MIME 4690/5690 Final exam Spring 2003

______

Solve all problems. Each problem counts 25 points. Sign the honor pledge:

I have not received nor given any help in this exam

Name ______Signature ______

1)The time to failure of a pump is a uniformly distributed random variable between 200 and 1000 hours as shown in the figure.

a)Find and sketch the cumulative probability distribution function of the time to failure, FT(t).

b)Find and sketch the reliability function, R(t).

c)Find and sketch the hazard function, h(t).

2)The stress in a component is uniformly distributed between 15,000 and 31,000 psi, as shown in the figure. The ultimate strength is also uniformly distributed between 30,000 and 40,000 psi. Derive an equation for the probability of failure assuming that the stress and the strength are statistically independent and use the equation to calculate the probability of failure.

3)The reliability diagram below shows a system with three components that fail independently. All components have reliability R=0.95. Find the reliability of the system.

4)Answer the questions. You do not need to justify your answers to true-false questions (marked T-F), just say if a statement is true or false. However, you can write something if you think a question is vague or ambiguous. Your answers to the remaining questions should be short (about 50 words).

1)The probability density function of the time to failure of a system can be greater than 1. (T-F)

2)The cumulative probability distribution function of the time to failure tends to zero as time tends to infinity. (T-F)

3)The hazard function of a drive shaft is approximately 0.210-4 failures/km. If we start with 100 shafts at 0 km then, approximately, 4 drive shafts fail on average between 0 and 20,000 km. (T-F)

4)The time to failure of a system follows the Weibull distribution with shape parameter, . The higher the value of  the steeper is the increase in the hazard rate as a function of time. (T-F)

5)A structural component, whose strength is Su, is subjected to a stress, S. The probability of failure is:

(T-F)

6)The standard deviation of the sum of two variables is equal to the sum of the standard deviations of these variables. (T-F)

7)The failure probability of a series system consisting of 10 nominally identical, independent components with failure probability 0.05 is 0.5. (T-F)

8)The hazard rate of a series system consisting of independent components is equal to the sum of the hazard rates of these components. (T-F)

9)Availability of a system is the probability of the system performing satisfactorily at any point in time. (T-F)

10)A system has mean time to failure 120 days and mean repair time 2 days. Then the steady state availability is . (T-F)