
|
||||||
cycle list name={matplotlib},
|
|
||||||
/utils/exec={\setschemecolorsmanually{Mpl1}{Mpl2}{Mpl3}{Mpl4}},
|
|
||||||
},
|
|
||||||
}
|
|
||||||
|
|
||||||
@@ -1,4 +0,0 @@
|
|||||||
FROM alpine:3.19
|
|
||||||
|
|
||||||
RUN apk update && apk upgrade
|
|
||||||
RUN apk add make texlive texmf-dist-pictures
|
|
||||||
@@ -1,8 +0,0 @@
|
|||||||
FROM archlinux:latest
|
|
||||||
|
|
||||||
RUN pacman-key --init
|
|
||||||
RUN pacman-key --populate archlinux
|
|
||||||
RUN pacman -Sy archlinux-keyring --noconfirm && pacman -Su --noconfirm
|
|
||||||
|
|
||||||
RUN pacman -Syu --noconfirm
|
|
||||||
RUN pacman -S make perl texlive texlive-binextra texlive-pictures --noconfirm
|
|
||||||
@@ -1,6 +0,0 @@
|
|||||||
FROM ubuntu:22.04
|
|
||||||
|
|
||||||
ARG DEBIAN_FRONTEND=noninteractive
|
|
||||||
|
|
||||||
RUN apt update -y && apt upgrade -y
|
|
||||||
RUN apt install make texlive latexmk texlive-pictures -y
|
|
||||||
@@ -1,103 +0,0 @@
|
|||||||
\documentclass{article}
|
|
||||||
|
|
||||||
% Packages necessary for common.tex
|
|
||||||
\usepackage{amsmath}
|
|
||||||
\usepackage{pgfplots}
|
|
||||||
\pgfplotsset{compat=newest}
|
|
||||||
|
|
||||||
% Other packages
|
|
||||||
\usepackage{float}
|
|
||||||
\usepackage{subcaption}
|
|
||||||
\usepackage[a4paper, total={5in, 9in}]{geometry}
|
|
||||||
\usetikzlibrary{positioning}
|
|
||||||
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
%%%%%% Set common options %%%%%%%
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
|
|
||||||
|
|
||||||
\input{common.tex}
|
|
||||||
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
%%%%%%% Actual Document %%%%%%%%%
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
|
|
||||||
|
|
||||||
\title{Colorschemes}
|
|
||||||
\author{}
|
|
||||||
\date{}
|
|
||||||
|
|
||||||
|
|
||||||
\begin{document}
|
|
||||||
|
|
||||||
\maketitle
|
|
||||||
|
|
||||||
\foreach \x in {cel, rocket, viridis, mako, cividis, matplotlib}{
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
% Set colorscheme
|
|
||||||
|
|
||||||
\pgfplotsset{colorscheme/\x}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
% Preview colorscheme
|
|
||||||
|
|
||||||
\noindent\begin{minipage}{\textwidth}
|
|
||||||
\colorbox{gray!30}{\texttt{colorscheme/\x}}
|
|
||||||
\begin{figure}[H]
|
|
||||||
\centering
|
|
||||||
|
|
||||||
\tikzstyle{colornode} = [draw, inner sep=0pt, minimum width=1cm, minimum height=0.5cm]
|
|
||||||
\pgfplotsset{scaled y ticks=false}
|
|
||||||
|
|
||||||
\begin{subfigure}[c]{0.45\textwidth}
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
domain=0:10,
|
|
||||||
view={0}{90},
|
|
||||||
yticklabel=\empty,xticklabel=\empty,
|
|
||||||
width=\textwidth,
|
|
||||||
height=0.75\textwidth,
|
|
||||||
]
|
|
||||||
\addplot3+[surf]
|
|
||||||
{x + y};
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
\end{subfigure}%
|
|
||||||
\begin{subfigure}[c]{0.45\textwidth}
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
domain=0:10,
|
|
||||||
ymin=-1e4,ymax=1,
|
|
||||||
legend pos=south west,
|
|
||||||
yticklabel=\empty,xticklabel=\empty,
|
|
||||||
width=\textwidth,
|
|
||||||
height=0.75\textwidth,
|
|
||||||
]
|
|
||||||
\foreach \i in {1,2,4,8}{
|
|
||||||
\addplot+ [mark=none, line width=1pt]
|
|
||||||
{-\i*exp(x)};
|
|
||||||
%\expandafter\addlegendentry\expandafter{$e^{-\text{\i} x}$}
|
|
||||||
\addlegendentryexpanded{$e^{-\i x}$}
|
|
||||||
}
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
\end{subfigure}%
|
|
||||||
\begin{subfigure}[c]{0.1\textwidth}
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\node[colornode, fill=scol3] (color3) {};
|
|
||||||
\node[colornode, fill=scol2, above=1mm of color3] (color2) {};
|
|
||||||
\node[colornode, fill=scol1, above=1mm of color2] (color1) {};
|
|
||||||
\node[colornode, fill=scol0, above=1mm of color1] (color0) {};
|
|
||||||
\end{tikzpicture}
|
|
||||||
\end{subfigure}%
|
|
||||||
\end{figure}
|
|
||||||
\vspace{\parskip}
|
|
||||||
\end{minipage}
|
|
||||||
}
|
|
||||||
|
|
||||||
|
|
||||||
\end{document}
|
|
||||||
|
|
||||||
@@ -1,177 +0,0 @@
|
|||||||
\documentclass{article}
|
|
||||||
|
|
||||||
% Packages necessary for common.tex
|
|
||||||
\usepackage{amsmath}
|
|
||||||
\usepackage{pgfplots}
|
|
||||||
\pgfplotsset{compat=newest}
|
|
||||||
|
|
||||||
% Other packages
|
|
||||||
\usepackage{float}
|
|
||||||
\usepackage{subcaption}
|
|
||||||
\usepackage[a4paper, total={6.5in, 9in}]{geometry}
|
|
||||||
\usetikzlibrary{positioning}
|
|
||||||
\usepackage{ifthen}
|
|
||||||
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
%%%%%% Set common options %%%%%%%
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
|
|
||||||
|
|
||||||
\input{common.tex}
|
|
||||||
\pgfplotsset{colorscheme/rocket}
|
|
||||||
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
%%%%%%% Actual Document %%%%%%%%%
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
|
|
||||||
|
|
||||||
\title{Manipulation of CSV data}
|
|
||||||
\author{}
|
|
||||||
\date{}
|
|
||||||
|
|
||||||
|
|
||||||
\begin{document}
|
|
||||||
|
|
||||||
\maketitle
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
% 'discard if lt' & 'discard if gt'
|
|
||||||
|
|
||||||
\begin{figure}[H]
|
|
||||||
\centering
|
|
||||||
|
|
||||||
\begin{subfigure}[t]{0.49\textwidth}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%
|
|
||||||
% Crop along x axis
|
|
||||||
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
width=\textwidth,
|
|
||||||
height=0.55\textwidth,
|
|
||||||
xmin=-5, xmax=132,
|
|
||||||
ymin=-3, ymax=6,
|
|
||||||
]
|
|
||||||
% Plot all data as reference
|
|
||||||
\addplot+[mark=none, line width=1pt]
|
|
||||||
table[col sep=comma, x=x, y=y]
|
|
||||||
{res/random.csv};
|
|
||||||
\addlegendentry{All data}
|
|
||||||
|
|
||||||
% Crop and plot desired data
|
|
||||||
\addplot+[scol2, mark=none, line width=1pt]
|
|
||||||
table[col sep=comma, x=x, y=y, discard if lt={x}{40},
|
|
||||||
discard if gt={x}{80}]
|
|
||||||
{res/random.csv};
|
|
||||||
\addlegendentry{Cropped data}
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
||||||
\caption{\texttt{discard if lt/gt} used to crop along $x$-axis}
|
|
||||||
\end{subfigure}%
|
|
||||||
\hfill%
|
|
||||||
\begin{subfigure}[t]{0.49\textwidth}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%
|
|
||||||
% Crop along y axis
|
|
||||||
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
width=\textwidth,
|
|
||||||
height=0.55\textwidth,
|
|
||||||
xmin=-5, xmax=132,
|
|
||||||
ymin=-3, ymax=6,
|
|
||||||
]
|
|
||||||
% Plot all data as reference
|
|
||||||
\addplot+[mark=*]
|
|
||||||
table[col sep=comma, x=x, y=y]
|
|
||||||
{res/random.csv};
|
|
||||||
\addlegendentry{All data}
|
|
||||||
|
|
||||||
% Crop and plot desired data
|
|
||||||
\addplot+[scol2, only marks]
|
|
||||||
table[col sep=comma, x=x, y=y, discard if gt={y}{1},
|
|
||||||
discard if lt={y}{-1}]
|
|
||||||
{res/random.csv};
|
|
||||||
\addlegendentry{Cropped data}
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
||||||
\caption{\texttt{discard if lt/gt} used to crop along $y$-axis}
|
|
||||||
\end{subfigure}
|
|
||||||
|
|
||||||
\caption{\texttt{discard if lt} and \texttt{discard if gt}}
|
|
||||||
\end{figure}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
% 'discard if' & 'discard if not'
|
|
||||||
|
|
||||||
\begin{figure}[H]
|
|
||||||
\centering
|
|
||||||
|
|
||||||
\begin{subfigure}[t]{0.49\textwidth}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%
|
|
||||||
% Select single datastream
|
|
||||||
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
width=\textwidth,
|
|
||||||
height=0.5\textwidth,
|
|
||||||
ymin=-4,ymax=14,
|
|
||||||
]
|
|
||||||
% Plot all data as reference
|
|
||||||
\foreach \i in {0.0, 3.0, 6.0, 9.0} {
|
|
||||||
\addplot[scol0!20, mark=none, line width=1pt, forget plot]
|
|
||||||
table[col sep=comma, x=x, y=y, discard if not={mu}{\i}]
|
|
||||||
{res/random_multiple.csv};
|
|
||||||
}
|
|
||||||
\addlegendimage{scol0!20, mark=none, line width=1pt, forget plot}
|
|
||||||
\addlegendentry{All data}
|
|
||||||
|
|
||||||
% Select and plot desired datastream
|
|
||||||
\addplot+[scol2, mark=none, line width=1pt]
|
|
||||||
table[col sep=comma, x=x, y=y, discard if not={mu}{3.0}]
|
|
||||||
{res/random_multiple.csv};
|
|
||||||
\addlegendentry{$\mu=3.0$}
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
||||||
\caption{\texttt{discard if not} used to select single datastream}
|
|
||||||
\end{subfigure}%
|
|
||||||
\begin{subfigure}[t]{0.49\textwidth}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%
|
|
||||||
% Discard single datastream
|
|
||||||
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
width=\textwidth,
|
|
||||||
height=0.5\textwidth,
|
|
||||||
ymin=-4,ymax=14,
|
|
||||||
]
|
|
||||||
% Plot all data as reference
|
|
||||||
\addplot+[scol0!20, only marks, point meta=\thisrow{mu}]
|
|
||||||
table[col sep=comma, x=x, y=y]
|
|
||||||
{res/random_multiple.csv};
|
|
||||||
\addlegendentry{All data}
|
|
||||||
|
|
||||||
% Discard datastream and plot desired data
|
|
||||||
\addplot+[scol1, only marks, point meta=\thisrow{mu}]
|
|
||||||
table[col sep=comma, x=x, y=y, discard if={mu}{6.0}]
|
|
||||||
{res/random_multiple.csv};
|
|
||||||
\addlegendentry{All except discarded}
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
||||||
\caption{\texttt{discard if} used to discard single datastream}
|
|
||||||
\end{subfigure}
|
|
||||||
|
|
||||||
\caption{\texttt{discard if} and \texttt{discard if not}}
|
|
||||||
\end{figure}
|
|
||||||
|
|
||||||
|
|
||||||
\end{document}
|
|
||||||
|
|
||||||
@@ -1,129 +0,0 @@
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|||||||
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|
||||||
84,10.070420121063762,9.0
|
|
||||||
85,9.818026166427849,9.0
|
|
||||||
86,9.62052913054561,9.0
|
|
||||||
87,7.672546779908393,9.0
|
|
||||||
88,10.039376452210002,9.0
|
|
||||||
89,8.116452134089174,9.0
|
|
||||||
90,7.401718198709535,9.0
|
|
||||||
91,9.400630644985931,9.0
|
|
||||||
92,8.587826634796986,9.0
|
|
||||||
93,9.421895244583938,9.0
|
|
||||||
94,8.940202946618973,9.0
|
|
||||||
95,9.029233202841228,9.0
|
|
||||||
96,7.9296331459850675,9.0
|
|
||||||
97,7.869659112262588,9.0
|
|
||||||
98,9.002048250570477,9.0
|
|
||||||
99,7.724892216479145,9.0
|
|
||||||
100,10.257550444614596,9.0
|
|
||||||
101,9.127328661421817,9.0
|
|
||||||
102,8.131108654637583,9.0
|
|
||||||
103,9.542262876538535,9.0
|
|
||||||
104,8.428802031233461,9.0
|
|
||||||
105,9.838799017809936,9.0
|
|
||||||
106,8.58253330489674,9.0
|
|
||||||
107,8.04945140126139,9.0
|
|
||||||
108,9.838666202697212,9.0
|
|
||||||
109,7.222210710864647,9.0
|
|
||||||
110,9.020797988316389,9.0
|
|
||||||
111,11.198755284208564,9.0
|
|
||||||
112,9.902311943504731,9.0
|
|
||||||
113,8.268455052341995,9.0
|
|
||||||
114,9.196679340508988,9.0
|
|
||||||
115,10.085221416887418,9.0
|
|
||||||
116,8.789187643097703,9.0
|
|
||||||
117,9.05252474696093,9.0
|
|
||||||
118,10.207225802526317,9.0
|
|
||||||
119,8.456518725841255,9.0
|
|
||||||
120,10.706072071243309,9.0
|
|
||||||
121,6.971235017869722,9.0
|
|
||||||
122,6.87664578474583,9.0
|
|
||||||
123,8.046183170871416,9.0
|
|
||||||
124,8.010525164368774,9.0
|
|
||||||
125,8.590253197994269,9.0
|
|
||||||
126,9.089547459354861,9.0
|
|
||||||
127,8.961471781668173,9.0
|
|
||||||
|
0
src/2025-11-07/presentation.bib
Normal file
0
src/2025-11-07/presentation.bib
Normal file
@@ -1,8 +1,4 @@
|
|||||||
\ifdefined\ishandout
|
|
||||||
\documentclass[de, handout]{CELbeamer}
|
|
||||||
\else
|
|
||||||
\documentclass[de]{CELbeamer}
|
\documentclass[de]{CELbeamer}
|
||||||
\fi
|
|
||||||
|
|
||||||
%
|
%
|
||||||
%
|
%
|
||||||
@@ -137,26 +133,24 @@
|
|||||||
\begin{align*}
|
\begin{align*}
|
||||||
\Omega &= \mleft\{(i,j): i,j \in \mleft\{
|
\Omega &= \mleft\{(i,j): i,j \in \mleft\{
|
||||||
1,\ldots, 6 \mright\}\mright\} \\
|
1,\ldots, 6 \mright\}\mright\} \\
|
||||||
A &= \mleft\{ (1,1),(1,2), \ldots, (6,6) \mright\}
|
A &= \mleft\{ (1,1),(2,2), \ldots, (6,6) \mright\}
|
||||||
\end{align*}
|
\end{align*}
|
||||||
\vspace*{-12mm}
|
\vspace*{-12mm}
|
||||||
\end{lightgrayhighlightbox}
|
\end{lightgrayhighlightbox}
|
||||||
\vspace*{0mm}
|
\vspace*{0mm}
|
||||||
\end{columns}\pause
|
\end{columns}\pause
|
||||||
\item Laplace'sches Zufallsexperiment
|
\item Laplace'sches Zufallsexperiment
|
||||||
% tex-fmt: off
|
|
||||||
\begin{gather*}
|
\begin{gather*}
|
||||||
\text{Voraussetzungen: }\hspace{5mm} \left\{
|
\text{Voraussetzungen: }\hspace{5mm} \left\{
|
||||||
\begin{array}{l}
|
\begin{array}{l}
|
||||||
\lvert\Omega\rvert \text{ endlich}\\
|
\lvert\Omega\rvert \text{ endlich}\\
|
||||||
P(\omega_i) = \frac{1}{\lvert\Omega\rvert}
|
P(\omega_i) = \frac{1}{\lvert\Omega\rvert}
|
||||||
\end{array}
|
\end{array}
|
||||||
\right.\\[1em]
|
\right.\\[1em]
|
||||||
P(A) = \frac{\lvert A \rvert}{\lvert \Omega \rvert} =
|
P(A) = \frac{\lvert A \rvert}{\lvert \Omega \rvert} =
|
||||||
\frac{\text{Anzahl ``günstiger''
|
\frac{\text{Anzahl ``günstiger''
|
||||||
Möglichkeiten}}{\text{Anzahl Möglichkeiten}}
|
Möglichkeiten}}{\text{Anzahl Möglichkeiten}}
|
||||||
\end{gather*}
|
\end{gather*}
|
||||||
% tex-fmt: on
|
|
||||||
\end{itemize}
|
\end{itemize}
|
||||||
\end{frame}
|
\end{frame}
|
||||||
|
|
||||||
@@ -275,17 +269,17 @@
|
|||||||
\item mindestens ein Ass hat?\pause
|
\item mindestens ein Ass hat?\pause
|
||||||
\begin{gather*}
|
\begin{gather*}
|
||||||
P(\text{mindestens ein Ass}) = 1 - P(\text{kein Ass})
|
P(\text{mindestens ein Ass}) = 1 - P(\text{kein Ass})
|
||||||
= 1 - \frac{\binom{4}{0}\binom{48}{5}}{\binom{52}{5}} \approx 0{,}341
|
= 1 - \frac{\binom{4}{0}\binom{48}{5}}{\binom{52}{5}} \approx 0.341
|
||||||
\end{gather*}\pause\vspace*{-5mm}
|
\end{gather*}\pause\vspace*{-5mm}
|
||||||
\item genau ein Ass hat?\pause
|
\item genau ein Ass hat?\pause
|
||||||
\begin{gather*}
|
\begin{gather*}
|
||||||
P(\text{genau ein Ass}) = \frac{\binom{4}{1}\binom{48}{4}}{\binom{52}{5}} \approx 0{,}299
|
P(\text{genau ein Ass}) = \frac{\binom{4}{1}\binom{48}{4}}{\binom{52}{5}} \approx 0.299
|
||||||
\end{gather*}\pause
|
\end{gather*}\pause
|
||||||
\item mindestens zwei Karten der gleichen Art (“Paar”) hat?\pause
|
\item mindestens zwei Karten der gleichen Art (“Paar”) hat?\pause
|
||||||
\begin{align*}
|
\begin{align*}
|
||||||
P(\text{mindestens zwei gleiche Karten}) &= 1 - P(\text{alle Karten unterschiedlich}) \\
|
P(\text{mindestens zwei gleiche Karten}) &= 1 - P(\text{alle Karten unterschiedlich}) \\
|
||||||
&= 1 - \frac{\text{Anzahl Möglichkeiten mit nur unterschiedlichen Karten}}{\text{Anzahl Möglichkeiten}}\\
|
&= 1 - \frac{\text{Anzahl Möglichkeiten mit nur unterschiedlichen Karten}}{\text{Anzahl Möglichkeiten}}\\
|
||||||
&= 1 - \frac{\binom{13}{5}\cdot 4^5}{\binom{52}{5}} \approx 0{,}493
|
&= 1 - \frac{\binom{13}{5}\cdot 4^5}{\binom{52}{5}} \approx 0.493
|
||||||
\end{align*}
|
\end{align*}
|
||||||
\end{enumerate}
|
\end{enumerate}
|
||||||
% tex-fmt: on
|
% tex-fmt: on
|
||||||
@@ -326,8 +320,7 @@
|
|||||||
\mleft\{ A \mright\}, \mleft\{ B \mright\},
|
\mleft\{ A \mright\}, \mleft\{ B \mright\},
|
||||||
\mleft\{ C \mright\}, \mleft\{ A, B \mright\},\\
|
\mleft\{ C \mright\}, \mleft\{ A, B \mright\},\\
|
||||||
&\mleft\{ A, C \mright\},
|
&\mleft\{ A, C \mright\},
|
||||||
\mleft\{ B, C \mright\}, \mleft\{ A, B, C
|
\mleft\{ B, C \mright\}, \mleft\{ A, B, C \mright\} \}
|
||||||
\mright\} \}
|
|
||||||
\end{align*}%
|
\end{align*}%
|
||||||
\vspace*{-14mm}%
|
\vspace*{-14mm}%
|
||||||
\end{lightgrayhighlightbox}
|
\end{lightgrayhighlightbox}
|
||||||
@@ -372,7 +365,7 @@
|
|||||||
\begin{lightgrayhighlightbox}
|
\begin{lightgrayhighlightbox}
|
||||||
Beispiel:
|
Beispiel:
|
||||||
\begin{gather*}
|
\begin{gather*}
|
||||||
\Omega = \{A, B, C\}\\
|
\Omega = {A, B, C}\\
|
||||||
\Pi_N = \{ (A,B,C), (A,C,B), (B,A,C),\\
|
\Pi_N = \{ (A,B,C), (A,C,B), (B,A,C),\\
|
||||||
(B,C,A), (C,A,B), (C,B,A)\}
|
(B,C,A), (C,A,B), (C,B,A)\}
|
||||||
\end{gather*}
|
\end{gather*}
|
||||||
@@ -522,4 +515,10 @@
|
|||||||
% tex-fmt: on
|
% tex-fmt: on
|
||||||
\end{frame}
|
\end{frame}
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\section{Zusammenfassung}
|
||||||
|
|
||||||
|
% TODO: Do we even want this section?
