From d75e317fd4ad06ec0c8f1838c229194900629509 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Sat, 11 Jul 2026 12:31:14 +0200 Subject: [PATCH] [PS] More expected values and variances --- .../02_discrete-rv.tex | 2 +- .../03_distributions/01_geom.tex | 4 +++- .../parts/03_expected-value/05_variance.tex | 2 ++ .../probability-and-statistics-cheatsheet.pdf | Bin 1121981 -> 1122372 bytes 4 files changed, 6 insertions(+), 2 deletions(-) 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 a36dc29..b80f6ed 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 @@ -14,5 +14,5 @@ \subsubsection{Zusammenhang Verteilung, Verteilungsfunktion} -\shorttheorem $\cX$ disk. Z.V. wie oben, dann ist Verteilungsfunktion: $\forall x \in \R \; F_\cX(x) = \sum_{\elementstack{y \in W}{y \leq x}} p(y)$. +\shorttheorem $\cX$ disk. Z.V. wie oben, dann ist \hl{Verteilungsfunktion}: $\forall x \in \R \; F_\cX(x) = \sum_{\elementstack{y \in W}{y \leq x}} p(y)$. \textbf{Umgekehrt:} $p(x)$ ist die ``Sprunghöhe'' im Punkt $x \in W$, $W$ pos. Sprünge in $F_\cX$ diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/01_geom.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/01_geom.tex index 69e2294..aa5d902 100644 --- a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/01_geom.tex +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/01_geom.tex @@ -2,7 +2,9 @@ {\scriptsize Warten auf den ersten Erfolg (in $\8$ Folge von Bernoulli-Experimenten)} \shortdefinition $\cX \sim \text{Geom}(p)$ mit $W = \N \backslash \{0\}$ falls $\forall k \in W$\\ -$\P[\cX = k] = \P[\cX > (k - 1)] \cdot p$ mit $\P[\cX > k] = (1 - p)^{k}$ +$\P[\cX = k] = (1 - p)^{k - 1} p$ + +\shortremark[Verteilung] $F_\cX(k) = 1 - (1 - p)^k$ \shortremark $\P[\cX = 1] = p$, da wir Konvetion $a^0 = 1$ verwenden. diff --git a/semester4/ps/ps-jh/parts/03_expected-value/05_variance.tex b/semester4/ps/ps-jh/parts/03_expected-value/05_variance.tex index 5989366..d99d0ea 100644 --- a/semester4/ps/ps-jh/parts/03_expected-value/05_variance.tex +++ b/semester4/ps/ps-jh/parts/03_expected-value/05_variance.tex @@ -26,5 +26,7 @@ Falls $\cX_k$ abhängig, dann gilt $\neq$ \item $\cX \sim \text{H}(n, r, m)$, $\V[\cX] = m\frac{rn - r^2}{n^2} \frac{n - m}{n - 1}$ \item $\cX \sim \text{Poisson}(\lambda)$, $\V[\cX] = \lambda = \E[\cX]$ \item $\cX \sim \cU([a, b])$, $\V[\cX] = \frac{(b - a)^2}{12}$ + \item $\cX \sim \text{Exp}(\lambda)$, $\V[\cX] = \frac{1}{\lambda^2}$ \item $\cX \sim \cN(\mu, \sigma^2)$, $\V[\cX] = \sigma^2$ + \item $\cX \sim \text{Cauchy}(x_0, \gamma)$: Existiert nicht \end{itemize} diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index 8daf9efbed09c4a234dd9a586a8473faa49a1430..77c9e89dad6bf4aa304001a6cab5113f13a58ba7 100644 GIT binary patch delta 31333 zcmV(*K;FN-y+p*oM6h!Zlktfwvz`!o0e_?fhg9<)Zq}=EaF#I@EUR5EDhL;re4m(^wgs!+E zEX%oR9>NnPC7kQJcmGKV!>RqSd{_;&_MTPnG`%~B8yMUJ=&6R)J+c#Y^Qk6U{|uj3 zpF%Gevj=Zy4}P+1+*;Lg)$L{hc7M{D1Z>l|QP?1J|NP%iFK6W@#*cW_*@9)z9S|Tc zGa@O&E-h&RENn=NYgDzdnE0eP->!biDDNdP9+I^>fq{N%Cbh7$TGZ3}Y3Bah%z|bW zz4TDEO9>?q!i#b&p})4MIU#p)a>U*gPiSXM;BfisJz75~w(XO!-1z_r9P(8q-0iPKuyjIW^VG#bY5;!6^9ujV@_(bASKwAg^dHxrC$(|UmJF|B7acuX7Y?1CwP zT(c0&7pwEj<>K-Bym(`gf0;O3a7dm3H`W?X5@RR!O#K)=ZH7S;nc;zhpEP>D@fd?B zk~rc5I^aD1p`0&})gr(rbpGZ9#Dnef;fd|>?Q*@oe$-{|O$Q4=RewgQmDqCW#GI!f zafIKrW+Z{#d-S8VG`JFWK{TvT&)6teSSiDe$~vR!&Zv*~7!o!G$y>G3tzNK=>jhY_ zBDn^}isCqsCZOgZH&F-Bzs1JNiBO1DO*6Om&k}UI-250Q6&M{<0P^cm+!*fhc`qhy zK=4ZA(&OZJ%bU}Ss(<4o#+VFbLEhRszW?6k(pz<=2?sG1C|fQ)F}3x*fJL{IvTF1V zpuE!29ucgdLpGwrNBBdPKk%>7$fo?Jl3IO96QP|5@S^IGdJr4Jf!Qp8sw*C@fsdbX zS~SUXr%|jTGk-n*Hnwmsq^iuVu{co~D;fDmAK4E6v5U{fZ5{R&pp%+HGL zsL?=8`wnJ%Cgp;Wa>3jW(B@96WdassSE{BGU2%{p0rHC6LmV`-&#b$F940en zOrX1=uqHVO>3_|q@NJ$`)%5N^sdcydVM`P&8+xK(*?S`kIvD<$bTeFI_! 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