# Conditional expectation of a product of two independent random variables

1. Oct 31, 2012

### 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.