From 27ed2b62a9e5fc716bbced000b8fcd47ffa3adbc Mon Sep 17 00:00:00 2001 From: RobinB27 Date: Mon, 20 Apr 2026 15:10:36 +0200 Subject: [PATCH] [PS] Ex. Notes --- semester4/ps/ps-rb/main.pdf | Bin 297346 -> 299923 bytes semester4/ps/ps-rb/parts/00_prereq.tex | 7 ++++++- .../ps/ps-rb/parts/04_joint-distributions.tex | 12 ++++++++++++ 3 files changed, 18 insertions(+), 1 deletion(-) diff --git a/semester4/ps/ps-rb/main.pdf b/semester4/ps/ps-rb/main.pdf index f3a4dfa17ae7f5041e86bacf81c545675b1d6fa9..6d6939d4917e362772d356bdc9919ac1d28fd8ce 100644 GIT binary patch delta 101237 zcmV(-K-|BAlM<7s5|AVTG?OqqDu3-E{ImZ(IcooQbB*ZTpx+y$^_mrHVGrGFAJQqqHUA%i;eYjlQ_z&vnn`3tL_HOypy889~ z?W$_(=8^wzUKSd{V!W_U&+Znp*N^_0 zciAg(E(P9l^%zt8kXHlCIVnBug znchtFqHZCgp%jFxld)LUN|aGQy3kg}gS{wAM$AbU%=m>Etg^Q=ub9o8&{0WDJbc=; z`I!`vg#<;RvAkA9sphcKV0C6DjV_esUU#0qEN7Sa;}_Ur;q2^}(DY{Jcf|ThUC+LO zCIy{IR>n$vHf|<0`hQe*cy(&T>O{V}cr?Ka7PiUaREJjA%Ns%pa~v~FdyX@1Y^o|Q z5tif4oE|euQB+~AnnnFQtX`VJVn&goY|%{_<6;UAIK@oqQJc(ax%X{y)7Mm+oT~63 zU^4)Asf7PfXkAO?-ySt@e*mhKM~h`Oh+@&#IAp|FcEA-68-Hmk^8)EvFGRjMDRn>viW($Yy%El_Gv2gMd;bC2EH@YYR$5EpeOp@V%(hyO4O0uPu&_|d6 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zy!IchZU|R9_^J`U-9KR30l9Xvn;+!GJNd%V0J|rSO*a7L!HNVe*a5qrHndnJKr>Pl ze1mS7ILbP;QBIdjBD`gc&N2Lx(!;Zd*nM4cYIeQI%?crLu%N0@^)Qn@;Il zMTQNTtp_ld(&nSI(UcOaM8;M7BunG_DE*t5*jWWzLp^voamHX;EhJ^03?-iNRc`V_ z)r1%6re7FxJ8{i+JQLZ-Te9U{#e3F*wH9V;5b2A31vk4p`uKYP<;&|mSVrdcCS~uf ye)3jY)U diff --git a/semester4/ps/ps-rb/parts/00_prereq.tex b/semester4/ps/ps-rb/parts/00_prereq.tex index 7821ad9..df30d9a 100644 --- a/semester4/ps/ps-rb/parts/00_prereq.tex +++ b/semester4/ps/ps-rb/parts/00_prereq.tex @@ -4,4 +4,9 @@ $$ s_n = a_1 \cdot \frac{1-q^n}{1-q} \qquad \underset{n\to\infty}{\lim} s_n = \frac{a_1}{1-q} $$ -\subtext{Wobei $a_i = a_1 \cdot q^{i-1}$} \ No newline at end of file +\subtext{Wobei $a_i = a_1 \cdot q^{i-1}$} + +\definition \textbf{Binomialkoeffizient} +$$ + \binom{n}{k} := \frac{n!}{k!\cdot(n-k)!} +$$ \ No newline at end of file diff --git a/semester4/ps/ps-rb/parts/04_joint-distributions.tex b/semester4/ps/ps-rb/parts/04_joint-distributions.tex index 9d92f45..9ea0e5e 100644 --- a/semester4/ps/ps-rb/parts/04_joint-distributions.tex +++ b/semester4/ps/ps-rb/parts/04_joint-distributions.tex @@ -28,6 +28,18 @@ $$ % good examples in script +\theorem \textbf{Randverteilung}\\ +\smalltext{Die einzelnen Verteilungen $p_X$ lassen sich extrahieren:}\\ +\subtext{$\forall z \in W_i$:} +$$ + \P[X_i = z] = \underset{x_1,\ldots,x_{i-1},z,x_{i+1},\ldots,x_n}{\sum} p\Bigl( x_1,\ldots,x_{i-1},z,x_{i+1},\ldots,x_n \Bigr) +$$ +\subtext{$X_1,\ldots,X_n$ diskret,$\quad$ gem. Vert. $p = \Bigl( p(x_1,\dots,x_n) \Bigr)_{x_1 \in W_1,\ldots,x_n \in W_n}$} + +{\footnotesize + \remark Nicht umgekehrt: Aus den Randverteilungen lässt sich nichts über die gem. Vert. schliessen. +} + \theorem \textbf{Erwartungswert}\\ $$ \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