This is a terminology question motivated by a review that I got on a paper. In the following I believe that $y$ would be considered to be distributed according to a mixture distribution:
$$y \sim \left\{ \begin{array}{ll} f_1(\theta_1) & \mbox{with } p ~ \mbox{probability} \\ f_2(\theta_2) & \mbox{with } (1 - p) ~ \mbox{probability} \end{array} \right. $$
where $f_1$ and $f_2$ are probability distributions. But would it be correct to say that $x$ is also distributed according to a mixture distribution?
$$ x \sim \text{min}(z_1,z_2) \\ z_1 \sim f_1(\theta_1) \\ z_2 \sim f_2(\theta_2)$$
If not, is there a better term? Or, at least, a general term for a distribution that is some combination of two other distributions?
(I mean, $x$ is certainly the result of "mixing together" $f_1$ and $f_2$ in some way, but I don't know if this is enough to call it a mixture... :)