Problem # 4. (40+5 bonus). A manufacturer of a certain electronic gadget offers a full refund in case of failure within the warranty period W. A consumer protection organization performed extensive tests and established that the lifetime of this gadget (time until failure) is a continuous random variable T, with the CDF (which explains the \"generous" warranty period) 0 t W. (For example, To could be the minimum lifetime that is believed acceptable to most consumers). b) (10) If the gadget was indeed still working after To, what is the probability that it will work for at least the next W (i.e., lifetime greater than To + W)? Is there some truth to the manufacturer's surprising claim that \"its reliability improves with age\"? c) (10) On the other hand, if you know that the gadget failed within 2T0, what is the probability that it was still working after To? d) (10) If you purchased three such units (which we assume fail independently) and they are all still working after To, what is the probability that at least one of them is still working by the end of To + W? e) (5) The manufacturer claims that \"on the average this system works forever.\" Is this extravagant claim laughable, or can it be mathematically justied? (As it ventures a bit beyond the oiciui scope of the midterm exam, this optional port is worth 5 bonus points.)