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Statistics / Fundamentals of Probability, with Stochastic Processes 3 / Chapter 7.3 / Problem 11

Fundamentals of Probability, with Stochastic Processes | 3rd Edition | ISBN: 9780131453401 | Authors: Saeed Ghahramani

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QUESTION:

In a factory, a certain machine operates for a period which is exponentially distributed with parameter . Then it breaks down and will be in repair shop for a period, which is also exponentially distributed with mean 1/. The operating and the repair times are independent. For this machine, we say that a change of state occurs each time that it breaks down, or each time that it is fixed. In a time interval of length t, find the probability mass function of the number of times a change of state occurs.

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QUESTION:

In a factory, a certain machine operates for a period which is exponentially distributed with parameter . Then it breaks down and will be in repair shop for a period, which is also exponentially distributed with mean 1/. The operating and the repair times are independent. For this machine, we say that a change of state occurs each time that it breaks down, or each time that it is fixed. In a time interval of length t, find the probability mass function of the number of times a change of state occurs.

ANSWER:

Step 1 of 3

Let be the number of times a change of state occurs in the time interval . Let the random variable represents the time until machine breakdown for the lst time, let be the repair time of the machine, let be the time until the machine breakdown again, and so on.

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