Can anyone give some advice on solving the following problem?
$X$ is distributed as $N(\mu_1, \sigma_1^2)$;
$Y \big| X > c$ is distributed as $N(\mu_2, \sigma_2^2)$;
$Z \big|\min(X, Y) >c$ is distributed as $N(g(X,Y), \sigma_3^2)$.
Here $\mu_1$, $\sigma_1$, $\mu_2$, $\sigma_2$, $\sigma_3$ are constants. How to find the variance of $Z$?
Thank you very much.