some changes
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3 changed files with 5 additions and 2 deletions
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@ -34,7 +34,7 @@ where $\mu = \bP X^{-1}$.
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\\&&
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+ \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)\\
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&=& \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)\\
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&\overset{\text{\autoref{fact:intsinxx}, dominated convergence}}{=}& \frac{1}{\pi} \int -\frac{\pi}{2} \One_{x < a} + \frac{\pi}{2} \One_{x > a }
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&\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 }
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- (- \frac{\pi}{2} \One_{x < b} + \frac{\pi}{2} \One_{x > b}) d\bP(x)\\
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&=& \frac{1}{2} \bP(\{a\} ) + \frac{1}{2} \bP(\{b\}) + \bP((a,b))\\
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&=& \frac{F(b) + F(b-)}{2} - \frac{F(a) - F(a-)}{2}
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@ -116,7 +116,7 @@ If $S_n \sim \Bin(n,p)$ and $[a,b] \subseteq \R$, we have
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$\bP[|S_n - np| \le 0.01 n] \le 0.05$.
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We have that
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\begin{IEEEeqnarray*}{rCl}
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&&\bP[|S_n - nü| \le 0.01n] \\
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&&\bP[|S_n - np| \le 0.01n] \\
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&=& \bP[ -0.01 n \le S_n - np \le 0.01n]\\
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&=& \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)}}\\
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&\approx& \Phi(0.01 \sqrt{\frac{n}{p(1-p)}}) - \Phi(-0.01 \sqrt{\frac{n}{p(1-p)}})\\
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@ -93,3 +93,6 @@
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\NewFancyTheorem[thmtools = { style = thmredmargin} , group = { big } ]{warning}
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\DeclareSimpleMathOperator{Var}
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\DeclareSimpleMathOperator{Exp}
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\DeclareSimpleMathOperator{Bin}
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\DeclareSimpleMathOperator{Ber}
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