From 1a0aedeca50c91d5bdc0fdb354490e2b7439ea22 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Wed, 25 Mar 2026 13:56:58 +0100 Subject: [PATCH] [PS] Start expected value --- .../parts/02_Sensors-Actuators/03_cameras.tex | 9 +++++++++ .../ps-jh/parts/03_expected-value/00_cont.tex | 13 +++++++++++++ .../ps-jh/parts/03_expected-value/01_disc.tex | 14 ++++++++++++++ .../probability-and-statistics-cheatsheet.pdf | Bin 341394 -> 343639 bytes .../probability-and-statistics-cheatsheet.tex | 6 ++++++ 5 files changed, 42 insertions(+) create mode 100644 electives/others/amr/parts/02_Sensors-Actuators/03_cameras.tex create mode 100644 semester4/ps/ps-jh/parts/03_expected-value/00_cont.tex create mode 100644 semester4/ps/ps-jh/parts/03_expected-value/01_disc.tex diff --git a/electives/others/amr/parts/02_Sensors-Actuators/03_cameras.tex b/electives/others/amr/parts/02_Sensors-Actuators/03_cameras.tex new file mode 100644 index 0000000..adbf8da --- /dev/null +++ b/electives/others/amr/parts/02_Sensors-Actuators/03_cameras.tex @@ -0,0 +1,9 @@ +\shortdefinition[Pinhole projection] +$\begin{bmatrix} + u \\ v + \end{bmatrix} + = \frac{f}{z} + \begin{bmatrix} + x \\ y + \end{bmatrix}$ + with $f$ the distance to the lens and $z$ the full distance diff --git a/semester4/ps/ps-jh/parts/03_expected-value/00_cont.tex b/semester4/ps/ps-jh/parts/03_expected-value/00_cont.tex new file mode 100644 index 0000000..ec9f7a2 --- /dev/null +++ b/semester4/ps/ps-jh/parts/03_expected-value/00_cont.tex @@ -0,0 +1,13 @@ +\subsection{Allgemeiner Erwartungswert} +\shortdefinition Für $\cX : \Omega \rightarrow \R_+$, $\E[\cX] = \int_{0}^{\8} (1 - F_\cX(x)) \dx x$ + +\shortremark $\E[\cX]$ immer definiert und endlich oder unendlich + +\shorttheorem $\cX$ n.-neg. Dann: $\E[\cX] \geq 0$. $=$, wenn $\cX = 0$ fast sicher + +\shortdefinition $\E[\cX] = \E[\cX_+] - \E[\cX_-]$ mit $\cX_-$ auch n.