Power-law random generator

Posted: June 2, 2008 | Author: ruvido | Filed under: Uncategorized |Leave a comment »

So you want to generate random numbers distributed along a power-law, uh?

Suppose your target distribution is:

$P(y) \sim y^{-n}$

you will use:

$y = x^{1/(1-n)}$

where x is a random variable distributed uniformly and n>1.
This formula comes from the differential equality:

$p(x)dx=p(y)dy$

which becomes:

$\frac{dx}{dy}=p(y)$

in case x is distributed uniformly. For example, suppose you want to generate numbers which are distributed exponentially, you will use:

$\frac{dx}{dy}=e^y$

$\Rightarrow x=e^y$

which finally gives:

$y=log(x)$

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Francesco Rao