Is there a name for the probability distribution with the following pdf?
$$f(x) = \begin{cases}-\log x, & x\in (0, 1] \\ 0,& \text{otherwise}\end{cases}$$
(Observe that $\lim_{t\to 0}\int_t^1 f(x)dx = 1$.)
The cdf is $$\int_0^x -\log(t) dt = 1 - \int_x^1 -\log(t)dt = x - x\log x$$
I am wary of this probability distribution because the density approaches infinity for $x$ close to zero. Is this acceptable? In general, I have not encountered probability distributions whose density approaches infinity anywhere in the space, except in cases where a Dirac delta is used to characterize a discrete distribution in continuous space, as discussed in Infinite probability density?.
On the other hand, the cdf is invertible (the expression involves the Lambert W function), so we can readily transform random uniform variables into realizations of this distribution, which suggests that it is not completely outlandish.