|
||||||
|
|
||||||
\end{document}
|
\end{document}
|
||||||
|
|
||||||
|
|||||||
@@ -1,497 +0,0 @@
|
|||||||
\ifdefined\ishandout
|
|
||||||
\documentclass[de, handout]{CELbeamer}
|
|
||||||
\else
|
|
||||||
\documentclass[de]{CELbeamer}
|
|
||||||
\fi
|
|
||||||
|
|
||||||
%
|
|
||||||
%
|
|
||||||
% CEL Template
|
|
||||||
%
|
|
||||||
%
|
|
||||||
|
|
||||||
\newcommand{\templates}{preambles}
|
|
||||||
\input{\templates/packages.tex}
|
|
||||||
\input{\templates/macros.tex}
|
|
||||||
|
|
||||||
\grouplogo{CEL_logo.pdf}
|
|
||||||
|
|
||||||
\groupname{Communication Engineering Lab (CEL)}
|
|
||||||
\groupnamewidth{80mm}
|
|
||||||
|
|
||||||
\fundinglogos{}
|
|
||||||
|
|
||||||
%
|
|
||||||
%
|
|
||||||
% Custom commands
|
|
||||||
%
|
|
||||||
%
|
|
||||||
|
|
||||||
\input{lib/latex-common/common.tex}
|
|
||||||
\pgfplotsset{colorscheme/rocket}
|
|
||||||
|
|
||||||
\newcommand{\res}{src/2025-11-21/res}
|
|
||||||
|
|
||||||
% \tikzstyle{every node}=[font=\small]
|
|
||||||
% \captionsetup[sub]{font=small}
|
|
||||||
|
|
||||||
%
|
|
||||||
%
|
|
||||||
% Document setup
|
|
||||||
%
|
|
||||||
%
|
|
||||||
|
|
||||||
\usepackage{tikz}
|
|
||||||
\usepackage{tikz-3dplot}
|
|
||||||
\usetikzlibrary{spy, external, intersections, positioning}
|
|
||||||
%\tikzexternalize[prefix=build/]
|
|
||||||
|
|
||||||
\usepackage{pgfplots}
|
|
||||||
\pgfplotsset{compat=newest}
|
|
||||||
\usepgfplotslibrary{fillbetween}
|
|
||||||
|
|
||||||
\usepackage{enumerate}
|
|
||||||
\usepackage{listings}
|
|
||||||
\usepackage{subcaption}
|
|
||||||
\usepackage{bbm}
|
|
||||||
\usepackage{multirow}
|
|
||||||
|
|
||||||
\usepackage{xcolor}
|
|
||||||
|
|
||||||
\title{WT Tutorium 2}
|
|
||||||
\author[Tsouchlos]{Andreas Tsouchlos}
|
|
||||||
\date[]{21. November 2025}
|
|
||||||
|
|
||||||
%
|
|
||||||
%
|
|
||||||
% Document body
|
|
||||||
%
|
|
||||||
%
|
|
||||||
|
|
||||||
\begin{document}
|
|
||||||
|
|
||||||
\begin{frame}[title white vertical, picture=images/IMG_7801-cut]
|
|
||||||
\titlepage
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\section{Aufgabe 1}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\subsection{Theorie Wiederholung}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Bedingte Wahrscheinlichkeiten \& Bayes}
|
|
||||||
|
|
||||||
\vspace*{-10mm}
|
|
||||||
|
|
||||||
\begin{columns}
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\begin{itemize}
|
|
||||||
\item Definition der bedingten Wahrscheinlichkeit
|
|
||||||
\begin{gather*}
|
|
||||||
P(A\vert B) = \frac{P(AB)}{P(B)}
|
|
||||||
\end{gather*}
|
|
||||||
\item Formel von Bayes
|
|
||||||
\begin{gather*}
|
|
||||||
P(A\vert B) = \frac{P(B\vert A) P(A)}{P(B)}
|
|
||||||
\end{gather*}
|
|
||||||
\end{itemize}
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\begin{figure}
|
|
||||||
\centering
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\node[rectangle, minimum width=8cm, minimum height=5cm,
|
|
||||||
draw, line width=1pt, fill=black!20] at (0,0) {};
|
|
||||||
\node [circle, minimum size = 4cm,
|
|
||||||
draw, line width=1pt, fill=KITgreen,
|
|
||||||
fill opacity = 0.5] at (1.25cm,0) {};
|
|
||||||
\draw[line width=1pt, fill=KITblue,
|
|
||||||
fill opacity = 0.5, rounded corners=5mm]
|
|
||||||
(-2.4cm, -2.25cm) -- (-2.4cm, 2.25cm) -- (1.1cm,0) -- cycle;
|
|
||||||
|
|
||||||
\node[left] at (4cm, 2cm) {\Large $\Omega$};
|
|
||||||
\node at (-1.8cm, 0) {$A$};
|
|
||||||
\node at (1.8cm, 0) {$B$};
|
|
||||||
\node at (0, 0) {$AB$};
|
|
||||||
\end{tikzpicture}
|
|
||||||
\end{figure}
|
|
||||||
\end{columns}
|
|
||||||
\vspace*{1cm}
|
|
||||||
\pause
|
|
||||||
\begin{columns}
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\begin{itemize}
|
|
||||||
\item Satz der totalen Wahrscheinlichkeit
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{gather*}
|
|
||||||
\text{Voraussetzungen: }\hspace{5mm} \left\{
|
|
||||||
\begin{array}{l}
|
|
||||||
A_1, A_2, \ldots \text{ disjunkt}\\
|
|
||||||
\displaystyle\sum_{n} A_n = \Omega
|
|
||||||
\end{array}
|
|
||||||
\right.\\[1em]
|
|
||||||
P(B) = \sum_{n} P(B\vert A_n)P(A_n)\\
|
|
||||||
\end{gather*}
|
|
||||||
% tex-fmt: on
|
|
||||||
\end{itemize}
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\begin{figure}
|
|
||||||
\centering
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\newcommand{\hordist}{1.2cm}
|
|
||||||
\newcommand{\vertdist}{2cm}
|
|
||||||
|
|
||||||
\node[circle, fill=KITgreen, inner sep=0pt,
|
|
||||||
minimum size=3mm] (root) at (0, 0) {};
|
|
||||||
\node[circle, fill=KITgreen, inner sep=0pt,
|
|
||||||
minimum size=3mm, below left=\vertdist and
|
|
||||||
2.4*\hordist of root] (n1) {};
|
|
||||||
\node[circle, fill=KITgreen, inner sep=0pt,
|
|
||||||
minimum size=3mm, below right=\vertdist and
|
|
||||||
2.4*\hordist of root] (n2) {};
|
|
||||||
\node[circle, fill=KITgreen, inner sep=0pt,
|
|
||||||
minimum size=3mm, below left=\vertdist and \hordist
|
|
||||||
of n1] (n11) {};
|
|
||||||
\node[circle, fill=KITgreen, inner sep=0pt,
|
|
||||||
minimum size=3mm, below right=\vertdist and \hordist
|
|
||||||
of n1] (n12) {};
|
|
||||||
\node[circle, fill=KITgreen, inner sep=0pt,
|
|
||||||
minimum size=3mm, below left=\vertdist and \hordist
|
|
||||||
of n2] (n21) {};
|
|
||||||
\node[circle, fill=KITgreen, inner sep=0pt,
|
|
||||||
minimum size=3mm, below right=\vertdist and \hordist
|
|
||||||
of n2] (n22) {};
|
|
||||||
|
|
||||||
\draw[-{Latex}, line width=1pt] (root) -- (n1);
|
|
||||||
\draw[-{Latex}, line width=1pt] (root) -- (n2);
|
|
||||||
\draw[-{Latex}, line width=1pt] (n1) -- (n11);
|
|
||||||
\draw[-{Latex}, line width=1pt] (n1) -- (n12);
|
|
||||||
\draw[-{Latex}, line width=1pt] (n2) -- (n21);
|
|
||||||
\draw[-{Latex}, line width=1pt] (n2) -- (n22);
|
|
||||||
|
|
||||||
\node[left] at ($(root)!0.4!(n1)$) {$P(A_1)$};
|
|
||||||
\node[right] at ($(root)!0.4!(n2)$) {$P(A_2)$};
|
|
||||||
|
|
||||||
\node[left] at ($(n1)!0.4!(n11)$) {$P(B\vert A_1)$};
|
|
||||||
\node[right] at ($(n1)!0.2!(n12)$) {$P(C\vert A_1)$};
|
|
||||||
\node[left] at ($(n2)!0.6!(n21)$) {$P(B\vert A_2)$};
|
|
||||||
\node[right] at ($(n2)!0.4!(n22)$) {$P(C\vert A_2)$};
|
|
||||||
|
|
||||||
\node[below] at (n11) {$P(BA_1)$};
|
|
||||||
\node[below] at (n12) {$P(CA_1)$};
|
|
||||||
\node[below] at (n21) {$P(BA_2)$};
|
|
||||||
\node[below] at (n22) {$P(CA_2)$};
|
|
||||||
\end{tikzpicture}
|
|
||||||
\end{figure}
|
|
||||||
\end{columns}
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Zusammenfassung}
|
|
||||||
|
|
||||||
\begin{columns}
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\begin{greenblock}{Bedingte Wahrscheinlichkeit}
|
|
||||||
\vspace*{-6mm}
|
|
||||||
\begin{gather*}
|
|
||||||
P(A\vert B) = \frac{P(AB)}{P(B)}
|
|
||||||
\end{gather*}
|
|
||||||
\end{greenblock}
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\begin{greenblock}{Formel von Bayes}
|
|
||||||
\vspace*{-6mm}
|
|
||||||
\begin{gather*}
|
|
||||||
P(A\vert B) = \frac{P(B\vert A) P(A)}{P(B)}
|
|
||||||
\end{gather*}
|
|
||||||
\end{greenblock}
|
|
||||||
\end{columns}
|
|
||||||
\begin{columns}
|
|
||||||
\column{\kitonecolumn}
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\begin{greenblock}{Satz der totalen Wahrscheinlichkeit}
|
|
||||||
\vspace*{-6mm}
|
|
||||||
\begin{gather*}
|
|
||||||
P(B) = \sum_{n} P(B\vert A_n)P(A_n)
|
|
||||||
\end{gather*}
|
|
||||||
\end{greenblock}
|
|
||||||
\column{\kitonecolumn}
|
|
||||||
\end{columns}
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\subsection{Aufgabe}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
|
|
||||||
\frametitle{Aufgabe 1: Bedingte Wahrscheinlichkeiten \\\& Bayes}
|
|
||||||
|
|
||||||
In einer Population von gelben Animationsfiguren, den Minions,
|
|
||||||
werden zwei Merkmale unterschieden: Augenzahl und Körpergröße. Es gilt:
|
|
||||||
\begin{itemize}
|
|
||||||
\item $80\%$ der Minions haben zwei Augen, $20\%$ nur eines.
|
|
||||||
\item Von den zweiäugigen Minions sind $20\%$ groß, $70\%$
|
|
||||||
mittelgroß und $10\%$ klein.
|
|
||||||
\item Von den einäugigen Minions sind $5\%$ groß, $60\%$
|
|
||||||
mittelgroß und $35\%$ klein.
|
|
||||||
\end{itemize}
|
|
||||||
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{enumerate}[a{)}]
|
|
||||||
\item Bestimmen Sie die Wahrscheinlichkeit, dass ein zufällig
|
|
||||||
ausgewähltes Minion klein, mittelgroß
|
|
||||||
oder groß ist.
|
|
||||||
\item Ein zufällig ausgewähltes Minion ist nicht klein. Mit
|
|
||||||
welcher Wahrscheinlichkeit ist es
|
|
||||||
einäugig?
|
|
||||||
\end{enumerate}
|
|
||||||
% tex-fmt: on
|
|
||||||
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
|
|
||||||
\frametitle{Aufgabe 1: Bedingte Wahrscheinlichkeiten \\\& Bayes}
|
|
||||||
|
|
||||||
In einer Population von gelben Animationsfiguren, den Minions,
|
|
||||||
werden zwei Merkmale unterschieden: Augenzahl und Körpergröße. Es gilt:
|
|
||||||
\begin{itemize}
|
|
||||||
\item $80\%$ der Minions haben zwei Augen, $20\%$ nur eines.
|
|
||||||
\item Von den zweiäugigen Minions sind $20\%$ groß, $70\%$
|
|
||||||
mittelgroß und $10\%$ klein.
|
|
||||||
\item Von den einäugigen Minions sind $5\%$ groß, $60\%$
|
|
||||||
mittelgroß und $35\%$ klein.
|
|
||||||
\end{itemize}
|
|
||||||
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{enumerate}[a{)}]
|
|
||||||
\item Bestimmen Sie die Wahrscheinlichkeit, dass ein zufällig
|
|
||||||
ausgewähltes Minion klein, mittelgroß
|
|
||||||
oder groß ist.
|
|
||||||
\pause\begin{align*}
|
|
||||||
P(K) &= P(K\vert N_1)P(N_1) + P(K\vert N_2)P(N_2) = 0{,}35\cdot 0{,}2 + 0{,}1\cdot 0{,}8 = 0{,}15\\
|
|
||||||
P(M) &= P(M\vert N_1)P(N_1) + P(M\vert N_2)P(N_2) = \cdots = 0{,}68\\
|
|
||||||
P(G) &= P(G\vert N_1)P(N_1) + P(G\vert N_2)P(N_2) = \cdots = 0{,}17
|
|
||||||
\end{align*}
|
|
||||||
\item \pause Ein zufällig ausgewähltes Minion ist nicht klein. Mit
|
|
||||||
welcher Wahrscheinlichkeit ist es
|
|
||||||
einäugig?
