Find the probability that there are no bulb failures before time t. 331
Beginning at time t = O. we start using bulbs. one at a time, to illuminate a room. Bulbs are replaced immediately upon failure. Each new bulb is selected independently by an equally likely choice between a type-A bulb and a type-B bulb. The lifetime, X, of any particular bulb of a particular type is a random variable, independent of everything else, with the following PDF:
(a) Find the expected time until the first failure.
(b) Find the probability that there are no bulb failures before time t. 331
(c) Given that there are no failures until time t, determine the conditional probability that the first bulb used is a type-A bulb.
(d) Find the variance of the time until the first bulb failure.
(e) Find the probability that the 12th bulb failure is also the 4th type-A bulb failure.
(f) Up to and including the 12th bulb failure, what is the probability that a total of exactly 4 type-A bulbs have failed?
(g) Determine either the PDF or the transform associated with the time until the 12th bulb failure.
(h) Determine the probability that the total period of illumination provided by the first two type-B bulbs is longer than that provided by the first type-A bulb.
(i) Suppose the process terminates as soon as a total of exactly 12 bulb failures have occurred. Determine the expected value and variance of the total period of illumination provided by type-B bulbs while the process is in operation.
(j) Given that there are no failures until time t, find the expected value of the time until the first failure.
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Posted 25 May 2017 10:05 AM
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