Suppose I have two random variables $a$ and $b$. $a$ follows a normal distribution of parameters $u_1, s_1$. $b$ follows a normal distribution of parameters $u_2, s_2$. $u_1$ and $u_2$ are the expectations. $s_1$ and $s_2$ are the variances. If I randomly generate a number from $a$ with probability $p_a$, from $b$ with probability $1-p_a$.
I know the expectation will be $$u_1p_1 + u_2(1-p_1).$$ What about the variance? Can I use some method to approximate it?
Let you new random variable be $c$. Then $\mbox{Var}(c)=E(c^2)-(E(c))^2$ and you already know $E(c)$, so that leaves only $E(c^2)$ to be found.
Now it is either in your notes or text (and if not easily proven) that: $E(c^2)=p_1E(a^2)+(1-p_1)E(b^2)$ and the two expectations on the right can be found from the means and variances of $a$ and $b$.