MIME 5960
Due 2/12/2015
Homework 2
Spring 2015
Homework format (applies to all homework assignments):
Put the page with the problem statement first. Write your name on this page and the final
answer (only) to each problem on this page. If an assignment contains true-false questions
(marked T-F) circle the answers. Do not write anything else on the first page. The detailed
solutions should be in the next pages. Use a separate page for each problem but do not use a
page for problem with T-F questions. All pages must be stapled together.
1. The probability density function if the failure time of a pump is uniform (i.e. it has a
constant value, k) between 0 and 100 months and zero elsewhere.
a) Find k
b) Find and plot the reliability function, R(t)
c) Find and plot the hazard function, h(t)
d) Find the expected life of the pump
2. The hazard function of a system is: h(t)=1/(t+2)
a) Find the probability density function of the failure time, fT(t).
b) Find the cumulative distribution function of the failure time, FT(t).
3. Answer to the following true-false questions. You do not need to justify your answers
but you can do so if you think a question is vague.
a) Consider two designs. If the safety factor of design 1 is greater than that of design 2,
then the failure probability of design 1 is always smaller than that of design 2. (T-F)
b) Safety factor is the ratio of the design strength over the design load. (T-F)
c) Safety factor is the ratio of the mean strength over the mean load. (T-F)
d) Reliability-based design is easier than reliability analysis. (T-F)
e) Human error increases reliability. (T-F)
f) The probability density function of the failure time fT(t) is equal to the probability of
failure of the system at time t. (T-F)
g) The probability density function of failure time is always less than or equal to one.
h) The cumulative distribution function of the failure time, FT(t), is always less or equal to
one. (T-F)
i) The area under the cumulative distribution function is always one. (T-F)
4. The time to failure of a particular class of computer hard disks follows the exponential
probability distribution
x  xl

1  e   if x  x
FX ( x)  
l .
 0
if x  xl

Test results show that the probability that the time to failure is less than 2,000 hours of
operation is e 1 .
a. Calculate the mean time to failure and write expressions for the CDF and PDF of the
time to failure. Assume that xl  0.
b. Calculate the probability that a hard disk could fail before 1,000 hrs of operation.