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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.

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  • $\begingroup$ Are $h,g,f$ independent? $\endgroup$
    – angryavian
    Aug 9 '21 at 0:20
  • $\begingroup$ Yes....$h,g,f$ are i.i.d $\endgroup$
    – paru
    Aug 9 '21 at 0:24
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We don't even need the normality here.

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

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

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  • $\begingroup$ 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 ? $Var(\beta) = E(h^2)+2\alpha E(h) E(g) E(f) + \alpha^2 E(g^2) E(f^2)$ = $Var(h) + 2\cdot \alpha \cdot 0 \cdot 0 \cdot 0 + \alpha^2 Var(g) Var(f)$ = $2+0+\alpha^2 2\cdot 2$ = $2+\alpha^2 4$ $\endgroup$
    – paru
    Aug 9 '21 at 2:37
  • $\begingroup$ @ shang, Also what is meant by your comment "We don't even need the normality here" $\endgroup$
    – paru
    Aug 9 '21 at 2:46
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    $\begingroup$ 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. $\endgroup$ Aug 9 '21 at 3:12
  • $\begingroup$ Ok.... Thanks for your efforts and comments... $\endgroup$
    – paru
    Aug 9 '21 at 5:19

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