From 81ae2c5ee4e6b174f97964694ed196a0a7174474 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Wed, 22 Apr 2026 16:39:44 +0200 Subject: [PATCH] [PS] Limit theorems, law of large numbers complete --- .../01_weak-law-of-large-numbers.tex | 7 +++++++ .../02_strong-law-of-large-numbers.tex | 9 +++++++++ .../03_central-limit-theorem.tex | 11 +++++++++-- .../05_limit-theorems/04_chernoff-bounds.tex | 15 +++++++++++++++ .../probability-and-statistics-cheatsheet.pdf | Bin 386713 -> 392775 bytes .../probability-and-statistics-cheatsheet.tex | 1 + 6 files changed, 41 insertions(+), 2 deletions(-) create mode 100644 semester4/ps/ps-jh/parts/05_limit-theorems/04_chernoff-bounds.tex diff --git a/semester4/ps/ps-jh/parts/05_limit-theorems/01_weak-law-of-large-numbers.tex b/semester4/ps/ps-jh/parts/05_limit-theorems/01_weak-law-of-large-numbers.tex index e69de29..27591bc 100644 --- a/semester4/ps/ps-jh/parts/05_limit-theorems/01_weak-law-of-large-numbers.tex +++ b/semester4/ps/ps-jh/parts/05_limit-theorems/01_weak-law-of-large-numbers.tex @@ -0,0 +1,7 @@ +\subsection{Schwaches Gesetz der grossen Zahlen} +\shorttheorem[Schwaches Ges. der grossen Zahlen] Sei $K = \{ 1, 2, \ldots \}$ und $\forall k \in K : \cX_k$ unabh. Z.V. mit $\E[\cX_k] = \mu$; $\V[\cX_k] = \sigma^2$: +\[ + \overline{\cX}_n = \frac{1}{n} S_n = \frac{1}{n} \sum_{k = 1}^{n} \cX_k +\] +Dann konvergiert $\overline{\cX}_n$ für $n \rightarrow \8$ in Wahrscheinlichkeit gegen $\mu = \E[\cX_k]$, +also $\forall \varepsilon > 0$ gilt $\P[|\overline{\cX}_n - \mu| > \varepsilon] \overset{n \rightarrow \8}{\longrightarrow} 0$ diff --git a/semester4/ps/ps-jh/parts/05_limit-theorems/02_strong-law-of-large-numbers.tex b/semester4/ps/ps-jh/parts/05_limit-theorems/02_strong-law-of-large-numbers.tex index e69de29..e272d70 100644 --- a/semester4/ps/ps-jh/parts/05_limit-theorems/02_strong-law-of-large-numbers.tex +++ b/semester4/ps/ps-jh/parts/05_limit-theorems/02_strong-law-of-large-numbers.tex @@ -0,0 +1,9 @@ +\subsection{Starkes Gesetz der grossen Zahlen} +\shorttheorem[Starkes Ges. der grossen Zahlen] Für $\cX_1, \ldots$ mit $\cX_k$ unabhängig mit $\E[\cX_k]$ endlich. Für +\[ + \overline{\cX}_n = \frac{1}{n} S_n = \frac{1}{n} \sum_{k = 1}^{n} \cX_k +\] +gilt $\overline{\cX}_n \overset{n \rightarrow \8}{\longrightarrow} \mu \quad \P$-fast sicher, also +\[ + \P\left[ \{ \omega \in \Omega \divider \overline{X}_n(\omega) \overset{n \rightarrow \8}{\longrightarrow} \mu \} \right] = 1 +\] diff --git a/semester4/ps/ps-jh/parts/05_limit-theorems/03_central-limit-theorem.tex b/semester4/ps/ps-jh/parts/05_limit-theorems/03_central-limit-theorem.tex index ab93e25..827b931 100644 --- a/semester4/ps/ps-jh/parts/05_limit-theorems/03_central-limit-theorem.tex +++ b/semester4/ps/ps-jh/parts/05_limit-theorems/03_central-limit-theorem.tex @@ -15,5 +15,12 @@ $\cX_k$ i.i.d mit $\E[\cX_k] = \mu$, $\V[\cX_k] = \sigma^2$. Für Partialsummen \limit{n}{\8} \P \left[ \frac{S_n - n\mu}{\sigma \sqrt{n}} \leq x \right] = \Phi(x) \] -\shortremark $\E[S_n] = n \mu$, $\V[S_n] = n \sigma^2$; $S_n^* = \frac{S_n - \mu}{\sigma \sqrt{n}} \overset{\text{approx}}{\sim} \cN(0, 1)$ -für grosse $n$, mit $\overset{\text{approx}}{\sim}$ gespr. ``approx. gleichverteilt gemäss'' +\shortremark $\E[S_n] = n \mu$, $\V[S_n] = n \sigma^2$; $S_n^* = \frac{S_n - \mu}{\sigma \sqrt{n}} \! \overset{\text{approx}}{\sim} \! \cN(0, 