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z^_v$_oy@y%@55~R?_ggtHAA0!+OS3x9Tt5FGaxYYA-K}@&}EO)n-lU6wP2Qu6FPNJEJ6JvyhnBhPZB?@v5yU$Y~ckV z8t!kCQiWe8z?!_)rihye=BJ1~<))_+2>pg|3Qo!D3gzE((>H2RTICiHsRMUejRezP z&=6Oadzw)0dlo-zd^lL|EHJ{{@?e2upmENI~u?RW1AgOeXC6|aNarI zi!i7=Q@p8fzL@G|D9h~pTIQ?XB%nUhL8gbTO^Peb=>Vjl7OIe+Np+ge2Wnr^3ggM* zI+0VzgQ_0PG^GBXcKobqI$OSH_?dJ6T=Y3(q!3})Gm%_q$ruBQ1rIPOhr;EGz-`q! zX6sADGN~+8T1v!(tyEkmdNPst>WHfiM`J9?qev@B$tPt{z)IXtg(=JEI^-&rye$tU zI-KFJoz_Y`E706#LP}IBFqU6EADGj^)P7(Gz3Mkl8LSqLr2$`m78PD2k3>;D{KOCB6Rui%dH6ew+PjGSaTx1_^^I%Vc2Xk zt9p@38f|XDhKJ^*S5F#42>W=(jX(Y9G;gGcHDPE@Wt*OK 0$} -\lemma \textbf{Eigenschaften von} $\text{exp}$ +\lemma \textbf{Eigenschaften von} $T \sim \text{Exp}(\lambda)$ +\begin{enumerate} + \item $\forall t \geq 0:\quad \P[T > t] = e^{-\lambda t}$ + \item $\forall t,s \geq 0:\quad \P\Bigl[ T > t+s\ \Big|\ T > t \Bigr] = \P[T>s]$ +\end{enumerate} +\subtext{(2) wird \textit{Gedächtnislosigkeit} genannt.} \definition \textbf{Normalverteilung} $X \sim \mathcal{N}(m, \sigma^2)$ $$ diff --git a/semester4/ps/ps-rb/parts/04_joint-distributions.tex b/semester4/ps/ps-rb/parts/04_joint-distributions.tex index 9ea0e5e..0790506 100644 --- a/semester4/ps/ps-rb/parts/04_joint-distributions.tex +++ b/semester4/ps/ps-rb/parts/04_joint-distributions.tex @@ -11,7 +11,17 @@ $$ \theorem $\displaystyle\sum_{x_1\in W_1,\ldots,x_n\in W_n} p(x_1,\ldots,x_n) = 1$ -% Randverteilung, Erwartungswert ... +\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. +} \newpage \subsection{Gemeinsame Stetige Verteilung} @@ -26,20 +36,29 @@ $$ \theorem $\displaystyle\int_{-\infty}^\infty\cdots\int_{-\infty}^\infty f(x_1,\ldots,x_n)\ dx_n\ dx_1 = 1$\\ \subtext{Umgekehrt existiert für jedes solches $f$ ein Raum $(\Omega, \F, \P)$} -% good examples in script - -\theorem \textbf{Randverteilung}\\ -\smalltext{Die einzelnen Verteilungen $p_X$ lassen sich extrahieren:}\\ -\subtext{$\forall z \in W_i$:} -$$ - \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) -$$ -\subtext{$X_1,\ldots,X_n$ diskret,$\quad$ gem. Vert. $p = \Bigl( p(x_1,\dots,x_n) \Bigr)_{x_1 \in W_1,\ldots,x_n \in W_n}$} +{\scriptsize + \textbf{Beispiel:} Finde $c$ s.d. $f_{X,Y}$ eine Dichte ist, wobei + $ + f_{X,Y}=\begin{cases} + ce^{-x} & 0 < x < y \\ + 0 & \text{sonst} + \end{cases} + $ + \begin{align*} + \int_0^\infty \int_0^x ce^{-x}\ dy\ dx &= c\cdot \int_0^\infty xe^{-x}\ dx \\ + &= c\cdot\biggl( \Bigl[ -xe^{-x} \Bigr]_0^\infty + \int_0^\infty e^{-x}\ dx \biggr) \\ + &= c\cdot\biggl( 0 + \Bigl[ -e^{-x} \Bigr]_0^\infty \biggr) \\ + &= c + \end{align*} + Also gilt wegen dem vorherigen Satz: $c=1$. +} {\footnotesize - \remark Nicht umgekehrt: Aus den Randverteilungen lässt sich nichts über die gem. Vert. schliessen. + \remark Randdichten: Integration über übrige Variablen. } +% Bedingte Dichten fehlen noch, siehe HW5, F3 + \theorem \textbf{Erwartungswert}\\ $$ \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