Discrete & Continuous Random Variables - Statistics

Example: Sprinkler System FailureA warehouse has a sprinkler system. Given that it fails to activate, it may incur sucha permanently failed state equally likely at any time X 2 during a 12 month period.Given that there is a fire during these 12 months, the time of fire can occur equallylikely at any time point X 1 in that interval (single fire!).Given that we have a sprinkler system failure and a fire during that year, what is thechance that the sprinkler system will fail before the fire, i.e., will be useless?Assuming reasonably that the occurrence times X 1 and X 2 are independent,it seems intuitive that P(X 1 < X 2 ) = 0.5.Rigorous treatment =⇒ next slide.The same kind of problem arises with a computer hard drive and a backup drive,or with a flight control system and its backup system. (latent failures)51