vault backup: 2025-12-18 17:14:55

2025-12-18 17:14:55 -05:00
parent f360418710
commit a561aaf128
33 changed files with 342 additions and 288 deletions
@@ -0,0 +1,88 @@
+# Decrease in Sigma
+
+```tikz
+\usepackage{pgfplots}
+\pgfplotsset{compat=1.16}
+
+\begin{document}
+\begin{tikzpicture}
+    \begin{axis}[
+      width=13cm,
+      height=7cm,
+      axis lines=middle,
+      xlabel={$x$},
+      ylabel={$\varphi(x;\mu,\sigma)$},
+      xmin=-6, xmax=6,
+      ymin=0, ymax=0.85,
+      samples=400,
+      domain=-6:6,
+      legend style={draw=none, fill=none, at={(0.98,0.98)}, anchor=north east},
+      legend cell align=left,
+      ytick=\empty,
+    ]
+    
+    % Normal PDF: (1/(sigma*sqrt(2*pi))) * exp(-(x-mu)^2/(2*sigma^2))
+    
+    \addplot[thick]
+      { (1/(1.8*sqrt(2*pi))) * exp(-((x-0.8)^2)/(2*1.8^2)) };
+    \addlegendentry{$\mu=0,\ \sigma=1.8$}
+    
+    \addplot[thick, dashed]
+      { (1/(0.8*sqrt(2*pi))) * exp(-((x-0.8)^2)/(2*0.8^2)) };
+    \addlegendentry{$\mu=0,\ \sigma=0.8$}
+    
+    \end{axis}
+\end{tikzpicture}
+\end{document}
+```
+
+^pdf
+
+```tikz
+\usepackage{pgfplots}
+\pgfplotsset{compat=1.16}
+
+\pgfmathdeclarefunction{erfapprox}{1}{%
+  \pgfmathparse{%
+    % save sign and work with |x|
+    ( (#1<0) ? -1 : 1 )
+    * ( 1 - (1 + 0.278393*abs(#1) + 0.230389*abs(#1)^2 + 0.000972*abs(#1)^3 + 0.078108*abs(#1)^4)^(-4) )
+  }%
+}
+
+% Normal CDF using erf approximation:
+% F(x;mu,sigma) = 0.5*(1 + erf((x-mu)/(sigma*sqrt(2))))
+\pgfmathdeclarefunction{normcdf}{3}{%
+  \pgfmathparse{ 0.5*(1 + erfapprox((#1-#2)/(#3*sqrt(2)))) }%
+}
+
+\begin{document}
+\begin{tikzpicture}
+    \begin{axis}[
+      width=13cm,
+      height=7cm,
+      axis lines=middle,
+      xlabel={$x$},
+      ylabel={$F(x;\mu,\sigma)$},
+      xmin=-6, xmax=6,
+      ymin=0, ymax=1.05,
+      samples=400,
+      domain=-6:6,
+      legend style={draw=none, fill=none, at={(0.02,0.98)}, anchor=north west},
+      legend cell align=left,
+      ytick={0,0.5,1},
+    ]
+    
+    % Normal CDF: 0.5*(1 + erf((x-mu)/(sigma*sqrt(2))))
+    \addplot[thick]
+      { normcdf(x,0,1.8) };
+    \addlegendentry{$\mu=0,\ \sigma=1.8$}
+    
+    \addplot[thick, dashed]
+      { normcdf(x,0.8) };
+    \addlegendentry{$\mu=0,\ \sigma=0.8$}
+    
+    \end{axis}
+\end{tikzpicture}
+\end{document}
+```