|
|
||||||
\pause\begin{align*}
|
|
||||||
P(N_1 \vert \overline{K})
|
|
||||||
= \frac{P(\overline{K} \vert N_1)P(N_1)}{P(\overline{K})}
|
|
||||||
= \frac{\left[ 1 - P(K\vert N_1) \right] P(N_1)}{1 - P(K)}
|
|
||||||
= \frac{(1 - 0{,}35)\cdot 0{,}2}{1 - 0{,}15} \approx 0{,}153
|
|
||||||
\end{align*}
|
|
||||||
\end{enumerate}
|
|
||||||
% tex-fmt: on
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\section{Aufgabe 2}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\subsection{Theorie Wiederholung}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Zusätzliche Bedingungen und Unabhängigkeit}
|
|
||||||
|
|
||||||
\begin{itemize}
|
|
||||||
\item Erweiterte Definition der bedingten Wahrscheinlichkeit
|
|
||||||
\begin{gather*}
|
|
||||||
P(A\vert BC) = \frac{P(AB\vert C)}{P(B\vert C)}
|
|
||||||
\end{gather*}
|
|
||||||
\item Satz von Bayes mit zusätzlichen Bedingungen
|
|
||||||
\begin{gather*}
|
|
||||||
P(A\vert BC) = \frac{P(B\vert AC) P(A\vert C)}{P(B\vert C)}
|
|
||||||
\end{gather*}
|
|
||||||
\pause
|
|
||||||
\item Unabhängigkeit
|
|
||||||
\begin{gather*}
|
|
||||||
A,B \text{ Unabhängig} \hspace{5mm}
|
|
||||||
\Leftrightarrow\hspace{5mm} P(AB) = P(A) P(B)
|
|
||||||
\hspace{5mm} \Leftrightarrow \hspace{5mm} P(A\vert B) = P(A)
|
|
||||||
\end{gather*}
|
|
||||||
\end{itemize}
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Zusammenfassung}
|
|
||||||
|
|
||||||
\begin{columns}
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\begin{greenblock}{Bedingte Wahrscheinlichkeit}
|
|
||||||
\vspace*{-6mm}
|
|
||||||
\begin{gather*}
|
|
||||||
P(A\vert B) = \frac{P(AB)}{P(B)}
|
|
||||||
\end{gather*}
|
|
||||||
\end{greenblock}
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\begin{greenblock}{Formel von Bayes}
|
|
||||||
\vspace*{-6mm}
|
|
||||||
\begin{gather*}
|
|
||||||
P(A\vert B) = \frac{P(B\vert A) P(A)}{P(B)}
|
|
||||||
\end{gather*}
|
|
||||||
\end{greenblock}
|
|
||||||
\end{columns}
|
|
||||||
\begin{columns}
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\begin{greenblock}{Satz der totalen Wahrscheinlichkeit}
|
|
||||||
\vspace*{-6mm}
|
|
||||||
\begin{gather*}
|
|
||||||
P(B) = \sum_{n} P(B\vert A_n)P(A_n)
|
|
||||||
\end{gather*}
|
|
||||||
\end{greenblock}
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\begin{greenblock}{Unabhängigkeit von Ereignissen}
|
|
||||||
\vspace*{-6mm}
|
|
||||||
\begin{gather*}
|
|
||||||
P(AB) = P(A) P(B)
|
|
||||||
\end{gather*}
|
|
||||||
\end{greenblock}
|
|
||||||
\end{columns}
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\subsection{Aufgabe}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Aufgabe 2: Bayes \& Unabhängigkeit}
|
|
||||||
|
|
||||||
\vspace*{-18mm}
|
|
||||||
|
|
||||||
Bei einer Qualitätskontrolle können Werkstücke zwei Fehler
|
|
||||||
aufweisen: Fehler $A$, Fehler $B$, oder
|
|
||||||
beide Fehler gleichzeitig. Die folgenden Wahrscheinlichkeiten
|
|
||||||
sind bekannt:
|
|
||||||
\begin{itemize}
|
|
||||||
\item mit Wahrscheinlichkeit $0{,}05$ hat ein Werkstück den Fehler $A$
|
|
||||||
\item mit Wahrscheinlichkeit $0{,}01$ hat ein Werkstück beide Fehler
|
|
||||||
\item mit Wahrscheinlichkeit $0{,}03$ hat ein Werkstück nur den
|
|
||||||
Fehler $B$ und nicht Fehler $A$.
|
|
||||||
\end{itemize}
|
|
||||||
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{enumerate}[a{)}]
|
|
||||||
\item Berechnen Sie die Wahrscheinlichkeit für das Auftreten von
|
|
||||||
Fehler $B$ und dafür, dass ein
|
|
||||||
Werkstück fehlerfrei ist.
|
|
||||||
\item Ist das Auftreten von Fehler $A$ unabhängig von Fehler $B$?
|
|
||||||
\end{enumerate}
|
|
||||||
% tex-fmt: on
|
|
||||||
|
|
||||||
Bei der Kontrolle wird unerwartet ein zusätzlicher, dritter Fehler $C$
|
|
||||||
beobachtet. Der Fehler tritt
|
|
||||||
mit der Wahrscheinlichkeit $0{,}01$ ein, wenn weder Fehler $A$ noch $B$
|
|
||||||
eingetreten sind und mit der
|
|
||||||
Wahrscheinlichkeit $0{,}02$, wenn sowohl Fehler $A$ als auch $B$ eingetreten
|
|
||||||
sind. In allen anderen
|
|
||||||
Fällen tritt der Fehler $C$ nicht auf.
|
|
||||||
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{enumerate}[a{)}]
|
|
||||||
\setcounter{enumi}{2}
|
|
||||||
\item Berechnen Sie die Wahrscheinlichkeit für das Auftreten von
|
|
||||||
Fehler $C$.
|
|
||||||
\item Sie beobachten, dass ein Werkstück den Fehler $C$ hat. Mit
|
|
||||||
welcher Wahrscheinlichkeit hat es auch Fehler $A$?
|
|
||||||
\end{enumerate}
|
|
||||||
% tex-fmt: on
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Aufgabe 2: Bayes \& Unabhängigkeit}
|
|
||||||
|
|
||||||
\vspace*{-10mm}
|
|
||||||
|
|
||||||
Bei einer Qualitätskontrolle können Werkstücke zwei Fehler
|
|
||||||
aufweisen: Fehler $A$, Fehler $B$, oder
|
|
||||||
beide Fehler gleichzeitig. Die folgenden Wahrscheinlichkeiten
|
|
||||||
sind bekannt:
|
|
||||||
\begin{itemize}
|
|
||||||
\item mit Wahrscheinlichkeit $0{,}05$ hat ein Werkstück den Fehler $A$
|
|
||||||
\item mit Wahrscheinlichkeit $0{,}01$ hat ein Werkstück beide Fehler
|
|
||||||
\item mit Wahrscheinlichkeit $0{,}03$ hat ein Werkstück nur den
|
|
||||||
Fehler $B$ und nicht Fehler $A$.
|
|
||||||
\end{itemize}
|
|
||||||
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{enumerate}[a{)}]
|
|
||||||
\item Berechnen Sie die Wahrscheinlichkeit für das Auftreten von
|
|
||||||
Fehler $B$ und dafür, dass ein
|
|
||||||
Werkstück fehlerfrei ist.
|
|
||||||
\pause\begin{gather*}
|
|
||||||
P(B) = P(B\vert A)P(A) + P(B\vert \overline{A})P(\overline{A}) = P(AB) + P(\overline{A}B) = 0{,}01 + 0{,}03 = 0{,}04
|
|
||||||
\end{gather*}\pause
|
|
||||||
\vspace*{-15mm}\begin{gather*}
|
|
||||||
P(\overline{A}\cap \overline{B}) = 1 - P(A\cup B) = 1 - \left[P(A) + P(B) - P(A\cap B)\right] = 1 - \left(0{,}05 + 0{,}04 - 0{,}01\right) = 0{,}92
|
|
||||||
\end{gather*}
|
|
||||||
\vspace*{-12mm}\pause \item Ist das Auftreten von Fehler $A$ unabhängig von Fehler $B$?
|
|
||||||
\pause\begin{gather*}
|
|
||||||
\left. \begin{array}{l}
|
|
||||||
P(AB) = 0{,}01 \\
|
|
||||||
P(A)P(B) = 0{,}05\cdot 0{,}04 = 0{,}002
|
|
||||||
\end{array}\right\}
|
|
||||||
\hspace{5mm} \Rightarrow \hspace{5mm} P(AB) \neq P(A)P(B) \hspace{5mm}\Rightarrow\hspace{5mm}A,B \text{ nicht unabhängig}
|
|
||||||
\end{gather*}
|
|
||||||
\end{enumerate}
|
|
||||||
% tex-fmt: on
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Aufgabe 2: Bayes \& Unabhängigkeit}
|
|
||||||
|
|
||||||
\vspace*{-13mm}
|
|
||||||
|
|
||||||
Bei der Kontrolle wird unerwartet ein zusätzlicher, dritter Fehler $C$
|
|
||||||
beobachtet. Der Fehler tritt
|
|
||||||
mit der Wahrscheinlichkeit $0{,}01$ ein, wenn weder Fehler $A$ noch $B$
|
|
||||||
eingetreten sind und mit der
|
|
||||||
Wahrscheinlichkeit $0{,}02$, wenn sowohl Fehler $A$ als auch $B$ eingetreten
|
|
||||||
sind. In allen anderen
|
|
||||||
Fällen tritt der Fehler $C$ nicht auf.
|
|
||||||
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{enumerate}[a{)}]
|
|
||||||
\setcounter{enumi}{2}
|
|
||||||
\item Berechnen Sie die Wahrscheinlichkeit für das Auftreten von
|
|
||||||
Fehler $C$.
|
|
||||||
\pause\begin{align*}
|
|
||||||
P(C) &= P(C\vert AB)P(AB) + \overbrace{P(C\vert A \overline{B})}^{0}P(A \overline{B})
|
|
||||||
+ \overbrace{P(C\vert \overline{A}B)}^{0}P(\overline{A} B)
|
|
||||||
+ P(C\vert \overline{A}\overline{B})P(\overline{A}\overline{B}) \\
|
|
||||||
&= 0{,}02\cdot 0{,}01 + 0{,}01\cdot 0{,}92 = 0{,}0094
|
|
||||||
\end{align*}
|
|
||||||
\vspace*{-12mm}\pause \item Sie beobachten, dass ein Werkstück den Fehler $C$ hat. Mit
|
|
||||||
welcher Wahrscheinlichkeit hat es auch Fehler $A$?
|
|
||||||
\pause\hspace*{-5mm}\begin{minipage}{0.48\textwidth}
|
|
||||||
\centering
|
|
||||||
\begin{align*}
|
|
||||||
P(A\vert C) &= \frac{P(AC)}{P(C)}\\[5mm]
|
|
||||||
P(AC) &= P(ACB) + P(AC \overline{B})\\
|
|
||||||
&= P(C\vert AB)P(AB) + \overbrace{P(C\vert A \overline{B})}^{0}P(A \overline{B})\\
|
|
||||||
&= 0{,}02\cdot 0{,}01 = 0{,}0002\\[5mm]
|
|
||||||
P(A\vert C) &= \frac{0{,}0002}{0{,}0094} \approx 0{,}0213
|
|
||||||
\end{align*}
|
|
||||||
\end{minipage}%
|
|
||||||
\hspace*{-10mm}
|
|
||||||
\begin{minipage}{0.06\textwidth}
|
|
||||||
\centering
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\draw[line width=1pt] (0,0) -- (0,6cm);
|
|
||||||
\end{tikzpicture}
|
|
||||||
\end{minipage}%
|
|
||||||
\begin{minipage}{0.48\textwidth}
|
|
||||||
\centering
|
|
||||||
\begin{align*}
|
|
||||||
P(A\vert C) &= \frac{P(C\vert A)P(A)}{P(C)}\\[5mm]
|
|
||||||
P(C\vert A) &= P(C\vert AB)P(B\vert A)
|
|
||||||
+ \overbrace{P(C\vert \overline{A} B)}^{0}P(\overline{A}B) \\
|
|
||||||
&= P(C\vert AB)\frac{P(AB)}{P(A)} = 0{,}02 \cdot \frac{0{,}01}{0{,}05} = 0{,}004\\[5mm]
|
|
||||||
P(A\vert C) &= \frac{0{,}004\cdot 0{,}05}{0{,}0094} \approx 0{,}0213
|
|
||||||
\end{align*}
|
|
||||||
\end{minipage}
|
|
||||||
\end{enumerate}
|
|
||||||
% tex-fmt: on
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\end{document}
|
|
||||||
@@ -1,928 +0,0 @@
|
|||||||
\ifdefined\ishandout
|
|
||||||
\documentclass[de, handout]{CELbeamer}
|
|
||||||
\else
|
|
||||||
\documentclass[de]{CELbeamer}
|
|
||||||
\fi
|
|
||||||
|
|
||||||
%
|
|
||||||
%
|
|
||||||
% CEL Template
|
|
||||||
%
|
|
||||||
%
|
|
||||||
|
|
||||||
\newcommand{\templates}{preambles}
|
|
||||||
\input{\templates/packages.tex}
|
|
||||||
\input{\templates/macros.tex}
|
|
||||||
|
|
||||||
\grouplogo{CEL_logo.pdf}
|
|
||||||
|
|
||||||
\groupname{Communication Engineering Lab (CEL)}
|
|
||||||
\groupnamewidth{80mm}
|
|
||||||
|
|
||||||
\fundinglogos{}
|
|
||||||
|
|
||||||
%
|
|
||||||
%
|
|
||||||
% Custom commands
|
|
||||||
%
|
|
||||||
%
|
|
||||||
|
|
||||||
\input{lib/latex-common/common.tex}
|
|
||||||
\pgfplotsset{colorscheme/rocket}
|
|
||||||
|
|
||||||
\newcommand{\res}{src/2025-12-05/res}
|
|
||||||
|
|
||||||
% \tikzstyle{every node}=[font=\small]
|
|
||||||
% \captionsetup[sub]{font=small}
|
|
||||||
|
|
||||||
%
|
|
||||||
%
|
|
||||||
% Document setup
|
|
||||||
%
|
|
||||||
%
|
|
||||||
|
|
||||||
\usepackage{tikz}
|
|
||||||
\usepackage{tikz-3dplot}
|
|
||||||
\usetikzlibrary{spy, external, intersections, positioning}
|
|
||||||
%\tikzexternalize[prefix=build/]
|
|
||||||
|
|
||||||
\usepackage{pgfplots}
|
|
||||||
\pgfplotsset{compat=newest}
|
|
||||||
\usepgfplotslibrary{fillbetween}
|
|
||||||
|
|
||||||
\usepackage{enumerate}
|
|
||||||
\usepackage{listings}
|
|
||||||
\usepackage{subcaption}
|
|
||||||
\usepackage{bbm}
|
|
||||||
\usepackage{multirow}
|
|
||||||
|
|
||||||
\usepackage{xcolor}
|
|
||||||
|
|
||||||
\title{WT Tutorium 3}
|
|
||||||
\author[Tsouchlos]{Andreas Tsouchlos}
|
|
||||||
\date[]{5. Dezember 2025}
|
|
||||||
|
|
||||||
%
|
|
||||||
%
|
|
||||||
% Document body
|
|
||||||
%
|
|
||||||
%
|
|
||||||
|
|
||||||
\begin{document}
|
|
||||||
|
|
||||||
\begin{frame}[title white vertical, picture=images/IMG_7801-cut]
|
|
||||||
\titlepage
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\section{Aufgabe 1}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\subsection{Theorie Wiederholung}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Zufallsvariablen \& Verteilungen}
|
|
||||||
|
|
||||||
\vspace*{-10mm}
|
|
||||||
|
|
||||||
\begin{itemize}
|
|
||||||
\item Zufallsvariablen (ZV)
|
|
||||||
\begin{minipage}{0.33\textwidth}
|
|
||||||
\centering
|
|
||||||
\begin{gather*}
|
|
||||||
\text{Idee: ``Wegabstrahieren'' von Ergebnisraum
|
|
||||||
$\Omega$} \\[1cm]
|
|
||||||
X: \Omega \mapsto \mathbb{R} \\
|
|
||||||
\underbrace{P_X(x)}_\text{Verteilung} :=
|
|
||||||
P(\underbrace{X}_\text{ZV}=\underbrace{x}_\text{Realisierung})
|
|
||||||
\end{gather*}
|
|
||||||
\end{minipage}%
|
|
||||||
\hspace*{15mm}%
|
|
||||||
\begin{minipage}{0.6\textwidth}
|
|
||||||
\centering
|
|
||||||
\begin{lightgrayhighlightbox}
|
|
||||||
Beispiel: Würfeln mit zwei Würfeln
|
|
||||||
\begin{gather*}
|
|
||||||
X := \text{\normalfont``Summe beider Augenzahlen''}\\
|
|
||||||
X: \underbrace{\left\{(i, j) : i, j \in \left\{1, \ldots
|
|
||||||
, 6\right\}\right\}}_{\Omega} \mapsto
|
|
||||||
\underbrace{\left\{2,3,
|
|
||||||
\ldots, 12\right\}}_{\in \mathbb{R}}
|
|
||||||
\end{gather*}\\
|
|
||||||
\vspace*{5mm}
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\draw[line width=1pt] (0,0) -- (18cm,0);
|
|
||||||
\end{tikzpicture}
|
|
||||||
\vspace*{2mm}
|
|
||||||
\begin{gather*}
|
|
||||||
A = \text{\normalfont``Die Summe der
|
|
||||||
Augenzahlen ist 4''}
|
|
||||||
\end{gather*}
|
|
||||||
\begin{minipage}[t]{0.5\textwidth}
|
|
||||||
\centering
|
|
||||||
Direkter Weg
|
|
||||||
\begin{align*}
|
|
||||||
P(A) &= P(\mleft\{ (1,3), (2,2),
|
|
||||||
(3,1) \mright\}) \\
|
|
||||||
&= P( (1,3)) + P( (2, 2)) + P( (3,1)) \\
|
|
||||||
&= 3\cdot \frac{1}{36} = \frac{1}{12}
|
|
||||||
\end{align*}
|
|
||||||
\end{minipage}%
|
|
||||||
\begin{minipage}[t]{0.5\textwidth}
|
|
||||||
\centering
|
|
||||||
Über ZV
|
|
||||||
\begin{gather*}
|
|
||||||
P(A) = P_X(4) = \cdots
|
|
||||||
\end{gather*}
|
|
||||||
\end{minipage}
|
|
||||||
\end{lightgrayhighlightbox}
|
|
||||||
\end{minipage}
|
|
||||||
\end{itemize}
