  In this article we give a purely noncommutative criterion for the
characterization of two-state normal distribution. We prove that families of
two-state normal distribution can be described by relations which is similar to
the conditional expectation in free probability, but has no classical analogue.
We also show a generalization of Bozejko, Leinert and Speicher's formula
(relating moments and noncommutative cumulants).
