Monitoring (inspection) as increasing model of reliability of system in use

The article defines the basic problem of monitoring (inspection) of a system in use The system operating rime with no failure is the random variable X. The failure detection moment Y is introduced (random variable too) for which applies Y greater than or equal to X. The common distribution function for the random variables X and Y is F(X.Y) and it is observed in the rime Interval (0,t) where t less than or equal to infinity, it is assumed that the distribution function of the variable X is known, F(x), so that F(x) = F-t (x, infinity) can be applied. The basic problem how to define the function F-t for the given function of cost L(X,Y,F-t) has been solved so that E[L(X,Y:F-t)] where E(x) is mathematically expected random variable X (expected cost). Than some cases are considered where functions L(,Y,Y:F-t) can be given. For some special cases final results are also calculated.