diff --git a/semester4/ps/ps-jh/TODO.md b/semester4/ps/ps-jh/TODO.md new file mode 100644 index 0000000..bb9d2d7 --- /dev/null +++ b/semester4/ps/ps-jh/TODO.md @@ -0,0 +1,7 @@ +# TODOs +- [ ] Add many more notes +- [ ] Possibly: restrucutre to group definitions, then put theorems / remarks separately +- [ ] Tips & Tricks +- [ ] Better examples +- [ ] Shorten defs, etc +- [ ] Link together (decide on style for that) diff --git a/semester4/ps/ps-jh/parts/00_basics/00_probability-space.tex b/semester4/ps/ps-jh/parts/00_basics/00_probability-space.tex index f2c78b4..2052d98 100644 --- a/semester4/ps/ps-jh/parts/00_basics/00_probability-space.tex +++ b/semester4/ps/ps-jh/parts/00_basics/00_probability-space.tex @@ -39,6 +39,7 @@ Keine $\sigma$-Algebren sind bspw: \end{itemize} +\newpage \subsubsection{Wahrscheinlichkeitsraum} \shortdefinition[W.R] ein Tripel $(\Omega, \cF, \P)$ diff --git a/semester4/ps/ps-jh/parts/01_random-variables/00_definition.tex b/semester4/ps/ps-jh/parts/01_random-variables/00_definition.tex index 47f3654..3bec2e2 100644 --- a/semester4/ps/ps-jh/parts/01_random-variables/00_definition.tex +++ b/semester4/ps/ps-jh/parts/01_random-variables/00_definition.tex @@ -4,3 +4,8 @@ $f = \{ \omega \in \Omega \divider \cX(\omega) \leq a \} \in \cF$ (notwendinge Bedingung für Wohldefiniertheit von $\P[f]$) \inlinenotation Ohne $\omega$: $\{ X \leq a \} = \{ \omega \in \Omega \divider X(\omega) \leq a \}$, etc + +\shortdefinition[Indikatorfunk.] $A \in \cF$, $\bm{1_A(\omega)} = \begin{cases} + 0 & \text{ wenn } \omega \notin A \\ + 1 & \text{ wenn } \omega \in A +\end{cases}$ diff --git a/semester4/ps/ps-jh/parts/01_random-variables/01_distribution-function.tex b/semester4/ps/ps-jh/parts/01_random-variables/01_distribution-function.tex index 20d47be..2d49efe 100644 --- a/semester4/ps/ps-jh/parts/01_random-variables/01_distribution-function.tex +++ b/semester4/ps/ps-jh/parts/01_random-variables/01_distribution-function.tex @@ -1,9 +1,12 @@ % P23 \subsection{Verteilungsfunktion} +{\scriptsize Die untenstehende Funktion ist die Cumulative Distribution Function (cdf)} + \shortdefinition $F_\cX : \R \rightarrow [0, 1]$, def: $\forall a \in \R \quad F_\cX(a) = \P[\cX \leq a]$ \shorttheorem $a < b \in \R$. Dann: $\P[a < X \leq b] = F(b) - F(a)$ +{\scriptsize Nachfolgendes wird gebraucht, um zu Überprüfen, ob eine Funktion eine cdf ist}\\ \shorttheorem $\cX$ Z.V. auf $(\Omega, \cF, \P)$ und V.F. $F = F_\cX$. Eig.: \begin{enumerate} \item $F$ ist monoton wachsend diff --git a/semester4/ps/ps-jh/parts/01_random-variables/02_independence.tex b/semester4/ps/ps-jh/parts/01_random-variables/02_independence.tex index dcee58d..9a15e46 100644 --- a/semester4/ps/ps-jh/parts/01_random-variables/02_independence.tex +++ b/semester4/ps/ps-jh/parts/01_random-variables/02_independence.tex @@ -12,5 +12,8 @@ $\{ \cX_1 \in I_1 \}, \ldots, \{ \cX_n \in I_n \}$ unabhängig $Y_1 = \phi_1(\cX_1, \ldots, \cX_{i_1}), \ldots, Y_k = \phi_k(X_{i_{k - 1} + 1}, \ldots, X_{i_k})$ \subsubsection{Unabhängig identisch verteilte ZV} +% FIXME: REMOVE into shorter