# What is this distribution called?

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

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