From 404ea2cb6bd5bfeae059d69d05fc732afdb24113 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Thu, 19 Mar 2026 09:19:47 +0100 Subject: [PATCH] [PS] Finish continuous distributions --- .../02_discrete-rv.tex | 3 +- .../04_cont-dist.tex | 2 +- .../05_cont-distributions/00_eq.tex | 16 ++++++++++ .../05_cont-distributions/00_normal.tex | 5 ---- .../05_cont-distributions/01_exp.tex | 15 ++++++++++ .../05_cont-distributions/02_cauchy.tex | 6 ++++ .../05_cont-distributions/03_normal.tex | 28 ++++++++++++++++++ .../probability-and-statistics-cheatsheet.pdf | Bin 332850 -> 342932 bytes .../probability-and-statistics-cheatsheet.tex | 9 ++++-- 9 files changed, 75 insertions(+), 9 deletions(-) create mode 100644 semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/00_eq.tex delete mode 100644 semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/00_normal.tex create mode 100644 semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/01_exp.tex create mode 100644 semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/02_cauchy.tex create mode 100644 semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/03_normal.tex diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/02_discrete-rv.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/02_discrete-rv.tex index 4ea1dab..22b3577 100644 --- a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/02_discrete-rv.tex +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/02_discrete-rv.tex @@ -7,7 +7,8 @@ \shorttheorem $(p(x))_{x \in W} = \sum_{x \in W} p(x) = 1$ -\shortremark $\forall (p(x))_{x \in W} \; \exists$ eine Z.V. mit dieser Verteilung. Können desh. schreiben: ``Sei $\cX$ disk. Z.V. mit Verteilung $(p(x))_{x \in W}$'' +\shortremark $\forall (p(x))_{x \in W} \; \exists$ Z.V. mit dieser Verteilung. Können desh. schreiben: +``Sei $\cX$ disk. Z.V. mit Verteilung $(p(x))_{x \in W}$'' \subsubsection{Zusammenhang Verteilung, Verteilungsfunktion} diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/04_cont-dist.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/04_cont-dist.tex index 7eae8a6..c55f360 100644 --- a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/04_cont-dist.tex +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/04_cont-dist.tex @@ -5,7 +5,7 @@ $f_\cX$ ist Dichte (pdf) von $\cX$ % \shortdefinition[Stückw. st. diff. F.] TODO: Consider adding this, Slides Chapter 3, P56 -\shorttheorem Sei $F_\cX$ st. stückw. diff. auf Partition $-\8 = x_0 < x_1 \ldots < x_n = \8$. +\shorttheorem Sei $F_\cX$ st. stückw. diff. auf Partition $-\8 = x_0 < x_1 < \ldots < x_n = \8$. Dann $\cX$ stetig, mit $a_k$ beliebig und \[ f_\cX = \begin{cases} diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/00_eq.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/00_eq.tex new file mode 100644 index 0000000..9e4ad3b --- /dev/null +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/00_eq.tex @@ -0,0 +1,16 @@ +\subsection{Stetige Verteilungen} + +\subsubsection{Gleichverteilung} + +\shortdefinition $\cX \sim \cU([a, b])$, f +$f_\cX = \begin{cases} + \frac{1}{b - a} & x \in [a, b] \\ + 0 & \text{sonst} + \end{cases}$ + +\shortremark $\P[\cX \in [c, c + l]] = \frac{l}{b - a}$, +$F_\cX(x) = \begin{cases} + 0 & x < a \\ + \frac{x - a}{b - a} & a \leq x \leq b \\ + 1 & x > b + \end{cases}$ diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/00_normal.