For a certain type of halogen light bulb, an old bulb that has been in use for a while tends to have a longer life than a new bulb. Let x represent the life (in hours) of a new halogen light bulb and assume that x has an exponential distribution with mean θ = 250 hours. According to Microelectronics and Reliability (Jan. 1986), the "life" distribution of x is considered new better than used (NBU) if
P(x > a + b) ≤ P(x > a) P(x > b)
Alternatively, a "life" distribution is considered new worse than used (NWU) if
P(x > a + b) ≥ P(x > a)P(x > b)
a. Show that when a = 300 and b = 200, the exponential distribution is both NBU and NWU.
b. Choose any two positive numbers a and b and repeat part a.
c. Show that, in general, for any positive a and b, the exponential distribution with mean θ is both NBU and NWU. Such a "life" distribution is said to be new same as used, or memory less. Explain why.