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Verteilungen \& Verteilungsfunktionen}
|
|
||||||
|
|
||||||
\vspace*{-18mm}
|
|
||||||
|
|
||||||
\begin{itemize}
|
|
||||||
\item Verteilungsfunktionen diskreter ZV
|
|
||||||
\vspace*{-6mm}
|
|
||||||
\begin{columns}
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\begin{align*}
|
|
||||||
\overbrace{F_X(x)}^\text{Verteilungsfunktion} = P(X \le x)
|
|
||||||
&= \sum_{n:x_n \le x}
|
|
||||||
\overbrace{P_X(x)}^\text{Verteilung}\\
|
|
||||||
&= \sum_{n:x_n \le x} P(X=x)
|
|
||||||
\end{align*}
|
|
||||||
\begin{gather*}
|
|
||||||
P(a < X \le b) = F_X(b) - F_X(a)
|
|
||||||
\end{gather*}
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\begin{lightgrayhighlightbox}
|
|
||||||
Beispiel: Würfeln mit zwei Würfeln
|
|
||||||
\begin{gather*}
|
|
||||||
X := \text{\normalfont``Summe beider Augenzahlen''}
|
|
||||||
\end{gather*}
|
|
||||||
\vspace*{-10mm}
|
|
||||||
\begin{figure}[H]
|
|
||||||
\centering
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
xmin=2,xmax=12,
|
|
||||||
ymin=-0.2,ymax=1.2,
|
|
||||||
xlabel=$x$,
|
|
||||||
ylabel=$F_X(x)$,
|
|
||||||
width=12cm,
|
|
||||||
height=5cm,
|
|
||||||
]
|
|
||||||
\addplot+[mark=none, line width=1pt]
|
|
||||||
coordinates
|
|
||||||
{
|
|
||||||
(2 , 0.02777)
|
|
||||||
(3 , 0.02777)
|
|
||||||
(3 , 0.08333)
|
|
||||||
(4 , 0.08333)
|
|
||||||
(4 , 0.16666)
|
|
||||||
(5 , 0.16666)
|
|
||||||
(5 , 0.27777)
|
|
||||||
(6 , 0.27777)
|
|
||||||
(6 , 0.41666)
|
|
||||||
(7 , 0.41666)
|
|
||||||
(7 , 0.58333)
|
|
||||||
(8 , 0.58333)
|
|
||||||
(8 , 0.72222)
|
|
||||||
(9 , 0.72222)
|
|
||||||
(9 , 0.83333)
|
|
||||||
(10 , 0.83333)
|
|
||||||
(10, 0.91666)
|
|
||||||
(11, 0.91666)
|
|
||||||
(11, 0.97222)
|
|
||||||
(12, 0.97222)
|
|
||||||
(12, 1.00000)
|
|
||||||
};
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
\end{figure}
|
|
||||||
\vspace*{-10mm}
|
|
||||||
\end{lightgrayhighlightbox}
|
|
||||||
\end{columns}
|
|
||||||
\pause
|
|
||||||
\item Kenngrößen von Verteilungen
|
|
||||||
\vspace*{2mm}
|
|
||||||
\begin{columns}[t]
|
|
||||||
\column{\kittwocolumns}
|
|
||||||
\centering
|
|
||||||
\textbf{Erwartungswert}
|
|
||||||
\begin{gather*}
|
|
||||||
E(X) = \sum_{n=1}^{\infty} x_n P(X=x_n)
|
|
||||||
\end{gather*}%
|
|
||||||
\vspace*{-8mm}%
|
|
||||||
\begin{align*}
|
|
||||||
E(X + b) &= E(X) + b\\
|
|
||||||
E(X+Y) &= E(X) + E(Y)\\
|
|
||||||
E(aX) &= aE(X)
|
|
||||||
\end{align*}
|
|
||||||
\column{\kittwocolumns}
|
|
||||||
\centering
|
|
||||||
\textbf{Varianz}
|
|
||||||
\begin{gather*}
|
|
||||||
V(X) = E\left(\left(X - E(X)\right)^2\right)
|
|
||||||
\end{gather*}%
|
|
||||||
\vspace*{-8mm}
|
|
||||||
\begin{align*}
|
|
||||||
V(X) &= E(X^2) - \left(E(X)\right)^2\\
|
|
||||||
V(aX) &= a^2 V(x)\\
|
|
||||||
V(X+b) &= V(X)
|
|
||||||
\end{align*}
|
|
||||||
\column{\kittwocolumns}
|
|
||||||
\centering
|
|
||||||
\textbf{$p$-Quantil}
|
|
||||||
\begin{gather*}
|
|
||||||
x_p = \text{inf}\mleft\{ x\in \mathbb{R} : P(X
|
|
||||||
\le x) \ge p \mright\}
|
|
||||||
\end{gather*}
|
|
||||||
\vspace*{-8mm}
|
|
||||||
\begin{gather*}
|
|
||||||
p=0.5 \hspace{5mm} \rightarrow \hspace{5mm} x_p
|
|
||||||
\equiv \text{``Median''}
|
|
||||||
\end{gather*}
|
|
||||||
\end{columns}
|
|
||||||
\end{itemize}
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Beispiele von Verteilungen}
|
|
||||||
|
|
||||||
\vspace*{-18mm}
|
|
||||||
|
|
||||||
\begin{columns}[t]
|
|
||||||
\column{\kittwocolumns}
|
|
||||||
\centering
|
|
||||||
\textbf{Bernoulli Verteilung}\\
|
|
||||||
\vspace*{10mm}
|
|
||||||
$X$ kann nur die Werte $0$ oder $1$\\ annehmen
|
|
||||||
\rule{0.9\textwidth}{0.4pt}
|
|
||||||
\begin{gather*}
|
|
||||||
X \sim \text{Bernoulli}(p)
|
|
||||||
\end{gather*}
|
|
||||||
\begin{gather*}
|
|
||||||
P(X=0) = 1-p, \hspace{5mm} P(X=1) = p
|
|
||||||
\end{gather*}
|
|
||||||
\begin{align*}
|
|
||||||
E(X) &= p\\
|
|
||||||
V(X) &= p(1-p)
|
|
||||||
\end{align*}
|
|
||||||
\column{\kittwocolumns}
|
|
||||||
\centering
|
|
||||||
\textbf{Binomialverteilung}\\
|
|
||||||
\vspace*{10mm}
|
|
||||||
$X\equiv$ ``Zählen der Treffer bei $N$ unabhängigen Versuchen''
|
|
||||||
\rule{0.9\textwidth}{0.4pt}
|
|
||||||
\begin{gather*}
|
|
||||||
X \sim \text{Bin}(N,p)
|
|
||||||
\end{gather*}
|
|
||||||
\begin{gather*}
|
|
||||||
P_X(k) = \binom{N}{k} p^k (1-p)^{N-k}
|
|
||||||
\end{gather*}
|
|
||||||
\begin{align*}
|
|
||||||
E(X) &= Np\\
|
|
||||||
V(X) &= Np(1-p)
|
|
||||||
\end{align*}
|
|
||||||
\column{\kittwocolumns}
|
|
||||||
\centering
|
|
||||||
\textbf{Poisson Verteilung}\\
|
|
||||||
\vspace*{10mm}
|
|
||||||
Binomialverteilung für $N\rightarrow \infty$ mit
|
|
||||||
$pN=\text{const.}=: \lambda$
|
|
||||||
\rule{0.9\textwidth}{0.4pt}
|
|
||||||
\begin{gather*}
|
|
||||||
X \sim \text{Poisson}(\lambda)
|
|
||||||
\end{gather*}
|
|
||||||
\begin{gather*}
|
|
||||||
P_X(k) = \frac{\lambda^k}{k!}e^{-\lambda}
|
|
||||||
\end{gather*}
|
|
||||||
\begin{align*}
|
|
||||||
E(X) &= \lambda\\
|
|
||||||
V(X) &= \lambda
|
|
||||||
\end{align*}
|
|
||||||
\end{columns}
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Zusammenfassung}
|
|
||||||
|
|
||||||
\begin{columns}[t]
|
|
||||||
\column{\kittwocolumns}
|
|
||||||
\begin{greenblock}{Verteilungsfunktion (diskret)}
|
|
||||||
\vspace*{-6mm}
|
|
||||||
\begin{gather*}
|
|
||||||
F_X(x) = P(X \le x) = \sum_{n:x_n < x} P_X(x_n)
|
|
||||||
\end{gather*}
|
|
||||||
\end{greenblock}
|
|
||||||
\column{\kittwocolumns}
|
|
||||||
\begin{greenblock}{Erwartungswert}
|
|
||||||
\vspace*{-6mm}
|
|
||||||
\begin{gather*}
|
|
||||||
E(X) = \sum_{n=1}^{\infty} x_n P(X=x_n)
|
|
||||||
\end{gather*}%
|
|
||||||
\vspace*{-8mm}%
|
|
||||||
\begin{align*}
|
|
||||||
E(X + b) &= E(X) + b\\
|
|
||||||
E(X+Y) &= E(X) + E(Y)\\
|
|
||||||
E(aX) &= aE(X)
|
|
||||||
\end{align*}
|
|
||||||
\end{greenblock}
|
|
||||||
\column{\kittwocolumns}
|
|
||||||
\begin{greenblock}{Binomialverteilung}
|
|
||||||
\vspace*{-6mm}
|
|
||||||
\begin{gather*}
|
|
||||||
P_X(k) = \binom{N}{k} p^k (1-p)^{N-k}
|
|
||||||
\end{gather*}
|
|
||||||
\begin{align*}
|
|
||||||
E(X) &= Np\\
|
|
||||||
V(X) &= Np(1-p)
|
|
||||||
\end{align*}
|
|
||||||
\end{greenblock}
|
|
||||||
\end{columns}
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\subsection{Aufgabe}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Aufgabe 1: Diskrete Verteilungen}
|
|
||||||
|
|
||||||
\vspace*{-10mm}
|
|
||||||
|
|
||||||
Eine Polizistin führt $N = 6$ Radarkontrollen auf einer
|
|
||||||
Landstraße durch. Die Radarkontrollen
|
|
||||||
können als unabhängig angenommen werden und führen jeweils mit
|
|
||||||
der Wahrscheinlichkeit
|
|
||||||
$p = 0{,}2$ zu einem Strafzettel. Die diskrete Zufallsvariable $R :
|
|
||||||
\Omega \rightarrow \mathbb{R}$ beschreibt die Anzahl der
|
|
||||||
Strafzettel in $N = 6$ Kontrollen.
|
|
||||||
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{enumerate}[a{)}]
|
|
||||||
\item Geben Sie den Ergebnisraum $\Omega$ der diskreten Zufallsvariablen $R$ an
|
|
||||||
und bestimmen Sie deren Erwartungswert $E(R)$.
|
|
||||||
\item Wie groß ist die Wahrscheinlichkeit dafür, dass es bei $6$
|
|
||||||
Kontrollen genau $3$ Strafzettel gibt?
|
|
||||||
\item Skizzieren Sie die Verteilungsfunktion $F_R(r)$ der
|
|
||||||
Zufallsvariablen $R$.
|
|
||||||
\end{enumerate}
|
|
||||||
% tex-fmt: on
|
|
||||||
|
|
||||||
\vspace*{5mm}
|
|
||||||
|
|
||||||
\textit{Die folgenden Teilaufgaben können unabhängig von den
|
|
||||||
bisherigen Teilaufgaben bearbeitet werden.}
|
|
||||||
|
|
||||||
\vspace*{5mm}
|
|
||||||
|
|
||||||
Ein Autofahrer muss jeden Tag auf seinem Arbeitsweg über die
|
|
||||||
Landstraße und über die
|
|
||||||
Autobahn fahren. Die Wahrscheinlichkeit dafür, dass der
|
|
||||||
Autofahrer auf der Landstraße bzw.
|
|
||||||
auf der Autobahn zu schnell fährt und einen Strafzettel bekommt,
|
|
||||||
liegt bei $p_\text{L} = 0{,}2$ bzw. bei
|
|
||||||
$p_\text{A} = 0{,}3$.
|
|
||||||
|
|
||||||
\vspace*{5mm}
|
|
||||||
|
|
||||||
\textbf{Hinweis}: Es wird nur der einfache Weg (Hinweg) betrachtet.
|
|
||||||
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{enumerate}[a{)}]
|
|
||||||
\setcounter{enumi}{3}
|
|
||||||
\item Wie groß ist die Wahrscheinlichkeit dafür, dass der Autofahrer
|
|
||||||
an einem Tag $0$, $1$ oder $2$ Strafzettel bekommt?
|
|
||||||
\item Der Autofahrer fährt an $200$ unabhängigen Tagen im Jahr über
|
|
||||||
seinen Arbeitsweg zur Arbeit. Wie viele Strafzettel sammelt der
|
|
||||||
Autofahrer innerhalb eines Jahres?
|
|
||||||
\end{enumerate}
|
|
||||||
% tex-fmt: on
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Aufgabe 1: Diskrete Verteilungen}
|
|
||||||
|
|
||||||
\vspace*{-16mm}
|
|
||||||
|
|
||||||
Eine Polizistin führt $N = 6$ Radarkontrollen auf einer
|
|
||||||
Landstraße durch. Die Radarkontrollen
|
|
||||||
können als unabhängig angenommen werden und führen jeweils mit
|
|
||||||
der Wahrscheinlichkeit
|
|
||||||
$p = 0{,}2$ zu einem Strafzettel. Die diskrete Zufallsvariable $R :
|
|
||||||
\Omega \rightarrow \mathbb{R}$ beschreibt die Anzahl der
|
|
||||||
Strafzettel in $N = 6$ Kontrollen.
|
|
||||||
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{enumerate}[a{)}]
|
|
||||||
\item Geben Sie den Ergebnisraum $\Omega$ der diskreten Zufallsvariablen $R$ an
|
|
||||||
und bestimmen Sie deren Erwartungswert $E(R)$.
|
|
||||||
\pause\begin{gather*}
|
|
||||||
\Omega = \mleft\{ 0, 1\mright\}^6 \\
|
|
||||||
R \sim \text{Bin}(N=6, p=0{,}2)\hspace{5mm} \Rightarrow \hspace{5mm} E(R) = Np = 1{,}2
|
|
||||||
\end{gather*}
|
|
||||||
\vspace*{-10mm}\pause \item Wie groß ist die Wahrscheinlichkeit dafür, dass es bei $6$
|
|
||||||
Kontrollen genau $3$ Strafzettel gibt?
|
|
||||||
\pause \begin{gather*}
|
|
||||||
P(R=3) = \binom{N}{3}p^3 (1-p)^{N-3} = \binom{6}{3} \cdot 0{,}2^3\cdot 0{,}8^3 \approx 0{,}0819
|
|
||||||
\end{gather*}
|
|
||||||
\vspace*{-6mm}\pause \item Skizzieren Sie die Verteilungsfunktion $F_R(r)$ der
|
|
||||||
Zufallsvariablen $R$.
|
|
||||||
\end{enumerate}
|
|
||||||
% tex-fmt: on
|
|
||||||
|
|
||||||
\vspace*{2mm}
|
|
||||||
|
|
||||||
\pause
|
|
||||||
\begin{columns}
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\begin{gather*}
|
|
||||||
F_R(r) = \sum_{\widetilde{r} \le r}
|
|
||||||
\binom{N}{\widetilde{r}}p^{\widetilde{r}} (1-p)^{N-\widetilde{r}}
|
|
||||||
\end{gather*}
|
|
||||||
\begin{table}
|
|
||||||
\begin{tabular}{c|ccccccc}
|
|
||||||
$r$ & $0$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ \\ \hline
|
|
||||||
$F_R(r)$ & $0{,}262$ & $0{,}655$ & $0{,}901$ &
|
|
||||||
$0{,}983$ & $0{,}998$ & $0{,}999$ & $1$
|
|
||||||
\end{tabular}
|
|
||||||
\end{table}
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\begin{figure}[H]
|
|
||||||
\centering
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
xmin=0,xmax=6,
|
|
||||||
ymin=-0.2,ymax=1.2,
|
|
||||||
xlabel=$r$,
|
|
||||||
ylabel=$F_R(r)$,
|
|
||||||
width=12cm,
|
|
||||||
height=5cm,
|
|
||||||
]
|
|
||||||
\addplot+[mark=none, line width=1pt]
|
|
||||||
coordinates
|
|
||||||
{
|
|
||||||
(0,0.262)
|
|
||||||
(1,0.262)
|
|
||||||
(1,0.655)
|
|
||||||
(2,0.655)
|
|
||||||
(2,0.901)
|
|
||||||
(3,0.901)
|
|
||||||
(3,0.983)
|
|
||||||
(4,0.983)
|
|
||||||
(4,0.998)
|
|
||||||
(5,0.998)
|
|
||||||
(5,0.999)
|
|
||||||
(6,0.999)
|
|
||||||
(6,1)
|
|
||||||
};
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
\end{figure}
|
|
||||||
\end{columns}
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Aufgabe 1: Diskrete Verteilungen}
|
|
||||||
|
|
||||||
\vspace*{-16mm}
|
|
||||||
|
|
||||||
Ein Autofahrer muss jeden Tag auf seinem Arbeitsweg über die
|
|
||||||
Landstraße und über die Autobahn fahren. Die Wahrscheinlichkeit
|
|
||||||
dafür, dass der Autofahrer auf der Landstraße bzw. auf der
|
|
||||||
Autobahn zu schnell fährt und einen Strafzettel bekommt, liegt
|
|
||||||
bei $p_\text{L} = 0{,}2$ bzw. bei $p_\text{A} = 0{,}3$.
|
|
||||||
|
|
||||||
\vspace*{2mm}
|
|
||||||
|
|
||||||
\textbf{Hinweis}: Es wird nur der einfache Weg (Hinweg) betrachtet.
|
|
||||||
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{enumerate}[a{)}]
|
|
||||||
\setcounter{enumi}{3}
|
|
||||||
\item Wie groß ist die Wahrscheinlichkeit dafür, dass der Autofahrer
|
|
||||||
an einem Tag $0$, $1$ oder $2$ Strafzettel bekommt?
|
|
||||||
\pause\begin{gather*}
|
|
||||||
R := A + L
|
|
||||||
\end{gather*}%
|
|
||||||
\vspace*{-14mm}%
|
|
||||||
\begin{align*}
|
|
||||||
P(R = 0) &= P(A = 0 \text{ und } L = 0) &&\hspace{-24mm}= (1-p_A)(1-p_L) &&\hspace{-24mm}= 0{,}56\\
|
|
||||||
P(R = 1) &= P(A=1 \text{ und } L=0) + P(A=0 \text{ und } L=1) &&\hspace{-24mm}= p_A \cdot (1-p_L) + (1-p_A)\cdot p_L &&\hspace{-24mm}= 0{,}38 \\
|
|
||||||
P(R = 2) &= P(A=1 \text{ und } L=1) &&\hspace{-24mm}= p_A\cdot p_L &&\hspace{-24mm}= 0{,}06
|
|
||||||
\end{align*}
|
|
||||||
\vspace*{-10mm}\pause \item Der Autofahrer fährt an $200$ unabhängigen Tagen im Jahr über
|
|
||||||
seinen Arbeitsweg zur Arbeit. Wie viele Strafzettel sammelt der
|
|
||||||
Autofahrer innerhalb eines Jahres?
