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b/semester4/ps/ps-rb/main.tex index eba2867..a86183c 100644 --- a/semester4/ps/ps-rb/main.tex +++ b/semester4/ps/ps-rb/main.tex @@ -11,5 +11,8 @@ \subtext{Basiert auf dem Skript von V. Tassion} \section{Wahrscheinlichkeitsräume} \input{parts/00_intro.tex} +\newpage +\section{Zufallsvariablen} +\input{parts/01_variables.tex} \end{document} diff --git a/semester4/ps/ps-rb/parts/00_intro.tex b/semester4/ps/ps-rb/parts/00_intro.tex index fd62e80..298fa38 100644 --- a/semester4/ps/ps-rb/parts/00_intro.tex +++ b/semester4/ps/ps-rb/parts/00_intro.tex @@ -76,6 +76,8 @@ $$ \subtext{$(A_n), (B_n)$ sind monotone Folgen von Ereignissen} +\subsection{Bedingte Wahrscheinlichkeit} + \definition \textbf{Bedingte Wahrscheinlichkeit} $$ \P\bigl[A\ \big|\ B\bigr] := \frac{\P[A \cap B]}{\P[B]} @@ -103,6 +105,8 @@ $$ \newpage +\subsection{Unabhängigkeit} + \definition \textbf{Unabhängigkeit} $$ A, B \text{ unabhängig } \iffdef \P[A \cap B] = \P[A] \cdot \P[B] diff --git a/semester4/ps/ps-rb/parts/01_variables.tex b/semester4/ps/ps-rb/parts/01_variables.tex new file mode 100644 index 0000000..0fa97aa --- /dev/null +++ b/semester4/ps/ps-rb/parts/01_variables.tex @@ -0,0 +1,136 @@ +\definition \textbf{Zufallsvariable} +$$ + X: \Omega \to \R \qquad \text{s.d} \quad \forall a \in \R:\quad \Bigl\{ \omega \in \Omega\ \Big|\ X(\omega)\leq a \Bigr\} \in \F +$$ +\subtext{$(\Omega, \F, \P)$ ein Wahrscheinlichkeitsraum} + +\footnotesize +\definition \textbf{Indikatorfunktion} +$$ + \forall \omega \in \Omega:\quad \mathbb{I}_A(\omega) := \begin{cases} + 0 & \omega \notin A \\ + 1 & \omega \in A + \end{cases} + \color{gray} + \qquad A \in \F + \color{black} +$$ +\normalsize + +\notation Ereignisse bezüglich Zufallsvariablen +\begin{align*} + & \{ X \leq a \} &=\quad& \Bigl\{ \omega \in \Omega \ \Big|\ X(\omega) \leq a \Bigr\} \\ + & \{ a < X \leq b \} &=\quad& \Bigl\{ \omega \in \Omega \ \Big|\ a < X(\omega) \leq b \Bigr\} \\ + & \P[X \leq a] &=\quad& \P\Bigl[\{ X \leq a \}\Bigr] +\end{align*} + +\definition \textbf{Verteilungsfunktion} $F_X: \R \to [0,1]$ +$$ + \forall a \in \R:\qquad F_X(a) = \P[X \leq a] +$$ + +\theorem \textbf{Eigenschaftern der Verteilungsfunktion} + +\begin{tabular}{ll} + (i) & $F_X$ monoton \\ + (ii) & $F_X$ rechtsstetig \\ + (iii) & $\underset{a \to -\infty}{\lim} F_X(a) = 0 \quad\land\quad \underset{a \to \infty}{\lim}F_X(a)=1$ +\end{tabular} + +\definition \textbf{Unabhängigkeit}\\ +\smalltext{$X_1,\cdots,X_n \text{ unabhängig } \iffdef§ \forall x_1,\cdots,x_n \in \R:$} +$$ + \P[X_1 \leq x_q,\cdots, X_n \leq x_n] = \P[X_1 \leq x_1] \cdots \P[X_n \leq x_n] +$$ + +\theorem \textbf{Unabhängigkeit von Gruppierungen}\\ +\smalltext{$X_1,\cdots X_n$ sind unabhängig, dann sind auch $Y_1,\cdots,Y_k$ unabhängig:} +$$ + Y_1 = \phi_1(X_1,\cdots,X_{i_1}), \cdots , Y_k = \phi_k(X_{i_{k-1}+1},\cdots,X_{i_k}) +$$ +\subtext{$1 \leq i_1 < i_2 < \cdots < i_k \leq n \text{ sind Indizes},\quad \phi_1,\cdots,\phi_k \text{ sind Abbildungen}$} + +\newpage + +\definition \textbf{Folgen von Zufallsvariablen} + +\begin{tabular}{llll} + (i) & unabhängig & $\iffdef$ & $\forall n \geq 1: X_1,\cdots,X_n \text{ unabhängig}$ \\ + (ii) & i.i.d & $\iffdef$ & unabhängig, und $\forall i, j: F_{X_i} = F_{X_j}$ +\end{tabular} + +\subtext{i.i.d = Independent \& identically distributed} + +\subsection{Diskrete Zufallsvariablen} + +\definition \textbf{Diskrete Zufallsvariable} +$$ + X: \Omega \to \R \text{ diskret } \iffdef \exists (W \subset \R) \preceq \N:\quad \P[X \in W] = 1 +$$ +\subtext{bzw. die Werte von $X$ liegen f.s. in $W$} + +\lemma \textbf{Variablen diskreter Grundräume sind diskret} +$$ + \Omega \preceq \N \implies X: \Omega \to \R \text{ ist diskret} +$$ + +\definition \textbf{Verteilung diskreter Variablen} +$$ + \Bigl( p(x) \Bigr)_{x \in W} \text{ s.d. } p(x) := \P[X = x] \text{ heisst Verteilung} +$$ +\subtext{$X$ ist diskret mit $W \preceq \N\quad p(x)$ ist \textit{nicht} $F_X$} + +\theorem $\forall p(x):\quad \displaystyle\sum_{x \in W} p(x) = 1$ \subtext{$\quad p(x)$ ist eine diskrete Verteilung} + +\lemma \textbf{Zur diskreten Verteilung existiert eine Variable} +$$ + \forall \Bigl( p(x) \Bigr)_{x \in W} \in [0,1]:\quad \exists (\Omega, \F, \P), X \text{ mit Verteilung } p(x) +$$ +\subtext{D.h man kann sagen: "Sei $X$ eine Variable mit Verteilung $\bigl(p(x)\bigr)_{x \in W}$"} + +\theorem \textbf{Diskrete Verteilungsfunktion} +$$ + F_X(x) = \P[x \leq X] = \sum_{y \leq x} p(y) \qquad y \in W +$$ + +\newpage + +\subsection{Diskrete Verteilungen} + +\definition \textbf{Bernoulli-Verteilung} $X \sim \text{Ber}(p)$\\ +\smalltext{Intuitiv: Münzwurf} +$$ + \P[X=1] = p \qquad \P[X=0] = 1-p +$$ +\subtext{$0 \leq p \leq 1,\quad W = \{0,1\}$} + +\definition \textbf{Binomial-Verteilung} $X \sim \text{Bin}(n,p)$\\ +\smalltext{Intuitiv: Anzahl Erfolge bei wiederholtem Bernoulli-Experiment} +$$ + \P[X=k] = \binom{n}{k}p^k(1-p)^{n-k} +$$ +\subtext{$0 \leq p \leq 1,\quad n \in \N,\quad W = \{0,\ldots, n\}$} + +\theorem \textbf{Bernoulli-Summen sind Binomial-verteilt} +$$ + S_n := X_1 + \ldots + X_n \sim \text{Bin}(p, n) +$$ +\subtext{$0\leq p\leq n,\quad n \in N,\quad X_1,\ldots,X_n \sim \text{Ber}(p) \text{ unabhängig}$} + +\scriptsize +\lemma $\text{Bin}(1,p) = \text{Ber}(p)$ + +\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} $X \sim \text{Geom}(p)$\\ +\smalltext{Intuitiv: Bernoulli Experiment erfolgreich beim $k$-ten Versuch} +$$ + \P[X=k] = p\cdot(1-p)^{k-1} +$$ +\subtext{$0 < p \leq 1,\quad k \in \N_{\neq0}$} \ No newline at end of file