From b8883d7689307738a95ace976b5ebf2ab73ee823 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Fri, 10 Jul 2026 13:30:19 +0200 Subject: [PATCH] [PS] Start adding more expected values / variance defs --- .../ps-jh/parts/03_expected-value/01_disc.tex | 3 +++ .../parts/03_expected-value/05_variance.tex | 4 ++++ .../parts/03_expected-value/06_covariance.tex | 1 - .../probability-and-statistics-cheatsheet.pdf | Bin 1122887 -> 1121981 bytes 4 files changed, 7 insertions(+), 1 deletion(-) diff --git a/semester4/ps/ps-jh/parts/03_expected-value/01_disc.tex b/semester4/ps/ps-jh/parts/03_expected-value/01_disc.tex index 9ddfd18..56c5778 100644 --- a/semester4/ps/ps-jh/parts/03_expected-value/01_disc.tex +++ b/semester4/ps/ps-jh/parts/03_expected-value/01_disc.tex @@ -10,6 +10,9 @@ \begin{itemize} \item $\cX \sim \text{Ber}(p)$: $\E[\cX] = p$ ($\E[1_A] = \P[A]$) \item $\cX \sim \text{Bin}(n, p)$: $\E[\cX] = np$ + \item $\cX \sim \text{NBin}(r, p)$: $\E[\cX] = \frac{r(1 - p)}{p}$ + \item $\cX \sim \text{Geom}(p)$: $\E[\cX] = \frac{1}{p}$ + \item $\cX \sim \text{H}(n, r, m)$: $\E[\cX] = m\frac{n}{r}$ \item $\cX \sim \text{Poisson}(\lambda)$: $\E[\cX] = \lambda$ \end{itemize} 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 5495d52..5989366 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 @@ -16,10 +16,14 @@ $\V[\cX] = \E[\cX^2] - \E[\cX]^2 = a^2\E[1_\Omega] - a^2 = 0$ \shortproposition $\cX_k$ paarw. unabh. $\V\left[ \sum_{k = 1}^{n} \cX_k \right] = \sum_{k = 1}^{n} \V[\cX_k]$. Falls $\cX_k$ abhängig, dann gilt $\neq$ +\newpage \shortexample Varianz von bekannten Verteilungen \begin{itemize} \item $\cX \sim \text{Ber}(p)$, $\V[\cX] = p (1 - p)$ \item $\cX \sim \text{Bin}(n, p)$, $\V[\cX] = n p (1 - p)$ + \item $\cX \sim \text{NBin}(r, p)$, $\V[\cX] = \frac{r(1 - p)}{p^2}$ + \item $\cX \sim \text{Geom}(p)$, $\V[\cX] = \frac{1 - p}{p^2}$ + \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 $\cX \sim \cN(\mu, \sigma^2)$, $\V[\cX] = \sigma^2$ diff --git a/semester4/ps/ps-jh/parts/03_expected-value/06_covariance.tex b/semester4/ps/ps-jh/parts/03_expected-value/06_covariance.tex index 03943d7..98fe343 100644 --- a/semester4/ps/ps-jh/parts/03_expected-value/06_covariance.tex +++ b/semester4/ps/ps-jh/parts/03_expected-value/06_covariance.tex @@ -1,4 +1,3 @@ -\newpage \subsection{Kovarianz} \shortdefinition $\cov(\cX, \cY) = \E[(\cX - \E[\cX])(\cY - \E[\cY])]$ diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index ccae4533632bacf8f0f73fcb99f40dad457c9d28..8daf9efbed09c4a234dd9a586a8473faa49a1430 100644 GIT binary patch delta 53648 zcmV)DK*7Jq#6-QlM6eDLe>v=hfxO84U5=%#QQGVxk08#K-U-d`1lXQ?o6+{rbcCf7^dOT~Y#Vui+&T zRC79mb49t#0d?y~uq#YaK;acV$sBgLeh6K0 zMOc<|(>#PHN=i7_b?^R@5{6UzVfnBcYVAF%;Awhy5H~Qm2hdXut9xW8=;l*RwEh`B zuReudE@ltj&K~?^f7iIRs^zNN%>wMCGYQzHaig$7=KlG=pI*+&O^hG$slpgSNy zTxLX4hFx0H0$A9P7T2h1V=?hbalT#sl2P7EVmu^kbpiwZ)J$q&XSJxO_0!D#x0waa zDthUmYL^m9AcPqW!39WJdFUxDdeipxf^$CqZVPm23rC}DeiKlm4Y>U-7Qoh8Q<*=-Xgac<5Pb>bTW6vxl zoQd$r0t#!Hc}JE(8w$YG$Dn$EM*}`HQh2SPE5=C1rS`srPYnhBR(I|{u@Icv&&p@s zfoVWkW=D(j4@JEK0{V@TK(ByZJ9w|c=gt`}gz zisTv?D~jVlnt+;v+(aEf{}vl7Cqf}sHO<`KKTFWBI!>OpJ>2WGPXs;+pr20nhm zY0)Ilokp>W%>4EI+t}h!Niv1}DhEx)ejG)Ae+#6)ZmZbr70Ha41*@CHe!vh-C~6@- zstlcm_GW@gGZ{ak+u6cm_7nUo7g$^~;jK$|XCZ0L!CW$%qF=wSF~(%l%aPd1>tRc%V$ z?fSz=Y0w+u0Iki#&QfdBEpbrD5YV*brRGjdgcGzaHl@h*#vZ6*RsKc)c4ArXD{G4~ za9pWCOB>~mj{n`hfG0>QyyrLDx5e6Zf9X-oKnyEIFiU>FO>?TZ(NCkK49I3oenpDF zMzz;-KC-avny(Vqyg*eJzfUp-8lXD zCOQ7w;$ppfSg!R`R<8bU#dclne?IN@V50~o2-A0ks*Gi$w?u`$M5@w-UiGj2Ev?P% z3Os7$+UtJ|iVk!WbLh^ z&^xr3S_}MurIrbjTl1U8)+pK2e_#6UGwZv*TwbS~e^lf|pquU9`S*o8fB)z)?LAa7 zP<&0Vy{Bq3(fj8O5q1C3eat-`dPco}TIV60ugLT2M5@kV{V^ePWGPnxz5?A#sJwF0 zN4^DuBInsi8byDN%VfsPFIT%Hxp|z1Q-771IJ}Ea8~~3EBWwF~*!tI`77K7~u6C

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