|
|
||||||
\end{enumerate}
|
|
||||||
% tex-fmt: on
|
|
||||||
|
|
||||||
\vspace*{-6mm}
|
|
||||||
|
|
||||||
\pause
|
|
||||||
\begin{minipage}{0.48\textwidth}
|
|
||||||
\centering
|
|
||||||
\begin{align*}
|
|
||||||
E\left(\sum_{n=1}^{200} R_n\right) &= \sum_{n=1}^{200}
|
|
||||||
E\left(R_n\right) = \sum_{n=1}^{200} \left[1\cdot0{,}38 +
|
|
||||||
2\cdot 0{,}06\right]\\[2mm]
|
|
||||||
&= 200\cdot 0{,}5 = 100
|
|
||||||
\end{align*}
|
|
||||||
\end{minipage}%
|
|
||||||
\begin{minipage}{0.06\textwidth}
|
|
||||||
\centering
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\draw[line width=1pt] (0,0) -- (0,4cm);
|
|
||||||
\end{tikzpicture}
|
|
||||||
\end{minipage}%
|
|
||||||
\begin{minipage}{0.48\textwidth}
|
|
||||||
\centering
|
|
||||||
\begin{align*}
|
|
||||||
E\left(\sum_{n=1}^{200} R_n\right) &=
|
|
||||||
E\Big(\overbrace{\sum_{n=1}^{200} A_n}^{\sim
|
|
||||||
\text{Bin}(N=200, p=0{,}3)} + \overbrace{\sum_{n=1}^{200}
|
|
||||||
L_n}^{\sim \text{Bin}(N=200, p=0{,}2)}\Big)\\[2mm]
|
|
||||||
&= E\left(\sum_{n=1}^{200} A_n\right) +
|
|
||||||
E\left(\sum_{n=1}^{200} L_n\right) \\[2mm]
|
|
||||||
&= 200\cdot 0{,}3 + 200 \cdot 0{,}2 = 100
|
|
||||||
\end{align*}
|
|
||||||
\end{minipage}
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\section{Aufgabe 2}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\subsection{Theorie Wiederholung}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Weitere Kenngrößen von Verteilungen}
|
|
||||||
|
|
||||||
\vspace*{-10mm}
|
|
||||||
|
|
||||||
\vspace*{10mm}
|
|
||||||
\begin{columns}[t]
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\centering
|
|
||||||
\textbf{$k$-tes Moment}
|
|
||||||
\begin{gather*}
|
|
||||||
E(X^k) = \sum_{n=1}^{\infty} x_n^k P(X=x_n)
|
|
||||||
\end{gather*}%
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\centering
|
|
||||||
\textbf{$k$-tes zentrales Moment}
|
|
||||||
\begin{gather*}
|
|
||||||
E\left( \left(X - E(X)\right)^k \right) =
|
|
||||||
\sum_{n=1}^{\infty} \left(x_n - E(X)\right)^k P(X=x_n)
|
|
||||||
\end{gather*}%
|
|
||||||
\end{columns}
|
|
||||||
\vspace*{20mm}
|
|
||||||
\pause
|
|
||||||
\begin{columns}[t]
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\centering
|
|
||||||
\textbf{Charakteristische Funktion (diskret)}
|
|
||||||
\begin{gather*}
|
|
||||||
\phi_X(s) = E(e^{jsX}) = \sum_{n=1}^{\infty}
|
|
||||||
e^{jsx_n} P(X=x_n)\\[5mm]
|
|
||||||
E(X^k) = \frac{\phi_X^{(k)}(0)}{j^k}
|
|
||||||
\end{gather*}
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\centering
|
|
||||||
\textbf{Erzeugende Funktion}
|
|
||||||
\begin{gather*}
|
|
||||||
\text{Voraussetzung:} \hspace{5mm} x \in \mathbb{N}_0\\[5mm]
|
|
||||||
\psi(z) = E(z^x) = \sum_{n=1}^{\infty} z^n P(x=n)\\[5mm]
|
|
||||||
P(X=n) = \frac{\psi_X^{(n)}(0)}{n!}
|
|
||||||
\end{gather*}
|
|
||||||
\end{columns}
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Zusammenfassung}
|
|
||||||
|
|
||||||
\vspace*{-16mm}
|
|
||||||
|
|
||||||
\begin{columns}[t]
|
|
||||||
\column{\kittwocolumns}
|
|
||||||
\begin{greenblock}{Verteilungsfunktion (diskret)}
|
|
||||||
\vspace*{-6mm}
|
|
||||||
\begin{gather*}
|
|
||||||
F_X(x) = P(X \le x) = \sum_{n:x_n < x} P_X(x_n)
|
|
||||||
\end{gather*}
|
|
||||||
\end{greenblock}
|
|
||||||
\column{\kittwocolumns}
|
|
||||||
\begin{greenblock}{Varianz}
|
|
||||||
\vspace*{-6mm}
|
|
||||||
\begin{gather*}
|
|
||||||
V(X) = E\left(\left(X - E(X)\right)^2\right)
|
|
||||||
\end{gather*}%
|
|
||||||
\vspace*{-8mm}
|
|
||||||
\begin{align*}
|
|
||||||
V(X) &= E(X^2) - \left(E(X)\right)^2\\
|
|
||||||
V(aX) &= a^2 V(x)\\
|
|
||||||
V(X+b) &= V(X)
|
|
||||||
\end{align*}
|
|
||||||
\end{greenblock}
|
|
||||||
\column{\kittwocolumns}
|
|
||||||
\begin{greenblock}{$p$-Quantil}
|
|
||||||
\vspace*{-6mm}
|
|
||||||
\begin{gather*}
|
|
||||||
x_p = \text{inf}\mleft\{ x\in \mathbb{R} : P(X
|
|
||||||
\le x) \ge p \mright\}
|
|
||||||
\end{gather*}
|
|
||||||
\vspace*{-8mm}
|
|
||||||
\begin{gather*}
|
|
||||||
p=0.5 \hspace{5mm} \rightarrow \hspace{5mm} x_p
|
|
||||||
\equiv \text{``Median''}
|
|
||||||
\end{gather*}
|
|
||||||
\end{greenblock}
|
|
||||||
\end{columns}
|
|
||||||
\begin{columns}[t]
|
|
||||||
\column{\kittwocolumns}
|
|
||||||
\begin{greenblock}{$k$-tes Moment}
|
|
||||||
\vspace*{-6mm}
|
|
||||||
\begin{gather*}
|
|
||||||
E(X^k) = \sum_{n=1}^{\infty} x_n^k P(X=x_n)
|
|
||||||
\end{gather*}%
|
|
||||||
\end{greenblock}
|
|
||||||
\column{\kittwocolumns}
|
|
||||||
\begin{greenblock}{Charakt. Funktion (diskret)}
|
|
||||||
\vspace*{-6mm}
|
|
||||||
\begin{gather*}
|
|
||||||
\phi_X(s) = \sum_{n=1}^{\infty}
|
|
||||||
e^{jsx_n} P(X=x_n)\\[5mm]
|
|
||||||
E(X^k) = \frac{\phi_X^{(k)}(0)}{j^k}
|
|
||||||
\end{gather*}
|
|
||||||
\end{greenblock}
|
|
||||||
\column{\kittwocolumns}
|
|
||||||
\begin{greenblock}{Erzeugende Funktion}
|
|
||||||
\vspace*{-6mm}
|
|
||||||
\begin{gather*}
|
|
||||||
\psi(z) = \sum_{n=1}^{\infty} z^n P(X=n)\\[5mm]
|
|
||||||
P(X=n) = \frac{\psi_X^{(n)}(0)}{n!}
|
|
||||||
\end{gather*}
|
|
||||||
\end{greenblock}
|
|
||||||
\end{columns}
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\subsection{Aufgabe}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Aufgabe 2: Erzeugende \& Charakteristische\\ Funktion}
|
|
||||||
|
|
||||||
Gegeben ist folgende Verteilungsfunktion $F_X(x)$:
|
|
||||||
|
|
||||||
\begin{figure}
|
|
||||||
\centering
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
xmin=0,xmax=5.5,
|
|
||||||
ymin=0,ymax=1,
|
|
||||||
xtick={0,...,5},
|
|
||||||
ytick={0,0.2,...,1},
|
|
||||||
xlabel=$x$,
|
|
||||||
ylabel=$F_X(x)$,
|
|
||||||
width=12cm,
|
|
||||||
height=5cm,
|
|
||||||
]
|
|
||||||
\addplot+[mark=none, line width=1pt]
|
|
||||||
coordinates
|
|
||||||
{
|
|
||||||
(0,0)
|
|
||||||
(1,0)
|
|
||||||
(1,0.2)
|
|
||||||
(2,0.2)
|
|
||||||
(2,0.6)
|
|
||||||
(3,0.6)
|
|
||||||
(3,0.7)
|
|
||||||
(4,0.7)
|
|
||||||
(4,0.9)
|
|
||||||
(5,0.9)
|
|
||||||
(5,1)
|
|
||||||
(5.5,1)
|
|
||||||
};
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
\end{figure}
|
|
||||||
|
|
||||||
\vspace*{-10mm}
|
|
||||||
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{enumerate}[a{)}]
|
|
||||||
\item Stellen Sie die Verteilung $P_X(x)$ graphisch dar.
|
|
||||||
\item Geben Sie die erzeugende Funktion $\psi_X(z)$ und die
|
|
||||||
charakteristische Funktion $\phi_X(s)$ an. Berechnen Sie mit mithilfe
|
|
||||||
von $\phi_X(s)$ die Varianz $V(X)$.
|
|
||||||
\item Vergleichen Sie den Median und den Erwartungswert von $X$. Sind
|
|
||||||
beide Kenngrößen gleich? Begründen Sie, welche Eigenschaft einer
|
|
||||||
diskreten Verteilung ausschlaggebend ist, damit beide Werte gleich
|
|
||||||
sind.
|
|
||||||
\end{enumerate}
|
|
||||||
% tex-fmt: on
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Aufgabe 2: Erzeugende \& Charakteristische\\ Funktion}
|
|
||||||
|
|
||||||
Gegeben ist folgende Verteilungsfunktion $F_X(x)$:
|
|
||||||
|
|
||||||
\begin{figure}
|
|
||||||
\centering
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
xmin=0,xmax=5.5,
|
|
||||||
ymin=0,ymax=1,
|
|
||||||
xtick={0,...,5},
|
|
||||||
ytick={0,0.2,...,1},
|
|
||||||
xlabel=$x$,
|
|
||||||
ylabel=$F_X(x)$,
|
|
||||||
width=12cm,
|
|
||||||
height=5cm,
|
|
||||||
]
|
|
||||||
\addplot+[mark=none, line width=1pt]
|
|
||||||
coordinates
|
|
||||||
{
|
|
||||||
(0,0)
|
|
||||||
(1,0)
|
|
||||||
(1,0.2)
|
|
||||||
(2,0.2)
|
|
||||||
(2,0.6)
|
|
||||||
(3,0.6)
|
|
||||||
(3,0.7)
|
|
||||||
(4,0.7)
|
|
||||||
(4,0.9)
|
|
||||||
(5,0.9)
|
|
||||||
(5,1)
|
|
||||||
(5.5,1)
|
|
||||||
};
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
\end{figure}
|
|
||||||
|
|
||||||
\vspace*{-10mm}
|
|
||||||
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{enumerate}[a{)}]
|
|
||||||
\item Stellen Sie die Verteilung $P_X(x)$ graphisch dar.
|
|
||||||
\end{enumerate}
|
|
||||||
% tex-fmt: on
|
|
||||||
|
|
||||||
\pause
|
|
||||||
\begin{figure}
|
|
||||||
\centering
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
xmin=0,xmax=5.5,
|
|
||||||
ymin=0,ymax=0.5,
|
|
||||||
xtick={0,...,5},
|
|
||||||
ytick={0,0.1,...,0.5},
|
|
||||||
xlabel=$x$,
|
|
||||||
ylabel=$P_X(x)$,
|
|
||||||
width=12cm,
|
|
||||||
height=5cm,
|
|
||||||
]
|
|
||||||
\addplot+[ycomb,mark=*, line width=1pt]
|
|
||||||
coordinates
|
|
||||||
{
|
|
||||||
(1,0.2)
|
|
||||||
(2,0.4)
|
|
||||||
(3,0.1)
|
|
||||||
(4,0.2)
|
|
||||||
(5,0.1)
|
|
||||||
};
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
\end{figure}
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Aufgabe 2: Erzeugende \& Charakteristische\\ Funktion}
|
|
||||||
|
|
||||||
\vspace*{-12mm}
|
|
||||||
|
|
||||||
\begin{figure}
|
|
||||||
\centering
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
xmin=0,xmax=5.5,
|
|
||||||
ymin=0,ymax=0.5,
|
|
||||||
xtick={0,...,5},
|
|
||||||
ytick={0,0.1,...,0.5},
|
|
||||||
xlabel=$x$,
|
|
||||||
ylabel=$P_X(x)$,
|
|
||||||
width=12cm,
|
|
||||||
height=5cm,
|
|
||||||
]
|
|
||||||
\addplot+[ycomb,mark=*, line width=1pt]
|
|
||||||
coordinates
|
|
||||||
{
|
|
||||||
(1,0.2)
|
|
||||||
(2,0.4)
|
|
||||||
(3,0.1)
|
|
||||||
(4,0.2)
|
|
||||||
(5,0.1)
|
|
||||||
};
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
\end{figure}
|
|
||||||
|
|
||||||
\vspace*{-5mm}
|
|
||||||
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{enumerate}[a{)}]
|
|
||||||
\setcounter{enumi}{1}
|
|
||||||
\item Geben Sie die erzeugende Funktion $\psi_X(z)$ und die
|
|
||||||
charakteristische Funktion $\phi_X(s)$ an. Berechnen Sie mit mithilfe
|
|
||||||
von $\phi_X(s)$ die Varianz $V(X)$.
|
|
||||||
\pause\begin{align*}
|
|
||||||
\psi_X(z) &= \sum_{n=1}^{5} z^n P(X=n) = 0{,}2z + 0{,}4z^2 + 0{,}1z^3
|
|
||||||
+ 0{,}2z^4 + 0{,}1z^5 \\
|
|
||||||
\phi_X(s) &= \sum_{n=1}^{5} e^{jsx_n}P(X=n) = 0{,}2e^{js}
|
|
||||||
+ 0{,}4e^{j2s} + 0{,}1e^{j3s} + 0{,}2e^{j4s} + 0{,}1e^{j5s}
|
|
||||||
\end{align*}
|
|
||||||
\pause\begin{gather*}
|
|
||||||
\left.\begin{array}{c}
|
|
||||||
V(X) = E(X^2) - \left(E(X)\right)^2\\[3mm]
|
|
||||||
E(X) = \displaystyle\frac{\phi_X'(0)}{j}
|
|
||||||
= \sum_{n=1}^{5} nP(X=n) = 2{,}6\\[5mm]
|
|
||||||
E(X^2) = \displaystyle\frac{\phi_X''(0)}{j^2}
|
|
||||||
= \sum_{n=1}^{5} n^2 P(X=n) = 8{,}4
|
|
||||||
\end{array}\right\} \Rightarrow V(X) = 8{,}4 - 2{,}6^2 = 1{,}64
|
|
||||||
\end{gather*}
|
|
||||||
\end{enumerate}
|
|
||||||
% tex-fmt: on
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Aufgabe 2: Erzeugende \& Charakteristische\\ Funktion}
|
|
||||||
|
|
||||||
\vspace*{-5mm}
|
|
||||||
|
|
||||||
\begin{figure}
|
|
||||||
\centering
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
xmin=0,xmax=5.5,
|
|
||||||
ymin=0,ymax=1,
|
|
||||||
xtick={0,...,5},
|
|
||||||
ytick={0,0.2,...,1},
|
|
||||||
xlabel=$x$,
|
|
||||||
ylabel=$F_X(x)$,
|
|
||||||
width=12cm,
|
|
||||||
height=5cm,
|
|
||||||
]
|
|
||||||
\addplot+[mark=none, line width=1pt]
|
|
||||||
coordinates
|
|
||||||
{
|
|
||||||
(0,0)
|
|
||||||
(1,0)
|
|
||||||
(1,0.2)
|
|
||||||
(2,0.2)
|
|
||||||
(2,0.6)
|
|
||||||
(3,0.6)
|
|
||||||
(3,0.7)
|
|
||||||
(4,0.7)
|
|
||||||
(4,0.9)
|
|
||||||
(5,0.9)
|
|
||||||
(5,1)
|
|
||||||
(5.5,1)
|
|
||||||
};
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
\end{figure}
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{enumerate}[a{)}]
|
|
||||||
\setcounter{enumi}{2}
|
|
||||||
\item Vergleichen Sie den Median und den Erwartungswert von $X$. Sind
|
|
||||||
beide Kenngrößen gleich? Begründen Sie, welche Eigenschaft einer
|
|
||||||
diskreten Verteilung ausschlaggebend ist, damit beide Werte gleich
|
|
||||||
sind.
|
|
||||||
\pause\begin{align*}
|
|
||||||
x_{1/2} &= \text{inf}\mleft\{ x\in \mathbb{R}: F_X(x) \ge 1/2 \mright\} = 2\\
|
|
||||||
E(X) &= 2{,}6
|
|
||||||
\end{align*}
|
|
||||||
|
|
||||||
\vspace*{5mm}
|
|
||||||
\centering
|
|
||||||
\pause\begin{minipage}{0.7\textwidth}
|
|
||||||
Median und Erwartungswert sind gleich (bei einer diskreten
|
|
||||||
Verteilung mit ganzzahligen Stützstellen), wenn die Verteilung
|
|
||||||
symmetrisch um denselben Punkt $c$ ist, d.h.,
|
|
||||||
\begin{gather*}
|
|
||||||
P(c+k) = P(c-k) \hspace*{5mm} \forall k\in \mathbb{Z}.