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/00_normal.tex deleted file mode 100644 index 6fc1a8c..0000000 --- a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/00_normal.tex +++ /dev/null @@ -1,5 +0,0 @@ -\subsection{Stetige Verteilungen} -\shortdefinition[Gleichverteilung] $\cX \sim \cU([a, b])$, falls $f_\cX = \begin{cases} - \frac{1}{b - a} & x \in [a, b] \\ - 0 & \text{sonst} -\end{cases}$ diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/01_exp.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/01_exp.tex new file mode 100644 index 0000000..2ec82bf --- /dev/null +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/01_exp.tex @@ -0,0 +1,15 @@ +\subsubsection{Exponentialverteilung} +{\scriptsize Wie Geomemtrische Verteilung warten auf Erfolg} + +\shortdefinition $\cX \sim \text{Exp}(\lambda)$, falls +$\forall x \in \R f_\cX(x) = \begin{cases} + \lambda e^{-\lambda x} & x \geq 0 \\ + 0 & x < 0 + \end{cases}$ + +\shortremark[Gedächtnisl.] $\P[\cX > t + s | \cX > s] = \P[\cX > t]$ + +\shortremark[Verteilungsfunktion] $F_\cX(x) = \begin{cases} + 1 - e^{-\lambda x} & x \geq 0 \\ + 0 & x < 0 +\end{cases}$ diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/02_cauchy.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/02_cauchy.tex new file mode 100644 index 0000000..58a2b16 --- /dev/null +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/02_cauchy.tex @@ -0,0 +1,6 @@ +\subsubsection{Cauchy-Verteilung} +\shortdefinition $\cX \sim \text{Cauchy}(x_0, \gamma)$, falls $\displaystyle f_\cX(x) = \frac{1}{\pi} \frac{\gamma}{\gamma^2 + (x - x_0)^2}$ + +\shortremark[Verteilungsfunk.] $\displaystyle F_\cX(x) = \frac{1}{2} + \frac{1}{\pi} \arctan \left( \frac{x - x_0}{\gamma} \right)$ + +\shortdefinition[Langschwänzige Verteilung] für $|x| \rightarrow \8$ nur sehr langsam gegen $0$ (quadratisch vs. exponentiell bei Norm. V) diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/03_normal.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/03_normal.tex new file mode 100644 index 0000000..f9c4dc8 --- /dev/null +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/03_normal.tex @@ -0,0 +1,28 @@ +\subsubsection{Normalverteilung} +\shortdefinition $\cX \sim \cN(\mu, \sigma^2)$ falls $f_\cX(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{\frac{1}{2} \left( \frac{x - \mu}{\sigma} \right)^2}$,\\ +mit $\sigma$ Standardabweichung. Auch: Gauss'sche Verteilung + +\shortdefinition[Standardnormalverteilung] $\cX \sim \cN(0, 1)$:\\ +$f_\cX = \varphi$ und $\F_\cX = \Phi = \int_{-\8}^{x} \varphi(t) \dx t = \frac{1}{\sqrt{2\phi}} \int_{-\8}^{x} e^\frac{-t^2}{2} \dx t$ + +\shorttheorem $cX \sim \cN(\mu, \sigma^2)$, dann $\frac{\cX - \mu}{\sigma} \sim \cN(0, 1)$, also: +\[ + F_\cX(x) = \P[\cX \leq x] = \P\left[ \frac{\cX - \mu}{\sigma} \leq \frac{x - \mu}{\sigma} \right] = \Phi \left( \frac{x - \mu}{\sigma} \right) +\] + +\shortexample für Phänomene modellierbar mit Normalverteilung: +\begin{itemize} + \item Streuung von Messwerten um Mittelwert + \item Grösse und Gewicht der Bevölkerung + \item Renditen von Aktien +\end{itemize} + +\shortremark Für $\cX_i \sim \cN(\mu_i, \sigma_i^2)$ unabhängig gilt: +\[ + \cY := \mu_0 + \sum_{k = 1}^{n} a_k \cX_k \sim \cN\left( \mu_0 + \sum_{k = 1}^{n} a_k \mu_k, \sum_{k = 1}^{n} a_k^2 \sigma_k^2 \right) +\] + +\shortremark Für $\mu \in \R, \sigma^2 > 0$ und $\cZ \sim \cN(0, 1)$ gilt\\ +$\mu + \sigma \cZ \sim \cN(\mu, \sigma^2)$ (nützlich für Simulation) + +\shortremark $\P[|\cX - \mu| \geq 3\sigma] \leq 0.0027$ diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index 6800e8e4ea1908bcea16aabb820e442bd78cb743..775ea50bb057aaf06635bdac24f7f107734c9772 100644 GIT binary patch delta 222899 zcmV)3K+C_fs1%g<6tG7L7&$W_Fd%PYY6?6&3NK7$ZfA68AU8QSF_U2gD1YreU3c3y zmhbr$>YIEfLWyrMJ>A)4Ch4}bnbU64&F-dsP;AClBFhB)j-SvG(2_{Fz>d2#ad%YW}4Jbn4#9}g&WlFjIhYLaDUrZPj65VO8%m_gXOaWBf|h;1I^lG_yx3l!UVjv;%^DVOzB;{l zk57x$4hNobMwFf{SPlR=h;BA#Ckw9h{Kam$TV3x~@3G~V&m^n{CAow>!MP}+VY~%1 z#N=+g@0Od@HN4C8{NHYNAIsNF?!N|r5=|v60Vla|BVO!wAFdyL`Q`xyRYa#lZFZawlLY&fos)>5>v?dksGl zK{cl{ID?eS9MHUe1UtwSg#)k-o*Bv*pHX5I=L%PnNdxF#bx=Pzoju@^D47E`)DNL6 zt_aI=ZkmViL`exWpzhs&Qo?X*Uo0P1L#@4M6+BJv4&nv|_W*jTVRet}1l@e9iPoRt z^XgUTvrLo=y`ozC4r!@=uO+#3(oocn=R0=Eq@%1vWZYZ3&H0P%O7&xdG!m| zYnFvSQ9gGA2C%(eFl~CF5%Z_(; zgfkHySwLYeGw;YUXhQ**`WRFX@MyqmMhdSLbj29SxYXX8@T#G}3+vAPCl-QJ`&#+T zJFrbV2Y*f7cQvURfZy=E!vC`TsOOb)WG=nOwQOS1|F6Z{vFEfNpnFW~+3Y>04R%HZ zHUadSg>e3q$?V1I{BpT?ygo19oG>%rF5w|foH8lr&wwx6BgNPW94m7vz_C~ zYx-*)bUb_@za-E4PZ{U&7uDPc_~D6d{&u-uUw=R9viIiA1t2S<)XI0cbfU&nN?Px< zrXqpidlaO#CP-i0N<8Zu8zl=XXSh*WXH?x8_3?lq-ELG%-0B6}xL$w>E0Swqtth+$ zaRPeIiJ}gmzr@DMiBO1FO*1>LDW@8t+~uyvK&ilp;J{Ol=0=dtVnPyy_;XPBlK5o}VV#gKlAt>-;40Z!zfZ;LN0|`HSWz=SE4HIX10|C4 z7~T&Z%=S#m1taBxxew6iPO4=(0ksZSs-_ZMfsiQy5{n!l5SrO1*4{u8lNmE6(0|@g zT$3Ec^yXD~H_xeRdiS5y+FSjwB@C7gJz=oyy^#hT41Xl;jRE^)d$hNzO{u+Ie;6qb zdP5wbw|UrE>TS9u4+)Hl@h*#vZ6*RsKc)d16`aD{Ij)a9pXdh8}KwQx7~r zQsF(n*}g5-u1k+%24Yw#f?4wWb$^;uy^X#aC1pT1V}dJE1U9O@p7W9I#YlMM$o^Gy zf#&D2ot;NSAPQ=N&)G6KhdJ^~ffT}F3Mi88*IFBbV$icFkQ}6mzFqvPZ^G?2aK`uh z-nZ}Pz5CnJOm1^yC+muHr{kS1Ym$>x7T!IxUWoa-ggpo$w9qK=xva}?jDPS6LUAZG zzb~>lc-W*ZAWC4-)y>o~0pooGVg~&3>!V)04gmTT)I813 zhqI|#HuxLP@apnIX|%Nh1%+dpWi%pcfTRKI>pflu2HrUR_$oR6+u~xqdswdZQ&z72 zFU59U>^|-Wuu%jPgz39NRe#2^(OaTIpCeW2La+MQ{+8C}b_E_aa_#l+gQ5f7L|wFF z=XuIQicpIzA#qLg-Lp=aFxCm0O@MY{EhJojCGEwM+-gx1Dsmhu_kPoRVe`1m6EE>IRZ&qh*e`g2@3 zlYD-;N;rWePdU;M4E;r7;_zE^;sAJT7+D*t!`441)meaRbG2I)+g*H(WTO_3<@RxL zG0*2c`n5I9F^vZ<1M=n*l42OZ0PP8?=0c9^{G@D&hc6IC6TGwP?@@APVrFv$NH1^$Xx^6Dqcg z_M@|T&+!Do;y7693?p5?9z`3zmcC=}`

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zjwzFgMIunLru?r;{1`vkeZ&;GM=;|Y;)}6(FG>UQjX?ys@f=*~xN+!&-3xJ98g$nG ep0CpRU?E}C{rMo_5!hJ1cV)m)Qc5UE!u>xWid1R< diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex index 81c889e..9bb7e0f 100644 --- a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex +++ b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex @@ -9,6 +9,9 @@ \setupCheatSheet{Wahrscheinlichkeit und Statistik} \loadGerman +\setnumberingpreset{off} +\renewcommand{\definitionShortNamingDE}{Def} +\renewcommand{\remarkShortNamingDE}{Bem} \setsubsectionnumbering{section} \renewcommand{\examplenumbering}{off} @@ -47,8 +50,10 @@ \input{parts/02_discrete-continuous-rv/03_distributions/03_hyp-geom.tex} \input{parts/02_discrete-continuous-rv/03_distributions/04_poisson.tex} \input{parts/02_discrete-continuous-rv/04_cont-dist.tex} -\input{parts/02_discrete-continuous-rv/05_cont-distributions/00_normal.tex} -% \input{parts/02_discrete-continuous-rv/05_cont-distributions/} +\input{parts/02_discrete-continuous-rv/05_cont-distributions/00_eq.tex} +\input{parts/02_discrete-continuous-rv/05_cont-distributions/01_exp.tex} +\input{parts/02_discrete-continuous-rv/05_cont-distributions/02_cauchy.tex} +\input{parts/02_discrete-continuous-rv/05_cont-distributions/03_normal.tex} % \input{parts/02_discrete-continuous-rv/}