From 7722b7de5279a07ba77c8b8eb3ab9f7f91ade752 Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Tue, 26 May 2026 15:13:56 +0200 Subject: [PATCH] Start des Grenzwersatzbeweises --- content/Mathematischer_Hintergrund.tex | 31 +++++++++++++++++++++++++ galton.pdf | Bin 58337 -> 66975 bytes 2 files changed, 31 insertions(+) diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index ec97d39..03fbec0 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -32,5 +32,36 @@ Anschließend ist folgendes zu bemerken: %Ref Cosi Daraus folgt, dass das Galton-Brett mit der Zufallsvariable $S_n$ eine Binomialmodell darstellt und somit die Binomialverteilung nach Definition eine Zähldichte definiert. Daher gilt: $S_n \sim \mathcal{B}in_{n,p}$ \end{proof} +Anhand des Galton-Brettes lässt sich nun wie bereits beschrieben leicht erkennen, dass bei $n\rightarrow\infty$ Kugeln sich die Verteilung der Kugeln in der Auffangung der Verteilung der Glockenkurve der Normalverteilung annähert. %"Verteilung" Fachlick Korrekt? +\begin{satz}[Zentraler Grenzwertsatz von de Moivre-Laplace]{satz:moivrelaplace} + Sei $S_n \sim \mathcal{B}in_{n,p}$ die Anzahl der Erfolge bei $n$ unabhängigen Bernoulli-Versuchen mit Erfolgswahrscheinlichkeit $p \in (0, 1)$ und sei $q = 1-p$. Für große $n$ lässt sich die Wahrscheinlichkeit, dass $S_n$ genau den Wert $k$ annimmt, durch die Dichtefunktion der Normalverteilung annähern: + \[ + P(S_n = k) = \binom{n}{k} p^k q^{n-k} \approx \frac{1}{\sqrt{2\pi npq}} \exp\left(-\frac{(k-np)^2}{2npq}\right) + \] +\end{satz} +\begin{proof} + Der Beweis basiert im Wesentlichen auf drei Approximationen: Der Stirling-Formel für die Fakultäten, der Vereinfachung der Wurzelausdrücke für große $n$ sowie der Taylor-Entwicklung des natürlichen Logarithmus zur Herleitung der Exponentialfunktion. Diese 3 Approximationen werden in diesem Beweis als wahr angenommen, jedoch in der Folgenden Arbeit weiter analysiert und bewiesen. + + \textbf{1. Anwendung der Stirling-Formel:} \\ + Wir beginnen mit der Definition der Binomialwahrscheinlichkeit: + \[ + P(S_n = k) = \frac{n!}{k!(n-k)!} p^k q^{n-k} + \] + Nach der Stirling-Formel gilt für große Zahlen näherungsweise $n! \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n$. Ersetzen wir $n!, k!