Renewal Processes with uniform and exponential distributions
Suppose the lifetime of a component Ti in hours is uniformly distributed on [100, 200]. Components are replaced as soon as one fails and assume that this process has been going on long enough to reach equilibrium.
(a) What is the probability that the current component has been in operation for at least 50 hours?
(b) What is the probability that the current component will last for at least 50 hours more?
(c) What is the probability that the total lifetime of the current component will be at least 150 hours?
(d) Suppose that it is known that the current component has been in operation for exactly 90 hours. What is the probability that it will last at least 50 more hours?
Is this as simple as .75 for a), .75 for b), .50 for c), and .60 for d)?
Also, how would you do this for an exponential distribution with mean of 150?