  Dual Lukacs type characterizations of random variables in free probability
are studied here. First, we develop a freeness property satisfied by Lukacs
type transformations of free-Poisson and free-Binomial non-commutative
variables which are free. Second, we give a characterization of non-commutative
free-Poisson and free-Binomial variables by properties of first two conditional
moments, which mimic Lukacs type assumptions known from classical probability.
More precisely, our result is a non-commutative version of the following result
known in classical probability: if $U$, $V$ are independent real random
variables, such that $E(V(1-U)|UV)$ and $E(V^2(1-U)^2|UV)$ are non-random then
$V$ has a gamma distribution and $U$ has a beta distribution.