|
|
||||||
\end{gather*}
|
|
||||||
\end{minipage}
|
|
||||||
\end{enumerate}
|
|
||||||
% tex-fmt: on
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\end{document}
|
|
||||||
|
|
||||||
@@ -1,731 +0,0 @@
|
|||||||
\ifdefined\ishandout
|
|
||||||
\documentclass[de, handout]{CELbeamer}
|
|
||||||
\else
|
|
||||||
\documentclass[de]{CELbeamer}
|
|
||||||
\fi
|
|
||||||
|
|
||||||
%
|
|
||||||
%
|
|
||||||
% CEL Template
|
|
||||||
%
|
|
||||||
%
|
|
||||||
|
|
||||||
\newcommand{\templates}{preambles}
|
|
||||||
\input{\templates/packages.tex}
|
|
||||||
\input{\templates/macros.tex}
|
|
||||||
|
|
||||||
\grouplogo{CEL_logo.pdf}
|
|
||||||
|
|
||||||
\groupname{Communication Engineering Lab (CEL)}
|
|
||||||
\groupnamewidth{80mm}
|
|
||||||
|
|
||||||
\fundinglogos{}
|
|
||||||
|
|
||||||
%
|
|
||||||
%
|
|
||||||
% Custom commands
|
|
||||||
%
|
|
||||||
%
|
|
||||||
|
|
||||||
\input{lib/latex-common/common.tex}
|
|
||||||
\pgfplotsset{colorscheme/rocket}
|
|
||||||
|
|
||||||
\newcommand{\res}{src/2025-12-19/res}
|
|
||||||
|
|
||||||
% \tikzstyle{every node}=[font=\small]
|
|
||||||
% \captionsetup[sub]{font=small}
|
|
||||||
|
|
||||||
%
|
|
||||||
%
|
|
||||||
% Document setup
|
|
||||||
%
|
|
||||||
%
|
|
||||||
|
|
||||||
\usepackage{tikz}
|
|
||||||
\usepackage{tikz-3dplot}
|
|
||||||
\usetikzlibrary{spy, external, intersections, positioning}
|
|
||||||
%\tikzexternalize[prefix=build/]
|
|
||||||
|
|
||||||
\usepackage{pgfplots}
|
|
||||||
\pgfplotsset{compat=newest}
|
|
||||||
\usepgfplotslibrary{fillbetween}
|
|
||||||
|
|
||||||
\usepackage{enumerate}
|
|
||||||
\usepackage{listings}
|
|
||||||
\usepackage{subcaption}
|
|
||||||
\usepackage{bbm}
|
|
||||||
\usepackage{multirow}
|
|
||||||
|
|
||||||
\usepackage{xcolor}
|
|
||||||
|
|
||||||
\title{WT Tutorium 4}
|
|
||||||
\author[Tsouchlos]{Andreas Tsouchlos}
|
|
||||||
\date[]{19. Dezember 2025}
|
|
||||||
|
|
||||||
%
|
|
||||||
%
|
|
||||||
% Document body
|
|
||||||
%
|
|
||||||
%
|
|
||||||
|
|
||||||
\begin{document}
|
|
||||||
|
|
||||||
\begin{frame}[title white vertical, picture=images/IMG_7801-cut]
|
|
||||||
\titlepage
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\section{Aufgabe 1}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\subsection{Theorie Wiederholung}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Stetige Zufallsvariablen I}
|
|
||||||
|
|
||||||
\vspace*{-10mm}
|
|
||||||
|
|
||||||
\begin{lightgrayhighlightbox}
|
|
||||||
Erinnerung: Diskrete Zufallsvariablen
|
|
||||||
\begin{align*}
|
|
||||||
\text{\normalfont Verteilung: }& P_X(x) = P(X = x) \\
|
|
||||||
\text{\normalfont Verteilungsfunktion: }& F_X(x) = P(X \le x) =
|
|
||||||
\sum_{n: x_n \le y} P_X(x)
|
|
||||||
\end{align*}
|
|
||||||
\vspace{-10mm}
|
|
||||||
\end{lightgrayhighlightbox}
|
|
||||||
|
|
||||||
\begin{columns}[t]
|
|
||||||
\pause\column{\kitthreecolumns}
|
|
||||||
\centering
|
|
||||||
\begin{itemize}
|
|
||||||
\item Verteilungsfunktion $F_X(x)$ einer stetigen ZV
|
|
||||||
\begin{gather*}
|
|
||||||
F_X(x) = P(X \le x)
|
|
||||||
\end{gather*}
|
|
||||||
\end{itemize}
|
|
||||||
\pause\column{\kitthreecolumns}
|
|
||||||
\centering
|
|
||||||
\begin{itemize}
|
|
||||||
\item Wahrscheinlichkeitsdichte $f_X(x)$ einer stetigen ZV
|
|
||||||
\begin{gather*}
|
|
||||||
F_X(x) = \int_{-\infty}^{x} f_X(u) du
|
|
||||||
\end{gather*}
|
|
||||||
\end{itemize}
|
|
||||||
\end{columns}
|
|
||||||
\begin{columns}[t]
|
|
||||||
\pause \column{\kitthreecolumns}
|
|
||||||
\centering
|
|
||||||
\begin{gather*}
|
|
||||||
\text{Eigenschaften:} \\[3mm]
|
|
||||||
\lim_{x\rightarrow -\infty} F_X(x) = 0 \\
|
|
||||||
\lim_{x\rightarrow +\infty} F_X(x) = 1 \\
|
|
||||||
x_1 \le x_2 \Rightarrow F_X(x_1) \le F_X(x_2)\\
|
|
||||||
F_X(x+) = \lim_{h\rightarrow 0^+} F_X (x+h) = F_X(x)
|
|
||||||
\end{gather*}
|
|
||||||
\pause \column{\kitthreecolumns}
|
|
||||||
\centering
|
|
||||||
\begin{gather*}
|
|
||||||
\text{Eigenschaften:} \\[3mm]
|
|
||||||
f_X(x) \ge 0 \\
|
|
||||||
\int_{-\infty}^{\infty} f_X(x) dx = 1
|
|
||||||
\end{gather*}
|
|
||||||
\end{columns}
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Stetige Zufallsvariablen II}
|
|
||||||
|
|
||||||
\begin{minipage}{0.6\textwidth}
|
|
||||||
\begin{itemize}
|
|
||||||
\item Wichtige Kenngrößen
|
|
||||||
\begin{align*}
|
|
||||||
\begin{array}{rlr}
|
|
||||||
\text{Erwartungswert: } \hspace{5mm} & E(X) =
|
|
||||||
\displaystyle\int_{-\infty}^{\infty} x f_X(x) dx
|
|
||||||
& \hspace{5mm} \big( = \mu \big) \\[3mm]
|
|
||||||
\text{Varianz: } \hspace{5mm} & V(X) = E\mleft(
|
|
||||||
\mleft( X - E(X) \mright)^2 \mright) \\[3mm]
|
|
||||||
\text{Standardabweichung: } \hspace{5mm} &
|
|
||||||
\sqrt{V(X)} & \hspace{5mm} \big( = \sigma \big)
|
|
||||||
\end{array}
|
|
||||||
\end{align*}
|
|
||||||
\end{itemize}
|
|
||||||
\end{minipage}
|
|
||||||
\begin{minipage}{0.38\textwidth}
|
|
||||||
\begin{lightgrayhighlightbox}
|
|
||||||
Erinnerung: Diskrete Zufallsvariablen
|
|
||||||
\begin{align*}
|
|
||||||
\text{\normalfont Erwartungswert: }& E(X) =
|
|
||||||
\sum_{n=1}^{\infty} x_n P_X(x) \\
|
|
||||||
\text{\normalfont Varianz: }& V(X) = E\mleft( \mleft(
|
|
||||||
X - E(X) \mright)^2 \mright)
|
|
||||||
\end{align*}
|
|
||||||
\end{lightgrayhighlightbox}
|
|
||||||
\end{minipage}
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Zusammenfassung}
|
|
||||||
|
|
||||||
\begin{columns}[t]
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\centering
|
|
||||||
\begin{greenblock}{Verteilungsfunktion (stetige ZV)}
|
|
||||||
\vspace*{-6mm}
|
|
||||||
\begin{gather*}
|
|
||||||
F_X(x) = P(X \le x)\\[4mm]
|
|
||||||
P(a < X \le b) = F_X(b) - F_X(a) \\[8mm]
|
|
||||||
\lim_{x\rightarrow -\infty} F_X(x) = 0 \\
|
|
||||||
\lim_{x\rightarrow +\infty} F_X(x) = 1 \\
|
|
||||||
x_1 \le x_2 \Rightarrow F_X(x_1) \le F_X(x_2)\\
|
|
||||||
F_X(x+) = \lim_{h\rightarrow 0^+} F_X (x+h) = F_X(x)
|
|
||||||
\end{gather*}
|
|
||||||
\end{greenblock}
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\centering
|
|
||||||
\begin{greenblock}{Wahrscheinlichkeitsdichte \phantom{()}}
|
|
||||||
\vspace*{-6mm}
|
|
||||||
\begin{gather*}
|
|
||||||
F_X(x) = \int_{-\infty}^{x} f_X(u) du \\[5mm]
|
|
||||||
f_X(x) \ge 0 \\
|
|
||||||
\int_{-\infty}^{\infty} f_X(x) dx = 1
|
|
||||||
\end{gather*}
|
|
||||||
\end{greenblock}
|
|
||||||
\end{columns}
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\subsection{Aufgabe}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Aufgabe 1: Stetige Verteilungen}
|
|
||||||
|
|
||||||
Die Zufallsvariable X besitze die Dichte
|
|
||||||
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{align*}
|
|
||||||
f_X (x) = \left\{
|
|
||||||
\begin{array}{ll}
|
|
||||||
C \cdot x e^{-ax^2}, & x \ge 0 \\
|
|
||||||
0, &\text{sonst}
|
|
||||||
\end{array}
|
|
||||||
\right.
|
|
||||||
\end{align*}
|
|
||||||
% tex-fmt: on
|
|
||||||
|
|
||||||
mit dem Parameter $a > 0$.
|
|
||||||
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{enumerate}[a{)}]
|
|
||||||
\item Bestimmen Sie den Koeffizienten $C$, sodass $f_X(x)$ eine
|
|
||||||
Wahrscheinlichkeitsdichte ist. Welche Eigenschaften muss eine
|
|
||||||
\textbf{Wahrscheinlichkeitsdichte} erfüllen? Skizzieren Sie
|
|
||||||
$f_X (x)$ für $a = 0{,}5$.
|
|
||||||
\item Welche Eigenschaften muss eine \textbf{Verteilungsfunktion}
|
|
||||||
erfüllen?
|
|
||||||
\item Berechnen und skizzieren Sie die Verteilungsfunktion $F_X (x)$.
|
|
||||||
\item Welche Wahrscheinlichkeit hat das Ereignis
|
|
||||||
$\{\omega : 1 < X(\omega) \le 2\}$?
|
|
||||||
\end{enumerate}
|
|
||||||
% tex-fmt: on
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Aufgabe 1: Stetige Verteilungen}
|
|
||||||
|
|
||||||
\vspace*{-15mm}
|
|
||||||
|
|
||||||
Die Zufallsvariable X besitze die Dichte
|
|
||||||
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{align*}
|
|
||||||
f_X (x) = \left\{
|
|
||||||
\begin{array}{ll}
|
|
||||||
C \cdot x e^{-ax^2}, & x \ge 0 \\
|
|
||||||
0, &\text{sonst}
|
|
||||||
\end{array}
|
|
||||||
\right.
|
|
||||||
\end{align*}
|
|
||||||
% tex-fmt: on
|
|
||||||
|
|
||||||
mit dem Parameter $a > 0$.
|
|
||||||
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{enumerate}[a{)}]
|
|
||||||
\item Bestimmen Sie den Koeffizienten $C$, sodass $f_X(x)$ eine
|
|
||||||
Wahrscheinlichkeitsdichte ist. Welche Eigenschaften muss eine
|
|
||||||
\textbf{Wahrscheinlichkeitsdichte} erfüllen? Skizzieren Sie
|
|
||||||
$f_X (x)$ für $a = 0{,}5$.
|
|
||||||
\pause\begin{columns}
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\begin{align*}
|
|
||||||
\text{Eigenschaften:} \hspace{5mm}
|
|
||||||
\left\{
|
|
||||||
\begin{array}{rl}
|
|
||||||
f_X(x) &\ge 0 \\[3mm]
|
|
||||||
\displaystyle\int_{-\infty}^{\infty} f_X(x) dx &= 1
|
|
||||||
\end{array}
|
|
||||||
\right.
|
|
||||||
\end{align*}
|
|
||||||
\pause\begin{gather*}
|
|
||||||
\int_{-\infty}^{\infty} f_X(x) dx
|
|
||||||
= \int_{0}^{\infty} C\cdot x e^{-ax^2} dx
|
|
||||||
= \frac{C}{-2a} \int_{0}^{\infty} (-2ax) e^{-ax^2} dx \\
|
|
||||||
= \frac{C}{-2a} \int_{0}^{\infty} (e^{-ax^2})' dx
|
|
||||||
= \frac{C}{-2a} \mleft[ e^{-ax^2} \mright]_0^{\infty} \overset{!}{=} 1 \hspace{10mm} \Rightarrow C = 2a
|
|
||||||
\end{gather*}
|
|
||||||
\centering
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\pause \begin{align*}
|
|
||||||
f_X(x) =
|
|
||||||
\left\{
|
|
||||||
\begin{array}{ll}
|
|
||||||
2ax \cdot e^{-ax^2}, & x\ge 0\\
|
|
||||||
0, & \text{sonst}
|
|
||||||
\end{array}
|
|
||||||
\right.
|
|
||||||
\end{align*}
|
|
||||||
\begin{figure}[H]
|
|
||||||
\centering
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
domain=0:5,
|
|
||||||
width=12cm,
|
|
||||||
height=5cm,
|
|
||||||
samples=100,
|
|
||||||
xlabel={$x$},
|
|
||||||
ylabel={$f_X(x)$},
|
|
||||||
]
|
|
||||||
\addplot+[mark=none, line width=1pt]
|
|
||||||
{x * exp(-0.5*x*x)};
|
|
||||||
% {x *exp(-a*x*x)};
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
\end{figure}
|
|
||||||
\end{columns}
|
|
||||||
\end{enumerate}
|
|
||||||
% tex-fmt: on
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Aufgabe 1: Stetige Verteilungen}
|
|
||||||
|
|
||||||
\vspace*{-20mm}
|
|
||||||
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{enumerate}[a{)}]
|
|
||||||
\setcounter{enumi}{1}
|
|
||||||
\item Welche Eigenschaften muss eine \textbf{Verteilungsfunktion}
|
|
||||||
erfüllen?
|
|
||||||
\pause\vspace{-10mm}\begin{columns}[t]
|
|
||||||
\column{\kitonecolumn}
|
|
||||||
\column{\kittwocolumns}
|
|
||||||
\centering
|
|
||||||
\begin{gather*}
|
|
||||||
\lim_{x\rightarrow -\infty} F_X(x) = 0\\
|
|
||||||
\lim_{x\rightarrow +\infty} F_X(x) = 1
|
|
||||||
\end{gather*}
|
|
||||||
\column{\kittwocolumns}
|
|
||||||
\centering
|
|
||||||
\begin{gather*}
|
|
||||||
x_1 \le x_2 \Rightarrow F_X(x_1) \le F_X(x_2) \\
|
|
||||||
F_X(x+) = \lim_{h\rightarrow 0^+} F_X (x+h) = F_X(x)
|
|
||||||
\end{gather*}
|
|
||||||
\column{\kitonecolumn}
|
|
||||||
\end{columns}
|
|
||||||
\pause\item Berechnen und skizzieren Sie die Verteilungsfunktion $F_X (x)$.
|
|
||||||
\begin{gather*}
|
|
||||||
f_X(x) = 2ax\cdot e^{-ax^2}, \hspace{5mm} x\ge 0
|
|
||||||
\end{gather*}
|
|
||||||
\pause \vspace*{-6mm}\begin{gather*}
|
|
||||||
F_X(x) = \int_{-\infty}^{x} f_X(u) du
|
|
||||||
= \left\{ \begin{array}{ll}
|
|
||||||
\displaystyle\int_{0}^{x} 2au\cdot e^{-au^2} du, & x\ge 0 \\
|
|
||||||
0, & x < 0
|
|
||||||
\end{array} \right.
|
|
||||||
\hspace{5mm} = \left\{ \begin{array}{ll}
|
|
||||||
\mleft[ -e^{-au^2} \mright]_0^{x}, & x\ge 0 \\
|
|
||||||
0, & x < 0
|
|
||||||
\end{array} \right.
|
|
||||||
\hspace{5mm} = \left\{ \begin{array}{ll}
|
|
||||||
1 - e^{-ax^2}, & x\ge 0\\
|
|
||||||
0, & x < 0
|
|
||||||
\end{array} \right.
|
|
||||||
\end{gather*}
|
|
||||||
\pause\begin{figure}[H]
|
|
||||||
\centering
|
|
||||||
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
domain=0:5,
|
|
||||||
width=14cm,
|
|
||||||
height=5cm,
|
|
||||||
xlabel={$x$},
|
|
||||||
ylabel={$F_X(x)$},
|
|
||||||
]
|
|
||||||
\addplot+[mark=none, line width=1pt]
|
|
||||||
{1 - exp(-0.5 * x*x)};
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
\end{figure}
|
|
||||||
\vspace*{-3mm}
|
|
||||||
\pause\item Welche Wahrscheinlichkeit hat das Ereignis
|
|
||||||
$\{\omega : 1 < X(\omega) \le 2\}$?