writing +{\scriptsize Abgekürzt u.i.v., oder i.i.d (independent \& identically distributed)} + \shortdefinition Eine Folge von ZV ist \bi{(1)} unabh. falls $X_i$ unabh. sind und \bi{(2)} uiv, falls unabh. und die ZV dieselbe Verteilungsf. haben, also: -$\forall i, j \quad F_{\cX_i} = F_{\cX_j}$ +$\forall i, j \ F_{\cX_i} = F_{\cX_j}$ diff --git a/semester4/ps/ps-jh/parts/06_estimators/02_max-likelihood.tex b/semester4/ps/ps-jh/parts/06_estimators/02_max-likelihood.tex index 8c0ebdb..e25ba41 100644 --- a/semester4/ps/ps-jh/parts/06_estimators/02_max-likelihood.tex +++ b/semester4/ps/ps-jh/parts/06_estimators/02_max-likelihood.tex @@ -30,7 +30,7 @@ Einfacher: Statt maximieren, Nullstellen von Ableitung nach $\vartheta$. % TODO: Maybe remark from slide 356 (= p33 in 7) \shortexample \bi{Verteilungen}\\ -\fbox{\bi{Bernoulli}} $\cX_i \sim \text{Ber}(p)$ i.i.d, hier $\vartheta = p$. Dabei: +\highlight{Bernoulli} $\cX_i \sim \text{Ber}(p)$ i.i.d, hier $\vartheta = p$. Dabei: $p_\cX(x; \vartheta) = \P_\vartheta[\cX = x] = \vartheta^x (1 - \vartheta)^{1 - x}$ mit $x \in \{0, 1\}$. LH-Func: \[ L(x_1, \ldots, x_n; \vartheta) = \vartheta^{\sum_{k = 1}^{n} x_k} (1 - \vartheta)^{n - \sum_{k = 1}^{n} x_k} diff --git a/semester4/ps/ps-jh/parts/06_estimators/03_properties.tex b/semester4/ps/ps-jh/parts/06_estimators/03_properties.tex index fc05f25..6da4fa1 100644 --- a/semester4/ps/ps-jh/parts/06_estimators/03_properties.tex +++ b/semester4/ps/ps-jh/parts/06_estimators/03_properties.tex @@ -21,7 +21,7 @@ $\chi^2_2 = \text{Exp}(\frac{1}{2})$ \] \shortremark $\cX \sim \cN(0, 1)$, $\cY \sim \chi^2_m$ unabh., dann: $\frac{\cX}{\sqrt{\frac{1}{m} \cY}} \sim t_m$ -Mit $m = 1$ Cauchy-V. mit $m \rightarrow \8$ asympt. $\cN(0, 1)$. $t_m$ symm. um $0$ wie $\cN(0, 1)$, aber \bi{langschänziger} +Mit $m = 1$ Cauchy-V. mit $m \rightarrow \8$ asympt. $\cN(0, 1)$. $t_m$ symm. um $0$ wie $\cN(0, 1)$, aber \bi{langschwänziger} \shorttheorem Für $\cX_k \sim \cN(\mu, \sigma^2)$ i.i.d. und \[ diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index 388bed9..18f764d 100644 Binary files a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf and b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf differ diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex index 9de09d6..0166f39 100644 --- a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex +++ b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex @@ -13,6 +13,11 @@ \renewcommand{\definitionShortNamingDE}{Def} \renewcommand{\propositionShortNamingDE}{Prop} \renewcommand{\remarkShortNamingDE}{Bem} +\renewcommand{\theoremShortNamingDE}{Thm} + +\renewcommand{\descriptorNameDisplay}[1]{\textbf{#1}} + +\newcommand{\highlight}[1]{\fbox{\bi{#1}}} \setsubsectionnumbering{section} \renewcommand{\examplenumbering}{off} @@ -24,7 +29,6 @@ \startDocument \noverticalspacing -% TODO: Short combinatorics summary % TODO: Most important sequences transforms % TODO: Logarithms (maybe including table), trick with -\ln(...) using \ln(1) - \ln(...) % TODO: Basic analysis stuff (some common integrals, fundamental theorem)