Are there any named probability distribution of $x$ like:
$$
Pr(x)=\frac{\Gamma(\alpha+\beta)} {\Gamma(\alpha)\Gamma(\beta)} e^{-\alpha x}(1-e^{-x})^{\beta -1}
$$
where $x\in(0,\infty), \alpha>0,\beta >0$. In particular I would like to have an analytical expression of the mean and variance of $Pr(x)$ in terms of $\alpha$ and $\beta$.
Note that the transformation $y(x)=e^{-x}$ yields a beta distribution with parameters $\alpha$ and $\beta$.
Best Answer
This is, unsurprisingly, called a beta exponential distribution (although I'm not sure it's a widely used distribution, so this name may not be as canonical as one might hope). Nadarajah and Kotz define this distribution given parameters $(\lambda,a,b)$ as having probability density function $$f(x)=\frac{\lambda}{B(a,b)}e^{-b\lambda x}\left(1-e^{-\lambda x}\right)^{a-1},$$ so your distribution has parameters $(1,\beta,\alpha)$. They show that, if $X$ is a random variable following this distribution, $$\mathbb E[X^n]=\frac{(-1)^n}{\lambda^n B(a,b)}\frac{\partial^n}{\partial p^n}B(a,1+p-a)\bigg|_{p=a+b-1},$$ which allows you to calculate the mean and variance. The linked article gives many more calculations of standard properties of this distribution, which may be of interest.