|
|
||||||
\pause \begin{gather*}
|
|
||||||
P(\mleft\{ \omega: 1 < X(\omega) \le 2 \mright\})
|
|
||||||
= P(1 < X \le 2) = F_X(2) - F_X(1) = e^{-a} - e^{-4a}
|
|
||||||
\end{gather*}
|
|
||||||
\end{enumerate}
|
|
||||||
% tex-fmt: on
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\section{Aufgabe 2}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\subsection{Theorie Wiederholung}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Die Normalverteilung}
|
|
||||||
|
|
||||||
\begin{columns}
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\centering
|
|
||||||
\begin{gather*}
|
|
||||||
X \sim \mathcal{N}\mleft( \mu, \sigma^2 \mright)
|
|
||||||
\end{gather*}%
|
|
||||||
\vspace{0mm}
|
|
||||||
\begin{align*}
|
|
||||||
f_X(x) &= \frac{1}{\sqrt{2\pi \sigma^2}} \exp\left(\frac{(x -
|
|
||||||
\mu)^2}{2 \sigma^2} \right) \\[2mm]
|
|
||||||
F_X(x) &=
|
|
||||||
\vcenter{\hbox{\scalebox{1.5}[2.6]{\vspace*{3mm}$\displaystyle\int$}}}_{\hspace{-0.5em}-\infty}^{\,x}
|
|
||||||
\frac{1}{\sqrt{2\pi
|
|
||||||
\sigma^2}} \exp\left(\frac{(u - \mu)^2}{2 \sigma^2} \right) du
|
|
||||||
\end{align*}
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\centering
|
|
||||||
\begin{figure}[H]
|
|
||||||
\centering
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
domain=-4:4,
|
|
||||||
xmin=-4,xmax=4,
|
|
||||||
width=15cm,
|
|
||||||
height=5cm,
|
|
||||||
samples=200,
|
|
||||||
xlabel={$x$},
|
|
||||||
ylabel={$f_X(x)$},
|
|
||||||
xtick={0},
|
|
||||||
xticklabels={\textcolor{KITblue}{$\mu$}},
|
|
||||||
ytick={0},
|
|
||||||
]
|
|
||||||
\addplot+[mark=none, line width=1pt]
|
|
||||||
{(1 / sqrt(2*pi)) * exp(-x*x)};
|
|
||||||
|
|
||||||
\addplot+ [KITblue, mark=none, line width=1pt]
|
|
||||||
coordinates {(-0.5, 0.15) (0.5, 0.15)};
|
|
||||||
\addplot+ [KITblue, mark=none, line width=1pt]
|
|
||||||
coordinates {(-0.5, 0.12) (-0.5, 0.18)};
|
|
||||||
\addplot+ [KITblue, mark=none, line width=1pt]
|
|
||||||
coordinates {(0.5, 0.12) (0.5, 0.18)};
|
|
||||||
\node[KITblue] at (axis cs: 0, 0.2) {$\sigma$};
|
|
||||||
|
|
||||||
% \addplot +[scol2, mark=none, line width=1pt]
|
|
||||||
% coordinates {(4.8, -1) (4.8, 2)};
|
|
||||||
% \addplot +[scol2, mark=none, line width=1pt]
|
|
||||||
% coordinates {(5.2, -1) (5.2, 2)};
|
|
||||||
% \node at (axis cs: 4.8, 3) {$S(1-\delta)$};
|
|
||||||
% \node at (axis cs: 5.2, 3) {$S(1+\delta)$};
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
\end{figure}
|
|
||||||
\begin{figure}[H]
|
|
||||||
\centering
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
domain=-4:4,
|
|
||||||
xmin=-4,xmax=4,
|
|
||||||
width=15cm,
|
|
||||||
height=5cm,
|
|
||||||
samples=200,
|
|
||||||
xlabel={$x$},
|
|
||||||
ylabel={$F_X(x)$},
|
|
||||||
xtick=\empty,
|
|
||||||
ytick={0, 1},
|
|
||||||
]
|
|
||||||
\addplot+[mark=none, line width=1pt]
|
|
||||||
{1 / (1 + exp(-(1.526*x*(1 + 0.1034*x))))};
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
\end{figure}
|
|
||||||
\end{columns}
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Rechnen mithilfe der Standardnormalverteilung}
|
|
||||||
|
|
||||||
\vspace*{-15mm}
|
|
||||||
|
|
||||||
\begin{itemize}
|
|
||||||
\item Die Standardnormalverteilung
|
|
||||||
\end{itemize}
|
|
||||||
|
|
||||||
\begin{minipage}{0.48\textwidth}
|
|
||||||
\centering
|
|
||||||
\begin{gather*}
|
|
||||||
X \sim \mathcal{N} (0,1) \\[4mm]
|
|
||||||
\Phi(x) := F_X(x) = P(X \le x) \\
|
|
||||||
\Phi(-x) = 1 - \Phi(x)
|
|
||||||
\end{gather*}
|
|
||||||
\end{minipage}%
|
|
||||||
\begin{minipage}{0.48\textwidth}
|
|
||||||
\centering
|
|
||||||
\begin{tabular}{|c|c||c|c||c|c|}
|
|
||||||
\hline
|
|
||||||
$x$ & $\Phi(x)$ & $x$ & $\Phi(x)$ & $x$ & $\Phi(x)$ \\
|
|
||||||
\hline
|
|
||||||
\hline
|
|
||||||
$0{,}00$ & $0{,}500000$ & $0{,}10$ & $0{,}539828$ &
|
|
||||||
$0{,}20$ & $0{,}579260$ \\
|
|
||||||
$0{,}02$ & $0{,}507978$ & $0{,}12$ & $0{,}547758$ &
|
|
||||||
$0{,}22$ & $0{,}587064$ \\
|
|
||||||
$0{,}04$ & $0{,}515953$ & $0{,}14$ & $0{,}555670$ &
|
|
||||||
$0{,}24$ & $0{,}594835$ \\
|
|
||||||
$0{,}06$ & $0{,}523922$ & $0{,}16$ & $0{,}563559$ &
|
|
||||||
$0{,}26$ & $0{,}602568$ \\
|
|
||||||
$0{,}08$ & $0{,}531881$ & $0{,}18$ & $0{,}571424$ &
|
|
||||||
$0{,}28$ & $0{,}610261$ \\
|
|
||||||
\hline
|
|
||||||
\end{tabular}\\
|
|
||||||
\end{minipage}
|
|
||||||
|
|
||||||
\pause
|
|
||||||
\begin{itemize}
|
|
||||||
\item Standardisierte ZV
|
|
||||||
\begin{gather*}
|
|
||||||
\begin{array}{cc}
|
|
||||||
E(X) &= 0 \\
|
|
||||||
V(X) &= 1
|
|
||||||
\end{array}
|
|
||||||
\hspace{45mm}
|
|
||||||
\text{Standardisierung: } \hspace{5mm}
|
|
||||||
\widetilde{X} = \frac{X - E(X)}{\sqrt{V(X)}}
|
|
||||||
= \frac{X - \mu}{\sigma}
|
|
||||||
\end{gather*}
|
|
||||||
\end{itemize}
|
|
||||||
|
|
||||||
\vspace*{3mm}
|
|
||||||
|
|
||||||
\pause
|
|
||||||
\begin{lightgrayhighlightbox}
|
|
||||||
Rechenbeispiel
|
|
||||||
\begin{gather*}
|
|
||||||
X \sim \mathcal{N}(\mu = 1, \sigma^2 = 0{,}5^2) \\[2mm]
|
|
||||||
P\left(X \le 1{,}12 \right)
|
|
||||||
= P\left(\frac{X - 1}{0{,}5} \le \frac{1{,}12 - 1}{0{,}5}\right)
|
|
||||||
= P\big(\underbrace{\widetilde{X}}_{\sim
|
|
||||||
\mathcal{N}(0,1)} \le 0{,}24\big)
|
|
||||||
= \Phi\left(0{,}24\right) = 0{,}594835
|
|
||||||
\end{gather*}
|
|
||||||
\end{lightgrayhighlightbox}
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Zusammenfassung}
|
|
||||||
|
|
||||||
\vspace*{-15mm}
|
|
||||||
|
|
||||||
\begin{columns}[t]
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\centering
|
|
||||||
\begin{greenblock}{Standardnormalverteilung}
|
|
||||||
\vspace*{-10mm}
|
|
||||||
\begin{gather*}
|
|
||||||
X \sim \mathcal{N} (0,1) \\[4mm]
|
|
||||||
\Phi(x) := F_X(x) = P(X \le x) \\
|
|
||||||
\Phi(-x) = 1 - \Phi(x)
|
|
||||||
\end{gather*}
|
|
||||||
\end{greenblock}
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\centering
|
|
||||||
\begin{greenblock}{Standardisierung}
|
|
||||||
\vspace*{-10mm}
|
|
||||||
\begin{gather*}
|
|
||||||
\widetilde{X} = \frac{X - E(X)}{\sqrt{V(X)}}
|
|
||||||
= \frac{X - \mu}{\sigma}
|
|
||||||
\end{gather*}
|
|
||||||
\end{greenblock}
|
|
||||||
\end{columns}
|
|
||||||
|
|
||||||
\vspace{5mm}
|
|
||||||
|
|
||||||
\begin{table}
|
|
||||||
\centering
|
|
||||||
% \cdots
|
|
||||||
\begin{tabular}{|c|c||c|c||c|c||c|c|}
|
|
||||||
\hline
|
|
||||||
$x$ & $\Phi(x)$ & $x$ & $\Phi(x)$ & $x$ & $\Phi(x)$ & $x$
|
|
||||||
& $\Phi(x)$ \\
|
|
||||||
\hline
|
|
||||||
\hline
|
|
||||||
$1{,}40$ & $0{,}919243$ & $2{,}80$ & $0{,}997445$ &
|
|
||||||
$3{,}00$ & $0{,}998650$ & $4{,}20$ & $0{,}999987$ \\
|
|
||||||
$1{,}42$ & $0{,}922196$ & $2{,}82$ & $0{,}997599$ &
|
|
||||||
$3{,}02$ & $0{,}998736$ & $4{,}22$ & $0{,}999988$ \\
|
|
||||||
$1{,}44$ & $0{,}925066$ & $2{,}84$ & $0{,}997744$ &
|
|
||||||
$3{,}04$ & $0{,}998817$ & $4{,}24$ & $0{,}999989$ \\
|
|
||||||
$1{,}46$ & $0{,}927855$ & $2{,}86$ & $0{,}997882$ &
|
|
||||||
$3{,}06$ & $0{,}998893$ & $4{,}26$ & $0{,}999990$ \\
|
|
||||||
$1{,}48$ & $0{,}930563$ & $2{,}88$ & $0{,}998012$ &
|
|
||||||
$3{,}08$ & $0{,}998965$ & $4{,}28$ & $0{,}999991$ \\
|
|
||||||
\hline
|
|
||||||
\end{tabular}
|
|
||||||
% \cdots
|
|
||||||
\end{table}
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\subsection{Aufgabe}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Aufgabe 2: Normalverteilung}
|
|
||||||
|
|
||||||
In einem Produktionsprozess werden Ladegeräte für Mobiltelefone
|
|
||||||
hergestellt. Bevor die Ladegeräte mit den Mobiltelefonen zusammen
|
|
||||||
verpackt werden, wird die Ladespannung von jedem Ladegerät einmal
|
|
||||||
gemessen. Die Messwerte der Ladespannungen der verschiedenen
|
|
||||||
Ladegeräte genüge näherungsweise einer normalverteilten
|
|
||||||
Zufallsvariablen mit $\mu = 5$ Volt und $\sigma = 0,07$ Volt. Alle
|
|
||||||
Ladegeräte, bei denen die Messung um mehr als $4$ \% vom Sollwert
|
|
||||||
$S = 5$ Volt abweicht, sollen aussortiert werden.
|
|
||||||
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{enumerate}[a{)}]
|
|
||||||
\item Wie viel Prozent der Ladegeräte werden aussortiert?
|
|
||||||
\item Der Hersteller möchte seinen Produktionsprozess so verbessern,
|
|
||||||
dass nur noch halb so viele Ladegeräte wie in a) aussortiert
|
|
||||||
werden. Auf welchen Wert müsste er dazu $\sigma$ senken?
|
|
||||||
\item Durch einen Produktionsfehler verschiebt sich der Mittelwert
|
|
||||||
$\mu$ auf $5{,}1$ Volt ($\sigma$ ist $0{,}07$ Volt). Wie groß ist
|
|
||||||
jetzt der Prozentsatz, der aussortiert wird?
|
|
||||||
\end{enumerate}
|
|
||||||
% tex-fmt: on
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Aufgabe 2: Normalverteilung}
|
|
||||||
|
|
||||||
\vspace*{-10mm}
|
|
||||||
|
|
||||||
In einem Produktionsprozess werden Ladegeräte für Mobiltelefone
|
|
||||||
hergestellt. Bevor die Ladegeräte mit den Mobiltelefonen zusammen
|
|
||||||
verpackt werden, wird die Ladespannung von jedem Ladegerät einmal
|
|
||||||
gemessen. Die Messwerte der Ladespannungen der verschiedenen
|
|
||||||
Ladegeräte genüge näherungsweise einer normalverteilten
|
|
||||||
Zufallsvariablen mit $\mu = 5$ Volt und $\sigma = 0,07$ Volt. Alle
|
|
||||||
Ladegeräte, bei denen die Messung um mehr als $4$ \% vom Sollwert
|
|
||||||
$S = 5$ Volt abweicht, sollen aussortiert werden.
|
|
||||||
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{enumerate}[a{)}]
|
|
||||||
\item Wie viel Prozent der Ladegeräte werden aussortiert?
|
|
||||||
\begin{columns}[c]
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\centering
|
|
||||||
\pause \begin{gather*}
|
|
||||||
X \sim \mathcal{N} \mleft( \mu = 0{,}5, \sigma^2 = 0{,}07^2 \mright)
|
|
||||||
\end{gather*}
|
|
||||||
\begin{align*}
|
|
||||||
P(E_\text{a}) &= P \Big( \big( X < S(1-\delta) \big)
|
|
||||||
\cup \big( X > S(1 + \delta) \big) \Big) \\
|
|
||||||
&= P(X < S(1 - \delta)) + P(X > S(1 + \delta)) \\[2mm]
|
|
||||||
&\overset{\widetilde{X} := \frac{X - \mu}{\sigma} }{=\joinrel=\joinrel=\joinrel=} P\left(\widetilde{X} < \frac{S(1 - \delta) - \mu}{\sigma}\right)
|
|
||||||
+ P\left(\widetilde{X} > \frac{S(1 + \delta) - \mu}{\sigma}\right) \\[2mm]
|
|
||||||
&\approx \Phi(-2{,}86) + \left(1 - \Phi(2{,}86)\right) \\
|
|
||||||
&= 2 - 2\Phi(2{,}86) \approx 0{,}424\text{\%}
|
|
||||||
\end{align*}
|
|
||||||
\column{\kitthreecolumns}
|
|
||||||
\centering
|
|
||||||
\pause\begin{figure}[H]
|
|
||||||
\centering
|
|
||||||
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
domain=4.6:5.3,
|
|
||||||
xmin=4.7, xmax=5.3,
|
|
||||||
width=14cm,
|
|
||||||
height=6cm,
|
|
||||||
xlabel={$x$},
|
|
||||||
ylabel={$F_X (x)$},
|
|
||||||
samples=100,
|
|
||||||
xtick = {4.6,4.7,4.8,...,5.4}
|
|
||||||
]
|
|
||||||
\addplot+[mark=none, line width=1pt]
|
|
||||||
{1 / sqrt(2*3.1415*0.07*0.07) * exp(-(x - 5)*(x-5)/(2*0.07*0.07))};
|
|
||||||
|
|
||||||
\addplot +[KITblue, mark=none, line width=1pt] coordinates {(4.8, -1) (4.8, 2)};
|
|
||||||
\addplot +[KITblue, mark=none, line width=1pt] coordinates {(5.2, -1) (5.2, 2)};
|
|
||||||
\node at (axis cs: 4.8, 3) {$S(1-\delta)$};
|
|
||||||
\node at (axis cs: 5.2, 3) {$S(1+\delta)$};
|
|
||||||
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
\end{figure}
|
|
||||||
\end{columns}
|
|
||||||
\end{enumerate}
|
|
||||||
% tex-fmt: on
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\begin{frame}
|
|
||||||
\frametitle{Aufgabe 2: Normalverteilung}
|
|
||||||
|
|
||||||
\vspace*{-18mm}
|
|
||||||
|
|
||||||
% tex-fmt: off
|
|
||||||
\begin{enumerate}[a{)}]
|
|
||||||
\setcounter{enumi}{1}
|
|
||||||
\item Der Hersteller möchte seinen Produktionsprozess so verbessern,
|
|
||||||
dass nur noch halb so viele Ladegeräte wie in a) aussortiert
|
|
||||||
werden. Auf welchen Wert müsste er dazu $\sigma$ senken?
|
|
||||||
\pause\begin{gather*}
|
|
||||||
P(E_\text{b}) = \frac{1}{2} P(E_\text{a}) \approx 0{,}212\text{\%} \\
|
|
||||||
\end{gather*}
|
|
||||||
\vspace*{-18mm}
|
|
||||||
\begin{columns}
|
|
||||||
\pause\column{\kitthreecolumns}
|
|
||||||
\centering
|
|
||||||
\begin{align*}
|
|
||||||
P(E_\text{b}) &\overset{\text{a)}}{=} P\left(\widetilde{X} < \frac{S(1 - \delta) - \mu}{\sigma'}\right)
|
|
||||||
+ P\left(\widetilde{X} > \frac{S(1 + \delta) - \mu}{\sigma'}\right) \\[2mm]
|
|
||||||
&= P\left(\widetilde{X} < -\frac{0{,}2}{\sigma'}\right)
|
|
||||||
+ P\left(\widetilde{X} > \frac{0{,}2}{\sigma'}\right) \\[2mm]
|
|
||||||
&= \Phi\left(-\frac{0{,}2}{\sigma'}\right)
|
|
||||||
+ \left(1 - \Phi\left(\frac{0{,}2}{\sigma'} \right)\right) \\[2mm]
|
|
||||||
&= 2 - 2 \Phi\left(\frac{0{,}2}{\sigma'} \right)
|
|
||||||
\end{align*}
|
|
||||||
\pause\column{\kitthreecolumns}
|
|
||||||
\centering
|
|
||||||
\begin{gather*}
|
|
||||||
2 - 2\Phi\left(\frac{0{,}2}{\sigma'}\right) = 2{,}12\cdot 10^{-3} \\[2mm]
|
|
||||||
\Rightarrow \Phi\left(\frac{0{,}2}{\sigma'}\right) \approx 0{,}9989 \\[2mm]
|
|
||||||
\Rightarrow \sigma' \approx \frac{0{,}2}{\Phi^{-1}(0{,}9989)}
|
|
||||||
\approx \frac{0{,}2}{3{,}08} \approx 0{,}065
|
|
||||||
\end{gather*}
|
|
||||||
\end{columns}
|
|
||||||
\pause \vspace*{-5mm}\item Durch einen Produktionsfehler verschiebt sich der
|
|
||||||
Mittelwert $\mu$ auf $5{,}1$ Volt ($\sigma$ ist $0{,}07$ Volt).
|
|
||||||
Wie groß ist jetzt der Prozentsatz, der aussortiert wird?
