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For every positive integer $n$, $nx^{n-1}$ is a probability density function in $[0,1]$. What is it called?

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The distribution with pdf

$$ f(x) = \frac{\beta x^{\beta-1}}{\alpha^\beta} $$

is known as the "power distribution" (unfortunately, it’s a very google-unfriendly name). It is alternatively parametrized by $k = 1/\alpha$. It has the $(0, \alpha]$ support.

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