Mean and Variance in a normal distribution

Question

I am trying to find the mean and variance for the following case.

$\beta = h+\alpha g f$

What will be the mean, variance of $\beta$ if $h,g,f $ all are distributed as $x \sim \mathcal{C}{N}(0,\sigma_{x}^2)$, $x \in [h,g,f]$. Also, let us assume that $\sigma^2_x = 2$ and $\alpha$ is some constant.

Answer 1

We don't even need the normality here.

$E(\beta) = E(h) + \alpha E(g)E(f) = 0.$

$Var(\beta) = E(\beta^2) = E(h^2 + 2 \alpha h g f + \alpha^2 g^2 f^2) = Var(h) + \alpha^2 Var(g) Var(f) = 2 + 4\alpha^2$.