$N(a\mu_1 + b\mu_2, a^2\sigma_1^2 + b^2\sigma_2^2)$

$N(a\mu_1 + b\mu_2, a\sigma_1^2 + b\sigma_2^2)$

A linear combination of normal variables is always normal.

The variance of $Z$ is $a^2\sigma_1^2 + b^2\sigma_2^2$.