88 lines
2.0 KiB
Markdown
88 lines
2.0 KiB
Markdown
# Decrease in Sigma
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```tikz
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\usepackage{pgfplots}
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\pgfplotsset{compat=1.16}
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\begin{document}
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\begin{tikzpicture}
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\begin{axis}[
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width=13cm,
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height=7cm,
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axis lines=middle,
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xlabel={$x$},
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ylabel={$\varphi(x;\mu,\sigma)$},
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xmin=-6, xmax=6,
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ymin=0, ymax=0.85,
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samples=400,
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domain=-6:6,
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legend style={draw=none, fill=none, at={(0.98,0.98)}, anchor=north east},
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legend cell align=left,
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ytick=\empty,
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]
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% Normal PDF: (1/(sigma*sqrt(2*pi))) * exp(-(x-mu)^2/(2*sigma^2))
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\addplot[thick]
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{ (1/(1.8*sqrt(2*pi))) * exp(-((x-0.8)^2)/(2*1.8^2)) };
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\addlegendentry{$\mu=0,\ \sigma=1.8$}
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\addplot[thick, dashed]
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{ (1/(0.8*sqrt(2*pi))) * exp(-((x-0.8)^2)/(2*0.8^2)) };
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\addlegendentry{$\mu=0,\ \sigma=0.8$}
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\end{axis}
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\end{tikzpicture}
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\end{document}
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```
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^pdf
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```tikz
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\usepackage{pgfplots}
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\pgfplotsset{compat=1.16}
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\pgfmathdeclarefunction{erfapprox}{1}{%
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\pgfmathparse{%
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% save sign and work with |x|
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( (#1<0) ? -1 : 1 )
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* ( 1 - (1 + 0.278393*abs(#1) + 0.230389*abs(#1)^2 + 0.000972*abs(#1)^3 + 0.078108*abs(#1)^4)^(-4) )
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}%
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}
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% Normal CDF using erf approximation:
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% F(x;mu,sigma) = 0.5*(1 + erf((x-mu)/(sigma*sqrt(2))))
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\pgfmathdeclarefunction{normcdf}{3}{%
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\pgfmathparse{ 0.5*(1 + erfapprox((#1-#2)/(#3*sqrt(2)))) }%
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}
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\begin{document}
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\begin{tikzpicture}
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\begin{axis}[
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width=13cm,
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height=7cm,
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axis lines=middle,
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xlabel={$x$},
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ylabel={$F(x;\mu,\sigma)$},
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xmin=-6, xmax=6,
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ymin=0, ymax=1.05,
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samples=400,
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domain=-6:6,
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legend style={draw=none, fill=none, at={(0.02,0.98)}, anchor=north west},
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legend cell align=left,
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ytick={0,0.5,1},
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]
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% Normal CDF: 0.5*(1 + erf((x-mu)/(sigma*sqrt(2))))
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\addplot[thick]
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{ normcdf(x,0,1.8) };
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\addlegendentry{$\mu=0,\ \sigma=1.8$}
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\addplot[thick, dashed]
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{ normcdf(x,0.8) };
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\addlegendentry{$\mu=0,\ \sigma=0.8$}
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\end{axis}
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\end{tikzpicture}
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\end{document}
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``` |