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Does this distribution have a name:

$f(x) = yx^{y-1}$ for 0 < x<1 and $y>0$?

It looks like an exponential distribution. Or is it a nameless distribution?

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    This is a beta distribution with $\beta=1$: http://en.wikipedia.org/wiki/Beta_distribution2012-04-18

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As Byron Schmuland says, this is a beta distribution with the second parameter $\beta=1$.

It is sometimes called a "standard power-function distribution".