Two different definitions of survival probability in the decay of an unstable quantum system are considered and the equations for them are deduced from the equation for the density operator. The regime situation for the ensemble of unstable systems is studied and the density operator in such a situation is derived. The possible definitions of lifetime are discussed. If the adopted definition makes reference to all time including the early ones, then it is compulsory to specify exactly the state at the initial time. The correct equations for the survival probabilities in such conditions are derived. Two typical examples of decay experiments are considered and the influence of the initial preparation on the mean life is discussed. Numerical examples are given.