Download Exam 1, 2011

MIME 5690
Exam 1
Spring 2011
This is a take-home exam. Please return your exam by Tuesday, April 5, 2011 at 3:30
PM.
Sign the honor pledge: I have not given nor received any help in this exam.
Name ________________, Signature __________________
1) (70%) The times to failure of 100 light bulbs, in hours, are in the Excel file.
a) Estimate and plot the cumulative probability distribution of the time to failure,
reliability function, probability density function of the time to failure and hazard function
using both the following two approaches:
Approach A: The cumulative probability distribution function of the time to failure at
time t can approximated by the fraction of the sample that has failed before t or
i
exactly at t using equation, FˆT (t i )  , where i is the failure number and n is the
n
sample size. A better approximation is obtained by using the following equation for
(i  0.3)
estimating the cumulative probability distribution, FˆT (t i ) 
. The following
(n  0.4)
equations can be used to estimate the reliability, the density and the hazard functions:
Rˆ (ti )  1  FˆT (ti )
Rˆ (t i )  Rˆ (t i 1 )
fˆT (t i ) 
t i 1  t i
fˆ (t )
hˆ(t i )  T i .
Rˆ (t i )
Approach B: Assume that the time to failure follows the Weibull distribution with
location parameter equal to zero. Estimate the shape parameter, , and the scale
parameter,  , from the data in the Excel file. Then calculate the cumulative probability
distribution of the time to failure, reliability function, probability density function of the
time to failure and hazard function from the Weibull distribution.
b) Compare the results of the two approaches. Which approach yielded better results?
Explain why.
2) (30%) Answer the following questions. You do not need to justify your answers to
true-false questions or questions asking for a numerical answer. However, you can
write something if you think a question is vague or ambiguous.
A. The mean value of the sum of two random variables is equal to the sum of the
mean values. (T-F)
B. If we multiply a random variable by a constant c, then its variance is scaled by c2.
(T-F)
C. The PDF of random variable X is fX(x). Then the PDF of 2X is 2fX(x). (T-F)
D. The safety index of a component with strength Su, which is subjected to a stress S,
is equal to the ratio Su/S. (T-F)
E. The safety index of a component with strength Su, which is subjected to a stress S,
is equal to the ratio of the difference of the mean values of the strength and stress,
normalized by the standard deviation of this difference. (T-F)
F. The strength of a rod is Su and the applied stress is S. Both the stress and strength
are statistically independent, Gaussian random variables. The stress has a mean
value of 30,000 psi and a standard deviation of 1,000 psi. The strength has a mean
value of 40,000 psi and a standard deviation of 2,000 psi. Then the safety index of
the rod is: _______.
G. In the above problem, the probability of failure is: ________.
H. I drove my car 300 days last year. I did not use the car for 55 days because I did
not need to do so. In the remaining 10 days the car was in the garage for repairs.
Based on this information, the availability of the car is ___________ .
I. In the previous question, the operational readiness is __________.
J. If the hazard (or failure) rate of a system is constant, then the probability
distribution of the time to failure is uniform. (T-F)
K. The probability that an earthquake of magnitude greater than of equal to 9 on the
Richter scale occurring in one year in Southern California is 10-5. Assume that the
magnitudes of severe earthquakes are statistically independent. Then the
probability that an earthquake of magnitude greater than or equal to 9 on the
Richter scale will hit Southern California in the next 30 years is _____.
L. Two random variables are uniformly distributed. Then the sum of these variables
is also uniformly distributed. (T-F)