# Mean and Variance in a normal distribution

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.

• Are $h,g,f$ independent? Aug 9 '21 at 0:20
• Yes....$h,g,f$ are i.i.d
– paru
Aug 9 '21 at 0:24

We don't even need the normality here.

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

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

• Ok... i understood your answer related to mean of $\beta$. Can you please tell whether my understanding related to variance of $\beta$ obtained by you is correct ? $Var(\beta) = E(h^2)+2\alpha E(h) E(g) E(f) + \alpha^2 E(g^2) E(f^2)$ = $Var(h) + 2\cdot \alpha \cdot 0 \cdot 0 \cdot 0 + \alpha^2 Var(g) Var(f)$ = $2+0+\alpha^2 2\cdot 2$ = $2+\alpha^2 4$
– paru
Aug 9 '21 at 2:37
• @ shang, Also what is meant by your comment "We don't even need the normality here"
– paru
Aug 9 '21 at 2:46
• Yes. You are on target. The result holds even if $h, g, f$ are NOT normally distributed. We just need them to have mean of 0, variance of 2, and joint independence. Aug 9 '21 at 3:12