Conditional expectation of a product of two independent random variables

[Geert]

Suppose that α and β are independently distributed random variables, with means; μ_α, μ_b
and variances; δ_α^2, δ_β^2, respectively.
Further, let c=αβ+e, where e is independently distributed from α and β
with mean 0 and variance δ_e^2.



Does it hold that

E(αβ | c) = E(α|c) E(β|c)

If not; does it hold when we assume that α, β and e are Gaussian?

If not; does it hold when μ_β = 0?


More general, does it still hold when c = f(α,β) + e, with $f(,)$ some arbitrary function.

Thanks in Advance;

Geert

 

[lippyka]

.No, it does not hold in general that E(αβ | c) = E(α|c) E(β|c). This is not true in the case of non-Gaussian distributions, and not even when μ_β = 0. However, if α, β, and e are all Gaussian, then this equation holds.More generally, if c = f(α,β) + e, with $f(,)$ some arbitrary function, then the equation will not necessarily hold.