From 339fb0530779d8e62d180ad2707a305a7c9c70e5 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Sun, 14 May 2023 22:49:50 +0200 Subject: [PATCH] some changes --- inputs/lecture_10.tex | 2 +- inputs/lecture_11.tex | 2 +- wtheo.sty | 3 +++ 3 files changed, 5 insertions(+), 2 deletions(-) diff --git a/inputs/lecture_10.tex b/inputs/lecture_10.tex index 2a6f4ea..32d17d0 100644 --- a/inputs/lecture_10.tex +++ b/inputs/lecture_10.tex @@ -34,7 +34,7 @@ where $\mu = \bP X^{-1}$. \\&& + \lim_{T \to \infty} \frac{1}{2\pi} \int_{\R}\int_{-T}^T \frac{\sin(t ( x - b)) - \sin(t(x-a))}{-t} dt d\bP(x)\\ &=& \lim_{T \to \infty} \frac{1}{\pi} \int_\R \int_{0}^T \frac{\sin(t(x-a)) - \sin(t(x-b))}{t} dt d\bP(x)\\ - &\overset{\text{\autoref{fact:intsinxx}, dominated convergence}}{=}& \frac{1}{\pi} \int -\frac{\pi}{2} \One_{x < a} + \frac{\pi}{2} \One_{x > a } + &\overset{\substack{\text{\autoref{fact:intsinxx},}\\\text{dominated convergence}}}{=}& \frac{1}{\pi} \int -\frac{\pi}{2} \One_{x < a} + \frac{\pi}{2} \One_{x > a } - (- \frac{\pi}{2} \One_{x < b} + \frac{\pi}{2} \One_{x > b}) d\bP(x)\\ &=& \frac{1}{2} \bP(\{a\} ) + \frac{1}{2} \bP(\{b\}) + \bP((a,b))\\ &=& \frac{F(b) + F(b-)}{2} - \frac{F(a) - F(a-)}{2} diff --git a/inputs/lecture_11.tex b/inputs/lecture_11.tex index 67c1752..bc4fdb7 100644 --- a/inputs/lecture_11.tex +++ b/inputs/lecture_11.tex @@ -116,7 +116,7 @@ If $S_n \sim \Bin(n,p)$ and $[a,b] \subseteq \R$, we have $\bP[|S_n - np| \le 0.01 n] \le 0.05$. We have that \begin{IEEEeqnarray*}{rCl} - &&\bP[|S_n - nü| \le 0.01n] \\ + &&\bP[|S_n - np| \le 0.01n] \\ &=& \bP[ -0.01 n \le S_n - np \le 0.01n]\\ &=& \bP[-\frac{0.01 n}{\sqrt{n p (1-p)} } \le \frac{S_n - np}{\sqrt{n p (1-p)} } \le \frac{0.01 n}{\sqrt{n p (1-p)}}\\ &\approx& \Phi(0.01 \sqrt{\frac{n}{p(1-p)}}) - \Phi(-0.01 \sqrt{\frac{n}{p(1-p)}})\\ diff --git a/wtheo.sty b/wtheo.sty index 14a67cf..a494021 100644 --- a/wtheo.sty +++ b/wtheo.sty @@ -93,3 +93,6 @@ \NewFancyTheorem[thmtools = { style = thmredmargin} , group = { big } ]{warning} \DeclareSimpleMathOperator{Var} +\DeclareSimpleMathOperator{Exp} +\DeclareSimpleMathOperator{Bin} +\DeclareSimpleMathOperator{Ber}