# Conditional expectation of the product of two independent random variables

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 $$E(ab | c) = {\rm E}(a|c) {\rm E}(b|c)$$

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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