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Suppose that $a$ and $b$ are independently distributed random variables, with means; $\mu_a$, $\mu_b$ and variances; $\sigma_{a}^2$, $\sigma_b^2$, respectively. Further, let $c=ab + e$, where $e$ is independently distributed from $a$ and $b$ with mean $0$ and variance $\sigma_e^2$.

Does it hold that \begin{equation} E(ab | c) = {\rm E}(a|c) {\rm E}(b|c) \end{equation}

If not; does it hold if $a$, $b$ and $e$ are Gaussian?

If not; does it hold when $\mu_b = 0$?

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Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. If this is homework, please add the homework tag; people will still help, so don't worry. – Julian Kuelshammer Oct 30 '12 at 11:52
To which only the silence of the abysses responded... (What a pity, the question IS interesting, but it seems morally difficult to post an answer if the OP stays mute.) – Did Oct 31 '12 at 8:56

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