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zG8R~`mRAtD0vdY!d3n|^yK$`6YjhsV-8!4M91dgYYGbBOpH1=U{FM^9E|E%V?We+FcIGim?#vgW5^GX#>@)@Ic@DtuyEb=Nu STLEQ>!{A{C2F_kCuzvw^K0~em diff --git a/semester4/ps/ps-rb/parts/01_variables.tex b/semester4/ps/ps-rb/parts/01_variables.tex index 0fa97aa..f217c39 100644 --- a/semester4/ps/ps-rb/parts/01_variables.tex +++ b/semester4/ps/ps-rb/parts/01_variables.tex @@ -122,10 +122,6 @@ $$ \lemma $X_1 \sim \text{Bin}(n, p), X_2 \sim \text{Bin}(m, p) \implies X_1 + X_2 \sim \text{Bin}(n+m,p)$ -\lemma \textbf{Binomialverteilung erfüllt die Summenvoraussetzung} -$$ - \sum_{k=0}^{n} p(k) = \sum_{k=0}^{n} \binom{n}{k} p^k(n-p)^{n-k} = (p + 1 - p)^n = 1 -$$ \normalsize \definition \textbf{Geometrische Verteilung} $X \sim \text{Geom}(p)$\\ @@ -133,4 +129,101 @@ $$ $$ \P[X=k] = p\cdot(1-p)^{k-1} $$ -\subtext{$0 < p \leq 1,\quad k \in \N_{\neq0}$} \ No newline at end of file +\subtext{$0 < p \leq 1,\quad k \in \N_{\neq0}$} + +\definition \textbf{Negativbinomiale Verteilung} $X \sim \text{NBin}(r,p)$\\ +\smalltext{Intuitiv: Wartezeit zum $r$-ten Erfolg} +$$ + \P[X = k] = \binom{k-1}{r-1}p^r(1-p)^{k-r} +$$ +\subtext{$r \in \N,\quad p \in [0,1],\quad k \in \{r, r+1,\ldots \},\quad \text{Nur auf Folien}$} + +\theorem \textbf{Interpretation}\\ +\smalltext{Für $X_1, X_2, \ldots$ unabhängig s.d. $X_i \sim \text{Ber}(p)$:} +$$ + T_r := \text{inf}\Bigl\{ n \geq 1 \Big| \sum_{i=1}^{n}X_i=r \Bigr\} \sim \text{NBin}(r,p) +$$ + +\scriptsize + +\lemma $X_1,X_2,\ldots,X_r \sim \text{Geom}(p) \implies X := \sum_{i=1}^{r}X_i \sim \text{NBin}(r,p)$ + +\normalsize + + +\newpage + +\definition \textbf{Hypergeometrische Verteilung} $X \sim \text{H}(n, r, m)$ +$$ + \P[X = K] = \frac{\binom{r}{k}\binom{n-r}{m-k}}{\binom{n}{m}} +$$ +\subtext{$n \in \N,\quad m,r \in \{0,1,\ldots,n\},\quad k \in \{0,1,\ldots \min(m,r) \}$} + +\theorem \textbf{Interpretation}\\ +\smalltext{$n$ Objekte in einer Urne, $r$ von Typ 1, $n-r$ von Typ 2 \\ Ziehe $m$, Sei $X$ die Anzahl Objekte Typ 1, dann $X \sim \text{H}(n,r,m)$}\\ +\subtext{Das modelliert z.B. Lotto} + +\definition \textbf{Poisson-Verteilung}\\ +\smalltext{Intuitiv: Sehr seltene Ereignisse} +$$ + \P[X=k] = \frac{\lambda^k}{k!}e^{-\lambda} +$$ +\subtext{$\lambda \in \R_{>0},\quad k \in \N_0$} + +\theorem \textbf{Poisson-Approximation der Binomialverteilung}\\ +\smalltext{$X_n \sim \text{Bin}(\frac{\lambda}{n}),\quad N \sim \text{Poisson}(\lambda),\quad n \geq 1$} +$$ + \limn \P[X_n=k] = \P[N=k] +$$ +\subtext{Intuitiv: Für grosse $n$ haben $X_n$ und $N$ sehr ähnliche Eigenschaften} + +\scriptsize + +\remark Auch genannt: \textit{Konvergenz in Verteilung, Schwache Konvergenz}\\ +\color{gray} +Convergence in distribution/law, weak convergence +\color{black} + +\normalsize + +\newpage + +\subsection{Stetige Verteilungen} + +\definition \textbf{Stetig verteilte Zufallsvariable} $X: \Omega \to \R$\\ +\smalltext{Intuitiv: $f_X(t)dt$ ist Die W.-keit, dass $X$ in $[t, t+dt]$ ist} +$$ + \exists f_X \R \to \R_+ \quad\text{ s.d. }\quad F_X(x) = \int_{-\infty}^{x}f_X(t)\ dt +$$ +\subtext{Stetig, da $F_X$ somit eine stetige Funktion ist} + +{\scriptsize + \remark $f_X$ heisst \textit{Dichtefunktion} von $X$ +} + +\theorem \textbf{Dichtefunktion ist die Ableitung der Verteilung} +\smalltext{$F_X$ stetig, stückweise diff.-bar $-\infty = x_0 < x_1 < \ldots x_{n-1} < x_n = \infty$} +$$ + f_X = \begin{cases} + F_X'(x) & \exists k \in \{0,\ldots,n-1\} \text{ s.d. } x \in (x_k,x_{k+1}) \\ + a_k & x \in \{x_1,\ldots,x_{n-1}\} + \end{cases} +$$ + +{\scriptsize + \remark $f_X$ (Stetig) ist Intuitiv das Analogon zu $p_x$ (Diskret) +} + +\definition \textbf{Gleichverteilung} $X \sim \mathcal{U}([a,b])$\\ +\smalltext{Intuitiv: Zufälliger Punkt in $[a,b]$} +$$ + f_X(x) = \begin{cases} + \frac{1}{b-a} & x \in [a,b] \\ + 0 & \text{sonst} + \end{cases} +$$ + +\definition \textbf{Exponentialverteilung} +$$ + +$$ \ No newline at end of file diff --git a/semester4/ps/ps-rb/util/helpers.tex b/semester4/ps/ps-rb/util/helpers.tex index 60dac48..39bfe45 100644 --- 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