A quantum random variable is a function with values i . If , then the quantum expectation of relative to the quantum probability measure is the operator denoted by and defined by . In this thesis first we introduce positive operator valued measure and quantum random variables to defined the quantum expected value . In so doing we are led to theorems for a change of quantum measure as well as a change of quantum variables. We also introduce a quantum conditional expectation which results in quantum version of some standard identities for Radon- derivatives. This allows us to formulate and prove a quantum analogue of rule.