$ und $(n-k)!$ durch diese Näherung: + \begin{align} + P(S_n = k) &\approx \frac{\sqrt{2\pi n} \left(\frac{n}{e}\right)^n}{(\sqrt{2\pi k} \left(\frac{k}{e}\right)^k)(\sqrt{2\pi (n-k)} \left(\frac{(n-k)}{e}\right)^{(n-k)})} p^k q^{n-k} \nonumber \\ + &= \frac{n^n \sqrt{2\pi n}}{k^k \sqrt{2\pi k} (n-k)^{n-k} \sqrt{2\pi (n-k)}} p^k q^{n-k} \label{eq1:e}\\ + &= \frac{\sqrt{n} \cdot n^k \cdot n^{n-k}}{\sqrt{2\pi} \sqrt{k} \sqrt{(n-k)} k^k (n-k)^{n-k}} p^k q^{n-k} \label{eq1:aufteilen}\\ + &= \frac{1}{\sqrt{2\pi}}\sqrt{\frac{n}{k(n-k)}} \left(\frac{np}{k}\right)^k \left(\frac{nq}{n-k}\right)^{n-k} \label{eq1:zusammenfassen} + \end{align} + Im Folgenden werden die nummerierten Schritte der Gleichungsketten erläutert: + \begin{itemize} + \item[(\ref{eq1:e})] Hier heben sich die $e$-Potenzen heraus, da $\left(\frac{n}{e}\right)^n = n^n \cdot e^{-n}$ sowie $e^n / (e^k e^{n-k}) = 1$ gilt. + \item[(\ref{eq1:aufteilen})] Hier lässt sich die Gleichung intelligent aufteilen und etwas umstellen durch die Potenzgesetze, da gilt: $n^n = n^k \cdot n^{n-k}$ Außerdem lässt sich einmal $\sqrt{2\pi}$ Kürzen. + \item[(\ref{eq1:zusammenfassen})] Ist das Resultat nach dem Sortieren der Brüche anhand der Exponenten + \end{itemize} + %weiter + ... +\end{proof} diff --git a/galton.pdf b/galton.pdf index aa3557972c9d7fe6170cf84ba625fff1ab681248..689159cfab8aa20544aee695a558e88bfec9b6b9 100644 GIT binary patch delta 42147 zcmZ6xQm`wyiGPwr$(>@0mFlGv_MvI@c5N##)j4qo8?F zVD(z;z^qN$?2#aVjGTvL$&wSa$rA#N;$$-_Y_IX)KluAj8~w-T?JnFE*(i?ZkZGlm zT_lowiLL%lnHkm#3?a@BL}TfHA>T6UC`_BgJty;L@W^KjpwNg;Hp-5q8j-Rs8~ls0 zN$_WrSs$J<=y?Sw+8!%O3k@rgI+&n)ScR}K=IPFVeA&wYbh##r=IAyCvcF91ha~=H zJHm}rsq$xW8Ix-0hlu%eH7WFz+H8&0K0rlmULxU)L^*m959h8-9|0IA4bf7!jq0ZdP!XHEByogjD15| z2Vsnt+#=H!2Q;8yV2~#Vk~-3w*X4+uhl=}kaFi5m?+Ftw0)DX!QPud(`G@f7R$B+s;*TtEwK+4{n?;FO5M6Ij zx&YEwZkaKcDfk#AK?p%bT^`F2z&6SWK{gROfG0){AY{p zvWggcKU#{5Vf^q!SKN)2uFX|vXA5({t=>HB)fn?)LhB#L1@5>TzMu9EhWbUQqx}Q` z{J%Xane52^9M?L_&(~Uqgr`$?z=0-`$Nlm_(`X=TjIW( zq;{+jha&_$!}F>(24D22?@IK;H80AGS+*sO=LEgjIHzt=_HJck3b6IXy*1yPvo}ul z_9QQCGH&V{G6#Ug3#{;D_wT<2{j6mGv4zi=l#k+koZ4xSJadhnYL^j`17TK+17c$= zzx0ejmhJ?