# Conjugated priors (Pareto and Beta), name of the unconditional distribution?

$$F_X(x)=\begin{cases} \quad\dfrac{\alpha}{\alpha+\theta}\left(\dfrac x\omega \right)^\theta &\text{ if } x<\omega \\ \\ 1-\dfrac{\theta}{\alpha+\theta}\left(\dfrac\omega x\right)^{\alpha} &\text{ if } x>\omega \end{cases}\quad\text{ where } 0\le x< +\infty.$$

I derived it by having $F_{X|Y}(x|y)=\left(\dfrac{x}{y}\right)^\theta$, which is $Y$ with a multiplicative "shock/noise" give by a $\mathrm{Beta}(\theta,1)$ (max of $\theta$ uniforms) and $Y$ follows a Pareto distribution $F_Y(y)=1-\left(\dfrac{\omega}{y}\right)^\alpha$.

I was not able to find a classification for it. Does it have a name/family?

Rodney Kreps (a legend among casualty actuaries) calls this the "Split Simple Pareto" For example, see page 3 in Effects of Parameters of Tranformed Beta Distributions, (Venter 2003){pdf}. It is the Transformed Beta distribution as $\tau \to \infty$ (although Krep uses $\theta$ instead of $\omega$.
Some statistics: $$E(X) = \omega\frac{\alpha\beta}{\alpha-1}\frac{1}{\beta+1}\\ E(X^2) = \omega^2\frac{\alpha\beta}{\alpha-2}\frac{1}{\beta+2}\\$$