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z?V~s32u9edTr`GD0#)c(svTx581kDVKudmev^?ASXAeDyFiMVm;~>AY2o0Gb$y*-r zkRgTV!S3m$GKsJv*f%{p5ejB3Dn~3jI1%zoj`+lM6;kb23(?+-Yhu{=oC+bHnp)z^ zmJSO5O2-AT)rNJjcW4=xo>@+6dhi8t*U*RW7KWts|7PCF*}&2HoAt~ThJ%fjfgOgF KR76e`=6?Z#YZaOR literal 0 HcmV?d00001 diff --git a/powerlaw_info.tex b/powerlaw_info.tex new file mode 100644 index 0000000000000000000000000000000000000000..3a96f33a60128d220f1b1ff2242c0cd46adb2643 --- /dev/null +++ b/powerlaw_info.tex @@ -0,0 +1,26 @@ +\documentclass{article} +\begin{document} +\section{Continuous powerlaw with minimum cut-off} +Exponent $\tau > 1$. +\subsection{Also maximum cut-off} +Let $m$ be the minimum, $M$ be the maximum.\\ +PDF: $$f(x) = \frac{\tau-1}{m^{-(\tau-1)} - M^{-(\tau-1)}} x^{-\tau}$$ +CDF: $$F(x) = \frac{m^{-(\tau-1)} - x^{-(\tau-1)}}{m^{-(\tau-1)} - M^{-(\tau-1)}}$$ +Inverse: $$F^{-1}(y) = \left( (1-y) m^{-(\tau-1)} + y M^{-(\tau-1)} \right)^{\frac{-1}{\tau-1}}$$ +i.e. linear interpolate between $M^{-(\tau-1)} < m^{-(\tau-1)}$.\\ +$F^{-1}(0) = m$ and $F^{-1}(1) = M$.\\ + +For $m=1$ and $M$ steps of interpolation:\\ +$F^{-1}(1/M) = \left( 1-M^{-1} + M^{-\tau} \right)^{\frac{-1}{\tau-1}}$\\ +$F^{-1}(1-1/M) = \left( M^{-1} + M^{-(\tau-1)} - M^{-\tau} \right)^{\frac{-1}{\tau-1}}$ + +\subsection{No maximum cut-off} +For $M=\infty$ we have:\\ +PDF: $$f(x) = \frac{\tau-1}{m^{-(\tau-1)}} x^{-\tau} = \frac{\tau-1}{m} \left(\frac{x}{m}\right)^{-\tau}$$ +CDF: $$F(x) = \frac{m^{-(\tau-1)} - x^{-(\tau-1)}}{m^{-(\tau-1)}} = 1 - \left(\frac{x}{m}\right)^{-(\tau-1)}$$ +Inverse: $$F^{-1}(y) = \left( (1-y) m^{-(\tau-1)} \right)^{\frac{-1}{\tau-1}} = m \; \left( 1-y \right)^{\frac{-1}{\tau-1}}$$ +For interpolation:\\ +$F^{-1}(0) = m$\\ +$F^{-1}(1-\frac{1}{n}) = m\cdot n^{\frac{1}{\tau-1}}$ + +\end{document}