})h%es=P9u2=N9@O6230#eer>gpm;8sg)7J4Lplk39w3Q5-BB^*nE2t@ z#Mo%(W{MP&oB_DuQ;hS(yVz~hhwZ=|M=Oje@X2gCT@(nd%%2`Ygx2omo%0$%e|a01 z8VM=(?Gbx%#*}p`6w{=F(%h+eLosweB+^*&#^wvnl~6yv(s)7cxE8VA#(H9^hLw!o zOi^5QMM2$Dp~jG>C%np2Xt0xc0R1@gW^_e&D%n)&H6~~BE@sl>W!jre-jQYz0bEN=^}1_j!`)LQPm1ae=`(3Mf)ox zuExN-@=$x!`?6)<`jO?3Le^DZ3aoKyvG}W65Ne|ntCN(8e(O?Q!3`zcLV30E2P%5J zoI!G&U*0r>M_)Y~aW5aMnYn0H3IEByW5dh4q{VQAnKd^3_;_DVY#%!x(l6WT0M7&t z+jFt;k}^7P)lsRVe=`^$er#FRf4i)s4UU^@;SsC#~f`bdB;V7K5t4s9G;)9pNj{X!*pfZo5jyq9xW# zx#wq&E5vw_bthKY)gP;2M~SIH0edk_>SF#3nfcc@wSE0xCvp#7eZJM@?$1BFP-trH-kikgN-Xckg|j-j z_bXYuHsd#_RbHY@uMeq|VyvNPdQ+Cz1N3_3<8WCogCR22~s!p_Yd9g9wB<(5d3zM zXInZHqO0n!strb9t{x2|rj0)VAuQ-!ntu97xWBt4x$o*W0N15_&k=tP*FLI2y~zRp~r`-`Ia(UxZo7ML{}-mhzn zt}|h8$Rb7ii-H5m%8`)L(+0_a9I?s>;{{Utn>TBW$q@K?;q1LBXW>Tp>y7sN-iN&n z#ksdtu6XOo!m(8bDOyu)|6Bb31N)rdJW*!GBZ!&qq$PYfQ=TMALPB)L@Z$0IDr zG?7O*$(dgSEnO6bDSqa)IBDhxLiqv~n$98|k>vs>p0`n9_YIThnDvW{EDRLRn+Vy6 z{#iOS<^GO%!Tl2}JU$(q=sYzUUF?<9Dt$bid@(HlF&PR-?ySy6dgT-z zvOOR_eeCG12(tJT=Eep+27ZW69Aj~>FvqQmtyR{}@Pq^OWZ1?*-?i~6VRYU+{rh=x zt{0OoI_VQV0yZRhDD?U17a6d|F?n?==1mZD3obORYP`ZkP2B(=?1GFS!(*8+$sjf{ zT0B(-CM8>Bmf4Q__os-MLcVS06oBjGWIZ$g`$H2B(0L?8q6?+3y zjb;zKo#Ug3=&}_=^)2+`*kR8HJiVnCNxrhdReurT3hFo~dR}v|N5vQGYmY~Uk%^SC zg1~m;&MsCD_(@RLZFbQ_mbCz6u1#5}8!b4f^QPfX6=DWTaKu4+$9yA~f{qx`YbJN# zh>A!O5uR16tQ%Lhq4Cn=3)mPP(l<>$CF#h$AZ@drV%nv!;a-8u#G^u4b0fdsP0Q+l z0uKQCx~FdZ75-z$W63^d3@zPa_zW{+mqra7=d1Fg6KZ(#Q02u7*}=#ab>Shl<|0rB z(pzcy!G=U`7ON7%U3)VW7)E~;x8V70M3}KBfTd0X+dE28yG*$WmyR*$fFljDN1gvb zPfiMvG)Ilk;|#T!8@G@7WmKRW9q3>yc60*P9S56IfHL`+{_d1Ynxt$jDORWN>p1q$ zAF6!}`OO!$lO;LXQc6RHt%rPezO~ZLneBD+#Xa43071E~ntyb^fZ>HK1!emI78=Ka zOC6_z5b76GeV#MvVL`>O33(y{eb}Rel;scqGzuh<3}18SvavB_Oi~ z1}&)}(}YzgZ$|@dGS!4l6K9BNcAy6ohioEv4F`ojN_@ir#pOV1T7|fyM^YfZtw@%+ zhQQdT!=CY%AF3l2AW4u?PQSK-n-yQcSwaVkr8Y0I$!u&zfZ0AKV9y-UN*jHKgujv> z%r`U$oS$jBQ1g^tPc`RpmrYM`$#I&FUR^$yATkKQW!-d4jL_ihwC-1$n0x?i?GIe3 z0Nz6>a`)FXo1q+?UbMY=??SXq1+J-G-t0r5t$ukDDlx!;EIM_$*s~@6)hbh^$Z>H2 zeJA2oU5$UXdtAG6XoPmG3@CDpiJ52#Pj$+piu6raMFK67xEMa|ni+PFICVaXou-QK z$EMn6fHwjIReEK8z3$4~kMsaZvu;|Kp2>ooX4yE|UxqZX^?%5r!a(3%gLCR%4y?b9 zJN4OpPalx~sDpW{evLdTfI{+H=nfnKMec`ol~~_u{j+xcRCSwq(rX(>kPs+hHIM

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