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\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:\\

$F^{-1}(0) = m$\\

$F^{-1}(1-\frac{1}{n}) = m\cdot n^{\frac{1}{\tau-1}}$

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