josia-notes/s23-probability-theory
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Josia Pietsch 2023-05-14 22:49:50 +02:00
parent 41cb2f2094
commit 339fb05307
Signed by: josia
GPG key ID: E70B571D66986A2D
3 changed files with 5 additions and 2 deletions
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2
inputs/lecture_10.tex
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@ -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}{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)\\ &=& \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{\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{1}{2} \bP(\{a\} ) + \frac{1}{2} \bP(\{b\}) + \bP((a,b))\\
&=& \frac{F(b) + F(b-)}{2} - \frac{F(a) - F(a-)}{2} &=& \frac{F(b) + F(b-)}{2} - \frac{F(a) - F(a-)}{2}

2
inputs/lecture_11.tex
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@ -116,7 +116,7 @@ If $S_n \sim \Bin(n,p)$ and $[a,b] \subseteq \R$, we have
$\bP[|S_n - np| \le 0.01 n] \le 0.05$. $\bP[|S_n - np| \le 0.01 n] \le 0.05$.
We have that We have that
\begin{IEEEeqnarray*}{rCl} \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[ -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)}}\\ &=& \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)}})\\ &\approx& \Phi(0.01 \sqrt{\frac{n}{p(1-p)}}) - \Phi(-0.01 \sqrt{\frac{n}{p(1-p)}})\\

3
wtheo.sty
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@ -93,3 +93,6 @@
\NewFancyTheorem[thmtools = { style = thmredmargin} , group = { big } ]{warning} \NewFancyTheorem[thmtools = { style = thmredmargin} , group = { big } ]{warning}
\DeclareSimpleMathOperator{Var} \DeclareSimpleMathOperator{Var}
\DeclareSimpleMathOperator{Exp}
\DeclareSimpleMathOperator{Bin}
\DeclareSimpleMathOperator{Ber}