פנגייה

Homework 5

• זמן קריאה: 3 דק'

Student A
Full NameIdo Fang Bentov
IDCLASSIFIED
EmailCLASSIFIED

Question 1

Twelve machines run in parallel, manufacturing the same part, in one 8-hour shift per day. The times to failure (TTF) for the 12 machines, , have independent, identical exponential distributions with hours. Whenever a machine fails, it is shut down for the remainder of the 8-hour shift and repaired later by a maintenance crew. The times to failure for the repaired machines have the same, independent exponential distributions as the new machines. Every
morning all twelve machines are working; by the end of the 8-hour shift, zero to twelve machines are still working.

We denote as the total number of failures for all 12 machines in one 8-hour shift. Note that .

Part a

Regarding the distribution of , which of the following statement is true?

  1. Because the machines are shut down for the rest of the shift after failing,
  2. Because the machines are shut down for the rest of the shift after failing,
  3. , with mean
  4. has an approximately normal distribution, with mean
  5. , with mean

Solution:

The key insight is understanding how the shutdown mechanism affects the failure process. Each machine has an exponential distribution with mean 50 hours, giving a failure rate failures per hour.

In a pure Poisson process with 12 machines over 8 hours, the expected number of failures would be . However, because machines are shut down after failing, they cannot contribute additional failures during the same shift. This reduces the effective “machine-hours” as failures occur, making the actual expected number of failures less than 1.92.

Therefore, statement 1 is correct: Because the machines are shut down for the rest of the shift after failing, .

Part b

What is the probability that exactly 10 of the 12 machines are still working at the end of the first 8-hour shift of this week?

Solution:

If exactly 10 machines are still working, then exactly 2 machines have failed ().

For each machine during an 8-hour period:

  • Probability of survival:
  • Probability of failure:

This follows a binomial distribution: .

Part c

If we know that exactly 10 of the 12 machines are still working at the end of the first 8-hour shift of this week, and we randomly select a sample of four machines from these 12 machines, what is the probability that exactly two of the four machines selected are still working?

Solution:

This is a hypergeometric distribution problem. We have:

  • total machines
  • working machines
  • machines selected
  • working machines desired

Using the hypergeometric formula:

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