-neg. + +\shortremark $|\cX| = \cX_+ + \cX_-$. Für $\cX \geq 0$ ist $\E[\cX]$ immer definiert. +Falls $\cX$ kein konst. Vorzeichen, $\E[\cX]$ undef. + +\shortremark $\E[\cX] = \int_{0}^{\8} (1 - F_\cX(x)) \dx x - \int_{-\8}^{0} F_\cX(x)$ 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 new file mode 100644 index 0000000..5b302eb --- /dev/null +++ b/semester4/ps/ps-jh/parts/03_expected-value/01_disc.tex @@ -0,0 +1,14 @@ +\subsection{Diskrete Zufallsvariablen} +\shorttheorem Für $\cX$ mit Werten fast sicher in $W$: +\[ + \E[\cX] = \sum_{x \in W} x \cdot \P[\cX = x] = \sum_{x \in W} x \cdot p_\cX(x) +\] + +\shortremark $\E[\cX]$ wohldefiniert falls $(x \cdot p_\cX(x))_{x \in W}$ abs. konv. + +\subsubsection{Beispiele} +\begin{itemize} + \item $\cX \sim \text{Ber}(p)$: $\E[\cX] = p$ + \item $\cX \sim \text{Bin}(n, p)$: $\E[\cX] = np$ + \item $\cX \sim \text{Poisson}(\lambda)$: $\E[\cX] = \lambda$ +\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 120ee4d769877e01f53b6158027ad11948384b97..b6bc250945843fedf5f9589b7720b2e3f0916002 100644 GIT binary patch delta 71504 zcmV({K+?aG>lD}g6tG4K12QlxKIvYN)ci}00EmOQkyn;4UG^h=hutV>qKwO1PbWS5JYs> z4@vjbzj4>=6Uty-lAG!L=CZ2KK?`HKkc29UDEp5I4Drpxj`P4`tpv2i$M7G#MhyH@ zlQn<*`sIi7xBq%Nrv%zw!;eHz&FKWrAmy?IG~Ygg9b}5a0a!-Q3}uW@C^3q2g)7OV z0rams*giO&Jm8WjSpqg}A3|4L5muDkG!Nm4k`idZw)gOfgW=S^=pNQXErVw@JWcN& z;syr)0D9WO>L1w&`uWro-G0W;>sO(di^+e3x044y*fnmgs#|rtS%ArOA_3hrZWJ~s z{J-$~>E*<2VwxP(oy}MQ-2tw0SrADXc4^5FU|~ase51OJ#l$Do`FinFL3uAZ3XrVT z2@Lc@GpU)K)oeShA1404)g)?G)7Ai0yHryG(JW{TEflXy)llGt+s?x$7J^gzntK);*rtD-hi2P%J*h1Kzu|d>{}ts?&nxH1TzZeIYhuy= zuhrYJ=X5(j_n2;Hv-6lX*qIR61kh_1!}(Joix-RY%lYi_^1OO;!pw9%hlezC%A}k= z1HNpJ6k{iFxb*b<92Lr@$iBCl0EA`6`ZGERMQaPhbOlA+xdTTdHtx% z-kUdPfUJyCE8msUiyBWUX}!~$iUfuqP>|M|Abs&G@vLublq{^A;YVeiQFUk3#{-7+ zyHPK3s~2qJdjTe_NU4FfqVNvH3FtW|ih6+lk{YWdLLpu?&Fr|wO*KKe-L9uVsliC# zz|(-{CXmizLKcPibOoYU4vBw6c32Lg+37{yaTa3?h@_mi_KpwVyHW12-xR@5OI(cvTfQ0@c&8jWl!Y$~bMt~3$a3jr_c9%%rv zAspDv4CuPz@fw8q38zJqvh@1LA~EyV)2~yDM;}XD!&9(F z5CEex5$sp6qGozltw)UpN+jngydOH4?U}d*BW}U`2k3Jr^)j7+T8AsOr4n6%kSPHY ziyR;jn#m{D-arzQ1v7sp(%w*9lN`kK=2dvNEU9XG51(wcx9!80Fz6Zv!l3KDkp?{s zejFbn1Ar8>n0_=2pn{LU2LdJj&6uGLE6S>~l163{DU-X|R zmgT;(799h}l?rR<;U+Zoz!M}D!SkE-+iK~%3@BzKhB*<;lHY%?^PK8!^3^0M1G1SC zT#+KMQSJ4Vk8CeS!XroauaXNiJx}fIG$8^}P!mGVR=_zdkzWd=5FS%Nk!