MIME 5690 Final Exam 5/1/2011 ______This is a take home exam. Open book and notes. Submit your solutions by email to Prof. Nikolaidis ([email protected]) including this exam by 5 PM, on Friday, May 6, 2011.
Complete the honor pledge: I have not given nor received any help in this exam.
Print name ______, Date ______
1) (25 pts) The Weibull PDF is,
1 y Y y Y f (y) min exp min for y Y Y1 w Y w Y w Y min 1 min 1 min 1 min
where Ymin is the lower bound value of the distribution, w1 is the characteristic smallest value of X, and is the shape parameter. The PDF is equal to zero for y < Ymin.
The capacity of a structural member of an offshore platform follows a Weibull distribution with mean value 300,000 lb, and coefficient of variation 0.15. The lower bound is 100,000 lb. a) Calculate the lower bound, the characteristic lower value and the shape parameter of the distribution. b) Calculate the probability of failure of the structural member if the applied load is 130,000 lb. 2 Hint: The mean value, E(Y1), and variance, of the Weibull distribution are given Y1 below in terms of the lower bound, characteristic value and shape parameter,
1 E(Y ) Y (w Y )(1 ) 1 min 1 min
2 2 2 1 2 (w1 Ymin ) {(1 ) [(1 )] } Y1
2) (25 pts) The time to failure of a particular class of computer hard disks follows the exponential probability distribution.
tt l FT (t) 1 e if t tl 0 if t tl Test results show that the probability that the time to failure is less than 2,000 hours of operation is e1. a. Calculate the mean time to failure and write expressions for the CDF and PDF of the time to failure. Assume that tl 0. b. Calculate the probability that a hard disk could fail before 1,000 hrs of operation.
3) (25 pts) The applied load in problem 1 is Gaussian with mean value 130,000 psi and standard deviation 20,000 psi. Calculate the probability of failure plus 95% confidence bounds using Monte Carlo simulation. You can use either standard Monte Carlo simulation or simulation with importance sampling.
4) (25 pts) Calculate the autocorrelation of the random process
X t Acos0t
if A and 0 are constants and is a uniformly distributed random variable on the interval (0, 2 )