Download MIME 5690 Practice exam 1 3/3/01

MIME 5690
Practice Exam 1
3/3/01
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1) The probability density function of the time to failure of a pump is triangular. It starts
with a value of k at time 0 and drops linearly to 0 at time equal to 4 months.
a) Find the value of k so that the probability density function of the time to failure is a
valid probability density function.
b) Find the reliability function of the pump, R(t).
c) Find the conditional probability of failure during the fourth month given that the
pump survived the first three months.
d) Find the hazard function of the pump.
2) Rods are bought from two factories (called Factory 1 and Factory 2, respectively). The
rods from the two factories have identical geometry. The ultimate tensile stresses
follow the Weibull probability distribution with a truncation parameter of zero. The
scale parameters of the rods from factories 1 and 2 are 1 and 2, respectively, and
their standard deviations are 1 and 2, respectively. The ultimate stresses are
statistically independent. If we pick up a rod at random and load it tension with a
tensile load, P, which is normally distributed with a mean value of S and a standard
deviation of S, what is the probability of failure?
Assume that the area of the cross section of the rod, A, is deterministic. When we pick
up a rod at random there is a 50% probability that it is from Factory 1 and 50%
probability that it is from Factory 2.
3) A muffler of a car has 10 pits. The muffler fails when any of the pits becomes a hole.
The times to failure of the pits are exponentially distributed, statistically independent,
and the mean value of the time to failure of each pit is 5 years. Derive an expression
for the reliability function of the muffler. Plot this function.
4) Answer the following true-false questions. You do not need to justify your answers,
just say if a statement is true or false. However, you can write something if you think
a question is vague or ambiguous.
a) The reliability function is monotonically decreasing. (T-F)
b) The hazard function can be negative. (T-F)
c) If the time to failure of a component is normally distributed, the hazard function is
constant. (T-F)
d) If a random variable follows the normal distribution, then the natural logarithm of the
function follows the log normal distribution. (T-F)
e) The standard deviation of the sum of two random variables is equal to the sum of the
standard deviations of these variables. (T-F)