On the reals, a normal distribution with mean $\mu$ and standard deviation $\sigma$ could be considered somewhat natural (loose terminology, but a lot of stuff in nature occurs normally distributed, it has some really nice mathematical properties, etc). A Beta distribution with parameters $\alpha$, $\beta$ seems to play a similar role for the closed interval $[0,1]$. A sufficiently narrow beta distribution far away from either end of the interval can look close to a normal distribution, which makes sense.
Is there any equivalently natural two-parameter distribution for the one-sided interval $[0,\infty)$?