# Is the distribution of the minimum of two other distributions a mixture distribution? Or is there a better term?

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... :)

• I believe there is no general term for "some combination" of distributions simply because the concept is far too broad to be useful. – whuber Sep 10 '14 at 15:27
• I would say most latent variable models qualify for your "mixing" behaviour, but this does not help much because about everything is a latent variable model! – Xi'an Nov 25 '14 at 15:29

The $\min$ case doesn't seem to have the characteristics I'd normally understand as being a mixture.