-)#+7J|j zo+W|gAWih`;#YkWZoh#uzTfx0eLwHr-Y9@TWunr+$ zyl+6vfM0%n)Qi^GWCPWR8G+=#$$MeX* zo2DOMWygP8T`V^b-C93oZuNhu*2`-1aWjC8BA6gd-xYtVGM0_r5*7L!sY(}m)xY+) zv^KXZ@Tie%uYVsD9q11+3KIC9RoDKo@M zQ5u=hJG7Tt3;YgCEfb`)=C^>YQ8Lhj@4WYg_1<63uXFA{DoP^I(RT0t`^?>c^qBS@ zDj6ugrq_SoQ@7dE{}&9AbO6#J=AHmO)BZofXbdRQtykoE=<;icJz8e8HcpG2&j5b{ zB}~A_7uk1#vdC#Ml2X&3)54kL)5}H12_$*Wk%nOCFESH{-;xstz*ED>t)V(@{c~2G z1-MpMn?<$Wq}NC`YVlZZpB5L(a@wO`ThkoVc;J6x-e&!(-o<(a+Q;gwa(_Ktz#l+B z=WjQ)E^=Ni@p})QmVl~dOr^UX*!E_i2%OstSb1EcpKIHoH763+VHgu9edxW9D5YQf7DSu zd+~p4t!qz)uD$(c9zsJE0VT?Ll0*zr*eNHRS{_$7`nm5AQK2kyCr(jL<$s)wT}o|< z0G?s=AXh8{ir5#@sZ${;l5Y;pbWX7zsK>CJzl{EYhhYH@zEI`8qy8Q3rv zUwrSByDLsPV~8qguZ-rK?CFwg1~|YY*I<99gHm-Jav_wj7L|2ZBl@F&W4}26nk)%u zW~2RU7(JfVKz|FPzStTtVP@oczwL6SVU6r+iXv~iOz`i=Z{94HXFxUblQ(^V8riUz zZ8S)mHPZs*t)ubg!0Q8(CG2IM2?1nCDv-9InkyspMrm4 z#zi}wB8q2GnkdW=(F9aN2c6nsBjv`0oEajIy1yV4kJ?|(< zBxJ8xib2r}r5GSoinaop;tgkg*b%dL5RJGF+53mYyMTU_r`UV6N7%l@-ILpq8p9A{ z^!PCXU!mQ9Ykpf_y&+yo)5c0sk(7V-Kl6)aR!Cc9BN%^mxjMiFFj;3XMgA%~RB9Qx zu%~I4tl9*v`9@!-L(Mcu1e}yC6b)OHfR#i6TN5QQx{ne-H=6blg1RHjDGCWe``lrc zfkOF!`~66t*+m~s-kk)H@a$k1a$|#?2#|=2p|LA<^zmZrk$Y8lV$jbv6diwdTR15S z8)fWDL=z_)YvQbPFH$B>*n|80hP!6qwCuLmyxB11J&c=m_Xn6ZYqZQKHUuC#v*v%~ z!^Wj_fu2abU!E8-T`xX3X0c3*z~1|Oh%rvLjd5RP1Phf(s_6PaQcGS+q5;39GN322 z`|QHIlwDYsgi1Iq5`=f9Ud{l7?jG+BJ7V0_B*rTsj6{b{M>v1kZ2QyD}QP z)?Ouei5Cuo71AWg_wnw>)QYukjHe32qB3q%ZWsOC}gi=1jEsFY!8h`*mhO` za?!F6!XJ7d>;`|PvVw)?(}%gJ5nk^=rX)k{jUzesD~twJ?B2zGH(cyB^95>aEziBo zw{A4r&E9(dK#M|vpe_L0!RvLWGMi#J1b(D>&~&EL{Zl(y zAom2(_N9p?S%OD8&48cMp>DD6@H6be9?m#-VnkwhJjthd=Y3tEXlzb+2Wz&-vuoRC zOk&ch{}_J(t~9%IRmbGY)ycB+CCJ>LFl;oANhlSt{=mV!O2||orj-Jyuquhtv{$BL zfLp4scIUP)3S^&l!VF3f21`S^3+Dt?dxE4dgbf<0piu!l9+fzvjEZaP#F3%K))-Ne zZ-=y~HPLWZP!yNC?D4~E!i)}=iFKDrf(oLc{~>?NBngF{;$@2BHMV_mIEgZvep|*2 zNLw5ZO0!a-z*?~7y;OL&q(Y%VkyJaRLf>d_sjwO3AgQpy(3Dh&(M~dk^%e2eO^99XmgQNcl!Ya*D8Y9hT)&?}IM zev~WlHFJFGU+EnWg#CZ^{wHv*Q-WQ2h?&}lymp5n&jHgsX4BA{_ z)FpfZ&~RYk-S?l=vTcLMC!wVoQSldmZ$NZ$7K=fg^Q&TPf|R;uw+d3aGsF{J397*P~b3@HjJZo4RClskX+ z@|H7|hQ~K!c|f^;`ckG+r07QS4r;! 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