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Does this probability distribution have a name?

$$f_a(x) dx = \frac{a}{e^a-(a+1)} \left(e^{a x} - 1\right) dx, \quad 0 \le x \le 1.$$

Edit: I want $a$ to be positive. I don't think this is what is usually called a truncated exponential distribution.

Edit 2: Here's an example where $a=2$.

Edit 3: This seems to be an unnamed distribution.

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    $\begingroup$ Asking for a name is reasonable because it gives you a way to look up properties and uses of the distribution. But, assuming it does not have a name, is there something in particular you would like to know about it, such as its moments, cumulants, or parameter estimators? $\endgroup$ – whuber Aug 22 '12 at 18:29
  • $\begingroup$ $1-X$ could be viewed as a mixture of a truncated exponential and a uniform distribution, but with an unusual negative weight on the uniform distribution. $\endgroup$ – Douglas Zare Aug 22 '12 at 18:30
  • $\begingroup$ @whuber: Because you asked, I want to mix it with 1-X thereby defining a 2-parameter distribution ($a$ and the mixture parameter) and then I want to compare this distribution with the beta distribution, which is also a 2-parameter model on [0,1]. $\endgroup$ – gam Aug 22 '12 at 18:45
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    $\begingroup$ Exactly how do you want to compare the distributions? Such mixtures of your pdf will be u-shaped with finite values at the ends, whereas u-shaped beta distributions always have infinite values at the ends: there's going to be considerable divergence, then! You might be shocked at what moment-matching comparisons give you, too: e.g., the beta which matches the first two moments of $a=2$ with mixing parameter $1/2$ has parameters $\frac{1}{4} \left(3 e^2-11\right)$, $\frac{1}{4}\left(3 e^2-11\right)$, which is bell-shaped, not u-shaped. $\endgroup$ – whuber Aug 22 '12 at 19:01
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    $\begingroup$ @whuber: lol I only answered because you asked. I wasn't expecting the spanish inquisition. I realize that the distributions are very different and I am not shocked; if you are interested in the details of my plans then I could make a new post. $\endgroup$ – gam Aug 22 '12 at 19:13

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