Resultant mean and variance of gaussian distribution

Question

X be random Gaussian variable with mean u1 and variance v1. u1 itself is a random variable which is also gaussian distributed with mean u2 and variance v2. Then the distribution of X will be Guassian/uniform? with mean and variance =?

Answer 1

$X=u_1+\sqrt{v_1}Y$, where $Y\tilde{}N(0,1)$.
$X=u_2+\sqrt{v_2}Z+\sqrt{v_1}Y\\X=u_2+\sqrt{v_1+v_2}W$
where both Z and W are also N(0,1). So $X$ is Gaussian with mean $u_2$ and variance $v_1+v_2$.