For the random process of Problem 10.3.2, what is the conditional PMF of T2 given T1? If the technician finds the first oscillator in 3 minutes, what is E[T2|T1 = 3], the conditional expected value of the time of finding the second one-part-in-104 oscillator?
Problem 10.3.2
In a production line for 10 kHz oscillators, the output frequency of each oscillator is a random variable W uniformly distributed between 9980 Hz and 1020 Hz. The frequencies of different oscillators are independent. The oscillator company has an order for one part in 104 oscillators with frequency between 9999 Hz and 10, 001 Hz. A technician takes one oscillator per minute from the production line and measures its exact frequency. (This test takes one minute.) The random variable Tr minutes is the elapsed time at which the technician finds r acceptable oscillators.
(a) What is p, the probability that any single oscillator has one-part-in-104 accuracy?
(b) What is E[T1] minutes, the expected time for the technician to find the first one-part-in-104 oscillator?
(c) What is the probability that the technician will find the first one-part-in-104 oscillator in exactly 20 minutes?
(d) What is E[T5], the expected time of finding the fifth one-part-in-104 oscillator?