1)$ +für grosse $n$, mit $\overset{\text{approx}}{\sim}$ gespr. ``approx. gleichverteilt gemäss''. +Also ist für $\E[S_n^*] = 0$ und $\V[S_n^*] = 1$. + +Für $S_n$ also: $S_n \! \overset{\text{approx}}{\sim} \! \cN(n\mu, n\sigma^2)$, +bzw. $\overline{\cX}_n \! \overset{\text{approx}}{\sim} \! \cN \left( \mu, \frac{1}{n} \sigma^2 \right)$ + +\shortremark Für $S_n \sim \text{Bin}(n, p)$ ist $S_n \overset{\text{approx}}{\sim} \cN(np, np(1 - p))$ und +$\P[a < S_n \leq b] \approx \Phi \left( \frac{b + \frac{1}{2} - np}{\sqrt{np(1 - p)}} \right) - \Phi \left( \frac{a + \frac{1}{2} - np}{\sqrt{np(1 - p)}} \right)$ diff --git a/semester4/ps/ps-jh/parts/05_limit-theorems/04_chernoff-bounds.tex b/semester4/ps/ps-jh/parts/05_limit-theorems/04_chernoff-bounds.tex new file mode 100644 index 0000000..d6b43ca --- /dev/null +++ b/semester4/ps/ps-jh/parts/05_limit-theorems/04_chernoff-bounds.tex @@ -0,0 +1,15 @@ +\subsection{Chernoff-Schranken} +\shortdefinition[Momenterzeugende Funktion] von $\cX$ ist für $t \in \R$ $M_\cX(t) + \E[e^{t\cX}]$ + +\shortexample $\cX \sim \text{Ber}(p)$, dann $M_\cX(t) = 1 - p + p e^t$;\\ +$\cX \sim \text{Bin}(n, p)$, dann $M_\cX(t) = (1 - p + pe^t)^n$ + +\shorttheorem[Chernoff-Ungleichung] $\cX_k$ i.i.d. Z.V. mit jeweils $\forall t \in \R \; M_\cX(t)$ endl.; $\forall b \in \R$ +\[ + \P[S_n \geq b] \leq \exp \left( \inf_{t \in \R}(n \log(M_\cX(t) - tb)) \right) +\] + +\shorttheorem[Chernoff-Schranke] $\cX_k \sim \text{Ber}(p_k)$ unabhängig; $S_n\! = \! \sum_{k = 1}^{n} \cX_k$; $\mu_n = \E[S_n] = \sum_{k = 1}^{n} p_k$ und $\delta > 0$. Dann: +\[ + \P[S_n \geq (1 + \delta) \mu_n] \leq \left( \frac{e^\delta}{(1 + \delta)^{1 + \delta}} \right) +\] diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index 08618551acc3626d89c94384dbdcea46caacab12..c4f82e7d8fcb99bb4d808c91c1642cd1af342326 100644 GIT binary patch delta 119270 zcmV(&K;gfc%@@b~7qCtc0XLUWbOjZ&WDrUp12Zr=v#~EUCjv1zv*%XPE0foyHv%v= zlVJoXf7MvsZxlBWf8W2tkMK~h#~y!G5fTuRqP7y^0JVfX9ATTJggbP%DKvk5f3wN$ zWs}XFn<7;_ocC=1Jf1JljBP0@qpT(6JU&)g3|Nc*9{+7n&L(`ba=|IoMgW{$yZ1E8+wq zJ}UB-D~kmb3G9FrMFxMtDoPQnf>)FYs{%Tv!(iHAKN~=uq!%&m2)!sAV=1_@Y(mAt z5g4#Z287?_gb_fKC?FXnv%z8}cYAPv*QQbKI}|(i2+a zmJc@z=5Gu3dPw@Lr+PXR>;?;$e}ZkT~ z;i+E-qxD$x{RwLC5VReJjE~=`c<#1%Z--m7Jr~m<>BTKU7sHhocjlU&TDv`Thq0Xc zbm&rxe$RQIM7!pP5yxB%qx;7Vx;l{h)2NrK-aBm%g^Qh?s($tN=SI~prpHZHKb_Co zX4WoIH5x%y^~+{CKVKX*f6K0N>xO=9KAcXU%)hDw!6YV)n!{%Vv=|O6P+EY5)R33( 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z=ig;FyEH8_Ulz2m>y0#Ra$rvhE=^_~zvFO~QzLwIlv^Og163AHwQ#kXHtqa$H7i!%g3GxZj!8;wcwQeEG>OvAUbHn! z4vjt)H9?2OY2|C*UxsMP+nGM^f#p7oz)$Idaw7)Du9EU;`0Y*AMaMAd7EL6b-5QW4 zlAao8xy?w@V2>t}{_R3&IhuTR4<@6x1M^;(BZY$>#?9Dagm!X7R-R<1F2>DLY(xt% z((Bk~pyTFv6ov_nL`iZ(FS7MrYGBxtT=-phZ(M|;yE>{>0NuGpFW#W`;8?btV^J&L zQCHvuEGHk~B&3)nU0C;@)I(4}Z-xO8)Hf5{ljI`Dg>)*E+yQA}Kk7FWrD4)35b+0q T2$7eIgO?Ycib_&h3jTiqo1)=6 diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex index 7bce7e9..28951a8 100644 --- a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex +++ b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex @@ -87,6 +87,7 @@ \input{parts/05_limit-theorems/01_weak-law-of-large-numbers.tex} \input{parts/05_limit-theorems/02_strong-law-of-large-numbers.tex} \input{parts/05_limit-theorems/03_central-limit-theorem.tex} +\input{parts/05_limit-theorems/04_chernoff-bounds.tex} % \input{parts/05_limit-theorems/}