distribution of $X=(e^Y-1)^{1/\theta}$ where Y is exponential distributed

Question

Let $Y \sim \text{Exp}(\lambda)$ what's the NAME of the distribution of X such that
$$X \equiv (e^Y -1)^{1/\theta}$$ where $\theta$ is a positive constant.

Answer 1

You can show that the random variable X obeys the CDF

$F_X(x)=1-\frac{1}{(1+x^{\theta})^\lambda}$ , $x\in[0,\infty)$

which can subsequently be classified as a Pareto type IV with $\mu=0$ and $\sigma=1$.