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I googled this but no success:

If $a,b,c$ are three random variables with independent gaussian distributions $G(a_0,\sigma_a)$, $G(b_0,\sigma_b)$, $G(c_0,\sigma_c)$, what is the formula for $\text{cov}(ab,ac)$ ?

Similarly, if a and c are dependent, with known $\text{cov}(a,c)$, what is the formula for $\text{cov}(ab,c)$ ?

Thanks in advance,

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You can use the fact that $Cov(X,Y) = E[XY]-E[X]E[Y]$ and that if X and Y are independent then $E[XY] = E[X]E[Y]$. In particular,

1) a,b,c, are independet:

$Cov(ab,ac) = E[a^2bc]-E[ab]E[ac] = E[a^2]E[b]E[c] - E[a]^2E[b]E[c]$;

2) $Cov(a,c) \neq 0$:

$Cov(ab,c) = E[abc]-E[ab]E[c] = E[ac]E[b]-E[a]E[b]E[c]$.

You can compute $E[a^2]$ using the fact that $Var(X) = E[X^2]-E[X]^2$ while you can compute $E[ac]$ from the formula of the covariance I wrote above.

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