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+\input{parts/05_limits.tex} + \end{document} diff --git a/semester4/ps/ps-rb/parts/04_joint-distributions.tex b/semester4/ps/ps-rb/parts/04_joint-distributions.tex new file mode 100644 index 0000000..9d92f45 --- /dev/null +++ b/semester4/ps/ps-rb/parts/04_joint-distributions.tex @@ -0,0 +1,48 @@ +\subsection{Gemeinsame Diskrete Verteilung} + +\definition \textbf{Gemeinsame diskrete Verteilung}\\ +\smalltext{$X_1,\ldots,X_n$ diskret,$\quad W_i \cleq \N,\quad X_i \in W_i$ fast sicher} +$$ + p := \Bigl(p(x_1,\ldots,x_n)\Bigr)_{x_1\in W_1,\ldots x_n\in W_n} +$$ +$$ + p(x_1,\ldots,x_n) = \P\Bigl[ X_1=x_1,\ldots,X_n=x_n \Bigr] +$$ + +\theorem $\displaystyle\sum_{x_1\in W_1,\ldots,x_n\in W_n} p(x_1,\ldots,x_n) = 1$ + +% Randverteilung, Erwartungswert ... + +\newpage +\subsection{Gemeinsame Stetige Verteilung} + +\definition \textbf{Gemeinsame Stetige Verteilung}\\ +\smalltext{$X_1,\ldots,X_n: \Omega\to\R,\quad f:\R^n\to\R_+,\quad a_1,\ldots a_n \in \R$} +$$ + \P\Bigl[ X_1\leq a_1,\ldots,X_n\leq a_n \Bigr] = \int_{-\infty}^{a_1}\cdots\int_{-\infty}^{a_n} f(x_1,\ldots,x_n)\ dx_n\ldots dx_1 +$$ +\subtext{$f$ heisst \textit{gemeinsame Dichte} von $(X_1,\ldots,X_n)$} + +\theorem $\displaystyle\int_{-\infty}^\infty\cdots\int_{-\infty}^\infty f(x_1,\ldots,x_n)\ dx_n\ dx_1 = 1$\\ +\subtext{Umgekehrt existiert für jedes solches $f$ ein Raum $(\Omega, \F, \P)$} + +% good examples in script + +\theorem \textbf{Erwartungswert}\\ +$$ + \E\Bigl[ \phi(X_1,\ldots,X_n) \Bigr] = \int_{-\infty}^\infty\cdots\int_{-\infty}^\infty\phi(x_1,\ldots,x_n)\cdot f(x_1\cdots x_n)\ dx_n\ldots dx_1 +$$ + +% Randverteilungen + +\theorem \textbf{Unabhängigkeit} +$$ + X_1,\ldots,X_n \text{ stetig, } f(x_1,\ldots,x_n) = f_1(x_1)\cdots f_n(x_n) +$$ + +{\scriptsize + \remark Unabhängige stetige Variablen sind automatisch gemeinsam stetig +} + +% It's important to him to know \E[X]\E[Y] = \E[X\cdot Y] isn't enough for indep., requires \E[\phi(X)]\E[\psi(X)] instead \forall \phi,\psi + diff --git a/semester4/ps/ps-rb/parts/05_limits.tex b/semester4/ps/ps-rb/parts/05_limits.tex new file mode 100644 index 0000000..4b8e90a --- /dev/null +++ b/semester4/ps/ps-rb/parts/05_limits.tex @@ -0,0 +1,9 @@ +\theorem \textbf{Schwaches Gesetz der grossen Zahlen}\\ +\smalltext{$X_1,X_2,\ldots$ unabh.,$\quad\forall k: \E[X_k] = \mu,\V[X_k]=\sigma^2$} +$$ + \bar{X}_n = \frac{1}{n}S_n = \frac{1}{n}\sum_{i=1}^{n}X_n +$$ +Dann konvergiert $\bar{X}_n$ für $n\to\infty$ gegen $\mu$ +$$ + \P\Bigl[ |\bar{X}_n - \mu| < \epsilon \Bigr] \overset{n\to\infty}{\rightarrow} 0 +$$ \ No newline at end of file