|
|
||||||
\pause \begin{align*}
|
|
||||||
P(E_\text{c}) &\overset{\text{a)}}{=} P\left(\widetilde{X} < \frac{S(1 - \delta) - \mu}{\sigma}\right)
|
|
||||||
+ P\left(\widetilde{X} > \frac{S(1 + \delta) - \mu}{\sigma}\right) \\[2mm]
|
|
||||||
&\approx \Phi(-4{,}29) + (1 - \Phi(1{,}43)) \\
|
|
||||||
& = 2 - \Phi(4{,}29) - \Phi(1{,}43) \approx 7{,}78 \text{\%}
|
|
||||||
\end{align*}
|
|
||||||
\end{enumerate}
|
|
||||||
% tex-fmt: on
|
|
||||||
\end{frame}
|
|
||||||
|
|
||||||
\end{document}
|
|
||||||
|
|
||||||
File diff suppressed because it is too large
Load Diff
@@ -1,38 +0,0 @@
|
|||||||
import numpy as np
|
|
||||||
import matplotlib.pyplot as plt
|
|
||||||
from numpy.typing import NDArray
|
|
||||||
import argparse
|
|
||||||
|
|
||||||
|
|
||||||
def twodim_array_to_pgfplots_table_string(a: NDArray):
|
|
||||||
return (
|
|
||||||
" \\\\\n".join([" ".join([str(vali) for vali in val]) for val in a]) + "\\\\\n"
|
|
||||||
)
|
|
||||||
|
|
||||||
|
|
||||||
def main():
|
|
||||||
# Parse command line arguments
|
|
||||||
|
|
||||||
parser = argparse.ArgumentParser()
|
|
||||||
parser.add_argument("--correlation", "-c", type=np.float32, required=True)
|
|
||||||
parser.add_argument("-N", type=np.int32, required=True)
|
|
||||||
parser.add_argument("--plot", "-p", action="store_true")
|
|
||||||
|
|
||||||
args = parser.parse_args()
|
|
||||||
|
|
||||||
# Generate & plot data
|
|
||||||
|
|
||||||
means = np.array([0, 0])
|
|
||||||
cov = np.array([[1, args.correlation], [args.correlation, 1]])
|
|
||||||
|
|
||||||
x = np.random.multivariate_normal(means, cov, size=args.N)
|
|
||||||
|
|
||||||
print(twodim_array_to_pgfplots_table_string(x))
|
|
||||||
|
|
||||||
if args.plot:
|
|
||||||
plt.scatter(x[:, 0], x[:, 1])
|
|
||||||
plt.show()
|
|
||||||
|
|
||||||
|
|
||||||
if __name__ == "__main__":
|
|
||||||
main()
|
|
||||||
@@ -1,40 +0,0 @@
|
|||||||
import argparse
|
|
||||||
import numpy as np
|
|
||||||
import matplotlib.pyplot as plt
|
|
||||||
from scipy.special import binom
|
|
||||||
|
|
||||||
|
|
||||||
def array_to_pgfplots_table_string(a):
|
|
||||||
return " ".join([f"({k}, {val})" for (k, val) in enumerate(a)]) + f" ({len(a)}, 0)"
|
|
||||||
|
|
||||||
|
|
||||||
def P_binom(N, p, k):
|
|
||||||
return binom(N, k) * p**k * (1 - p) ** (N - k)
|
|
||||||
|
|
||||||
|
|
||||||
def main():
|
|
||||||
# Parse command line arguments
|
|
||||||
|
|
||||||
parser = argparse.ArgumentParser()
|
|
||||||
parser.add_argument("-N", type=np.int32, required=True)
|
|
||||||
parser.add_argument("-p", type=np.float32, required=True)
|
|
||||||
parser.add_argument("--show", "-s", action="store_true")
|
|
||||||
|
|
||||||
args = parser.parse_args()
|
|
||||||
|
|
||||||
# Generate and show data
|
|
||||||
|
|
||||||
N = args.N
|
|
||||||
p = args.p
|
|
||||||
|
|
||||||
bars = np.array([P_binom(N, p, k) for k in range(N + 1)])
|
|
||||||
|
|
||||||
print(array_to_pgfplots_table_string(bars))
|
|
||||||
|
|
||||||
if args.show:
|
|
||||||
plt.stem(bars)
|
|
||||||
plt.show()
|
|
||||||
|
|
||||||
|
|
||||||
if __name__ == "__main__":
|
|
||||||
main()
|
|
||||||
File diff suppressed because it is too large
Load Diff
File diff suppressed because it is too large
Load Diff
Binary file not shown.
@@ -1,3 +1,3 @@
|
|||||||
$pdflatex="pdflatex -shell-escape -interaction=nonstopmode -synctex=1 %O %S";
|
$pdflatex="pdflatex -shell-escape -interaction=nonstopmode -synctex=1 %O %S -cd ./../..";
|
||||||
$out_dir = 'build';
|
$out_dir = "build";
|
||||||
$pdf_mode = 1;
|
$pdf_mode = 1;
|
||||||
1
src/template/lib
Symbolic link
1
src/template/lib
Symbolic link
@@ -0,0 +1 @@
|
|||||||
|
/home/andreas/Documents/kit/wt-tut/presentations/lib
|
||||||
0
src/template/presentation.bib
Normal file
0
src/template/presentation.bib
Normal file
304
src/template/presentation.tex
Normal file
304
src/template/presentation.tex
Normal file
@@ -0,0 +1,304 @@
|
|||||||
|
\documentclass[de]{CELbeamer}
|
||||||
|
|
||||||
|
%
|
||||||
|
%
|
||||||
|
% CEL Template
|
||||||
|
%
|
||||||
|
%
|
||||||
|
|
||||||
|
\newcommand{\templates}{preambles}
|
||||||
|
\input{\templates/packages.tex}
|
||||||
|
\input{\templates/macros.tex}
|
||||||
|
|
||||||
|
\grouplogo{CEL_logo.pdf}
|
||||||
|
|
||||||
|
\groupname{Communication Engineering Lab (CEL)}
|
||||||
|
\groupnamewidth{80mm}
|
||||||
|
|
||||||
|
\fundinglogos{}
|
||||||
|
|
||||||
|
%
|
||||||
|
%
|
||||||
|
% Custom commands
|
||||||
|
%
|
||||||
|
%
|
||||||
|
|
||||||
|
\input{lib/latex-common/common.tex}
|
||||||
|
\pgfplotsset{colorscheme/rocket}
|
||||||
|
|
||||||
|
%TODO: Fix path
|
||||||
|
\newcommand{\res}{src/template/res}
|
||||||
|
|
||||||
|
% \tikzstyle{every node}=[font=\small]
|
||||||
|
% \captionsetup[sub]{font=small}
|
||||||
|
|
||||||
|
%
|
||||||
|
%
|
||||||
|
% Document setup
|
||||||
|
%
|
||||||
|
%
|
||||||
|
|
||||||
|
\usepackage{tikz}
|
||||||
|
\usepackage{tikz-3dplot}
|
||||||
|
\usetikzlibrary{spy, external, intersections}
|
||||||
|
%\tikzexternalize[prefix=build/]
|
||||||
|
|
||||||
|
\usepackage{pgfplots}
|
||||||
|
\pgfplotsset{compat=newest}
|
||||||
|
\usepgfplotslibrary{fillbetween}
|
||||||
|
|
||||||
|
\usepackage{listings}
|
||||||
|
\usepackage{subcaption}
|
||||||
|
\usepackage{bbm}
|
||||||
|
\usepackage{multirow}
|
||||||
|
|
||||||
|
\usepackage{xcolor}
|
||||||
|
|
||||||
|
\title{WT Tutorium 1}
|
||||||
|
\author[Tsouchlos]{Andreas Tsouchlos}
|
||||||
|
\date[]{\today}
|
||||||
|
|
||||||
|
%
|
||||||
|
%
|
||||||
|
% Document body
|
||||||
|
%
|
||||||
|
%
|
||||||
|
|
||||||
|
\begin{document}
|
||||||
|
|
||||||
|
\begin{frame}[title white vertical, picture=images/IMG_7801-cut]
|
||||||
|
\titlepage
|
||||||
|
\end{frame}
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\section{Aufgabe 1}
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\subsection{Theorie}
|
||||||
|
|
||||||
|
% TODO: Replace slide content with relevant stuff
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Relevante Theorie I}
|
||||||
|
|
||||||
|
\begin{columns}
|
||||||
|
\column{\kitthreecolumns}
|
||||||
|
\begin{greenblock}{Zufallsvariablen (ZV)}%
|
||||||
|
\vspace*{-6mm}
|
||||||
|
\begin{gather*}
|
||||||
|
f_X(x) := \frac{d}{dx} F_X(x) \\
|
||||||
|
P(X \le x) = F_X(x) = \int_{-\infty}^{x} f_X(t) dt \\
|
||||||
|
E(X) = \int_{-\infty}^{\infty} x\cdot f_X(x) dx
|
||||||
|
\end{gather*}
|
||||||
|
\end{greenblock}
|
||||||
|
|
||||||
|
\column{\kitthreecolumns}
|
||||||
|
\begin{greenblock}{Important Equations}%
|
||||||
|
\vspace*{-6mm}
|
||||||
|
\begin{gather*}
|
||||||
|
f_X(x) := \frac{d}{dx} F_X(x) \\
|
||||||
|
P(X \le x) = F_X(x) = \int_{-\infty}^{x} f_X(t) dt \\
|
||||||
|
E(X) = \int_{-\infty}^{\infty} x\cdot f_X(x) dx
|
||||||
|
\end{gather*}
|
||||||
|
\end{greenblock}
|
||||||
|
\end{columns}
|
||||||
|
|
||||||
|
\begin{greenblock}{Normalverteilung}
|
||||||
|
\begin{columns}
|
||||||
|
\column{\kitthreecolumns}
|
||||||
|
\begin{gather*}
|
||||||
|
\text{Normalverteilung:} \hspace{8mm}
|
||||||
|
f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}}
|
||||||
|
e^{-\frac{(x - \mu)^2}{2\sigma^2}}
|
||||||
|
\end{gather*}
|
||||||
|
|
||||||
|
\column{\kitthreecolumns}
|
||||||
|
\begin{figure}
|
||||||
|
\centering
|
||||||
|
\begin{tikzpicture}
|
||||||
|
\begin{axis}[
|
||||||
|
domain=-4:4,
|
||||||
|
samples=100,
|
||||||
|
width=11cm,
|
||||||
|
height=6cm,
|
||||||
|
ticks=none,
|
||||||
|
xlabel={$x$},
|
||||||
|
ylabel={$f_X(x)$}
|
||||||
|
]
|
||||||
|
\addplot+[mark=none, line width=1pt] {exp(-x^2)};
|
||||||
|
\end{axis}
|
||||||
|
\end{tikzpicture}
|
||||||
|
\end{figure}
|
||||||
|
\end{columns}
|
||||||
|
\end{greenblock}
|
||||||
|
\end{frame}
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\subsection{Aufgabe}
|
||||||
|
|
||||||
|
% TODO: Replace slide content with relevant stuff
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{2022H - Aufgabe 4}
|
||||||
|
|
||||||
|
Für die Planung und Konstruktion von Windkraftanlagen ist eine
|
||||||
|
statistische Modellierung der
|
||||||
|
Windgeschwindigkeit essentiell. Die absolute Windgeschwindigkeit
|
||||||
|
kann als Weibull-verteilte
|
||||||
|
Zufallsvariable V mit den Parametern $\beta > 0$ und $\theta > 0$
|
||||||
|
modelliert werden. Die zugehörige
|
||||||
|
Verteilungsfunktion ist%
|
||||||
|
%
|
||||||
|
\begin{gather*}
|
||||||
|
F_V(v) = 1 - exp\left( -\left( \frac{v}{\theta} \right)^\beta
|
||||||
|
\right), \hspace{3mm} v \ge 0
|
||||||
|
\end{gather*}
|
||||||
|
%
|
||||||
|
|
||||||
|
\begin{enumerate}
|
||||||
|
\item Berechnen Sie die Wahrscheinlichkeitsdichte $f_V(v)$
|
||||||
|
der Weibullverteilung.
|
||||||
|
\item Eine Windkraftanlage speist Strom in das Stromnetz ein,
|
||||||
|
wenn die absolute Windgeschwindigkeit größer als $4
|
||||||
|
m/s$, jedoch kleiner als $25 m/s$ ist. Berechnen Sie die
|
||||||
|
Wahrscheinlichkeit dafür, dass eine Windkraftanlage Strom
|
||||||
|
einspeist, wenn die Windgeschwindigkeit Weibull-verteilt
|
||||||
|
mit $\beta = 2,0$ und $\theta = 6,0$ ist.
|
||||||
|
\item Eine Zufallsvariable W genüge einer Weibullverteilung
|
||||||
|
mit $\beta = 1$ und $\theta = 3$. Ermitteln Sie den
|
||||||
|
Erwartungsvert $E(W)$.
|
||||||
|
\item Warum ist die Weibullverteilung für die Modellierung
|
||||||
|
der absoluten Windgeschwindigkeit besser geeignet als
|
||||||
|
eine Normalverteilung?
|
||||||
|
\end{enumerate}
|
||||||
|
|
||||||
|
\end{frame}
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\section{Aufgabe 2}
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\subsection{Theorie}
|
||||||
|
|
||||||
|
% TODO: Replace slide content with relevant stuff
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Relevante Theorie II}
|
||||||
|
|
||||||
|
\begin{gather*}
|
||||||
|
f_X(x) := \frac{d}{dx} F_X(x) \\
|
||||||
|
P(X \le x) = F_X(x) = \int_{-\infty}^{x} f_X(t) dt \\
|
||||||
|
E(X) = \int_{-\infty}^{\infty} x\cdot f_X(x) dx
|
||||||
|
\end{gather*}
|
||||||
|
|
||||||
|
\begin{figure}
|
||||||
|
\centering
|
||||||
|
|
||||||
|
\begin{subfigure}[c]{0.5\textwidth}
|
||||||
|
\centering
|
||||||
|
\begin{gather*}
|
||||||
|
\text{Normalverteilung:} \hspace{8mm}
|
||||||
|
f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}}
|
||||||
|
e^{-\frac{(x - \mu)^2}{2\sigma^2}}
|
||||||
|
\end{gather*}
|
||||||
|
\end{subfigure}%
|
||||||
|
\begin{subfigure}[c]{0.4\textwidth}
|
||||||
|
\centering
|
||||||
|
\begin{tikzpicture}
|
||||||
|
\begin{axis}[
|
||||||
|
domain=-4:4,
|
||||||
|
samples=100,
|
||||||
|
width=\textwidth,
|
||||||
|
height=0.5\textwidth,
|
||||||
|
ticks=none,
|
||||||
|
xlabel={$x$},
|
||||||
|
ylabel={$f_X(x)$}
|
||||||
|
]
|
||||||
|
\addplot+[mark=none, line width=1pt] {exp(-x^2)};
|
||||||
|
\end{axis}
|
||||||
|
\end{tikzpicture}
|
||||||
|
\end{subfigure}
|
||||||
|
\end{figure}
|
||||||
|
\end{frame}
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\subsection{Aufgabe}
|
||||||
|
|
||||||
|
% TODO: Replace slide content with relevant stuff
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{2022H - Aufgabe 4}
|
||||||
|
|
||||||
|
Für die Planung und Konstruktion von Windkraftanlagen ist eine
|
||||||
|
statistische Modellierung der
|
||||||
|
Windgeschwindigkeit essentiell. Die absolute Windgeschwindigkeit
|
||||||
|
kann als Weibull-verteilte
|
||||||
|
Zufallsvariable V mit den Parametern $\beta > 0$ und $\theta > 0$
|
||||||
|
modelliert werden. Die zugehörige
|
||||||
|
Verteilungsfunktion ist%
|
||||||
|
%
|
||||||
|
\begin{gather*}
|
||||||
|
F_V(v) = 1 - exp\left( -\left( \frac{v}{\theta} \right)^\beta
|
||||||
|
\right), \hspace{3mm} v \ge 0
|
||||||
|
\end{gather*}
|
||||||
|
%
|
||||||
|
|
||||||
|
\begin{enumerate}
|
||||||
|
\item Berechnen Sie die Wahrscheinlichkeitsdichte $f_V(v)$
|
||||||
|
der Weibullverteilung.
|
||||||
|
\item Eine Windkraftanlage speist Strom in das Stromnetz ein,
|
||||||
|
wenn die absolute Windgeschwindigkeit größer als $4
|
||||||
|
m/s$, jedoch kleiner als $25 m/s$ ist. Berechnen Sie die
|
||||||
|
Wahrscheinlichkeit dafür, dass eine Windkraftanlage Strom
|
||||||
|
einspeist, wenn die Windgeschwindigkeit Weibull-verteilt
|
||||||
|
mit $\beta = 2,0$ und $\theta = 6,0$ ist.
|
||||||
|
\item Eine Zufallsvariable W genüge einer Weibullverteilung
|
||||||
|
mit $\beta = 1$ und $\theta = 3$. Ermitteln Sie den
|
||||||
|
Erwartungsvert $E(W)$.
|
||||||
|
\item Warum ist die Weibullverteilung für die Modellierung
|
||||||
|
der absoluten Windgeschwindigkeit besser geeignet als
|
||||||
|
eine Normalverteilung?
|
||||||
|
\end{enumerate}
|
||||||
|
\end{frame}
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\section{Zusammenfassung}
|
||||||
|
|
||||||
|
% TODO: Replace slide content with relevant stuff
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Zusammenfassung}
|
||||||
|
|
||||||
|
\begin{gather*}
|
||||||
|
f_X(x) := \frac{d}{dx} F_X(x) \\
|
||||||
|
P(X \le x) = F_X(x) = \int_{-\infty}^{x} f_X(t) dt \\
|
||||||
|
E(X) = \int_{-\infty}^{\infty} x\cdot f_X(x) dx
|
||||||
|
\end{gather*}
|
||||||
|
|
||||||
|
\begin{figure}
|
||||||
|
\centering
|
||||||
|
|
||||||
|
\begin{subfigure}[c]{0.5\textwidth}
|
||||||
|
\centering
|
||||||
|
\begin{gather*}
|
||||||
|
\text{Normalverteilung:} \hspace{8mm}
|
||||||
|
f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}}
|
||||||
|
e^{-\frac{(x - \mu)^2}{2\sigma^2}}
|
||||||
|
\end{gather*}
|
||||||
|
\end{subfigure}%
|
||||||
|
\begin{subfigure}[c]{0.4\textwidth}
|
||||||
|
\centering
|
||||||
|
\begin{tikzpicture}
|
||||||
|
\begin{axis}[
|
||||||
|
domain=-4:4,
|
||||||
|
samples=100,
|
||||||
|
width=\textwidth,
|
||||||
|
height=0.5\textwidth,
|
||||||
|
ticks=none,
|
||||||
|
xlabel={$x$},
|
||||||
|
ylabel={$f_X(x)$}
|
||||||
|
]
|
||||||
|
\addplot+[mark=none, line width=1pt] {exp(-x^2)};
|
||||||
|
\end{axis}
|
||||||
|
\end{tikzpicture}
|
||||||
|
\end{subfigure}
|
||||||
|
\end{figure}
|
||||||
|
\end{frame}
|
||||||
|
|
||||||
|
\end{document}
|
||||||
|
|
||||||
1
src/template/src
Symbolic link
1
src/template/src
Symbolic link
@@ -0,0 +1 @@
|
|||||||
|
/home/andreas/Documents/kit/wt-tut/presentations/src
|
||||||
Reference in New Issue
Block a user