LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – APRIL 2012

ST 2957 – RELIABILITY THEORY

 

 

Date : 24-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION – A

 

Answer all the questions                                                                                                             (10×2=20)

1. Define Reliability R(t) of a system
2. Define Hazard function r(t)

3.

R_A=0.87, R_B=0.85, R_C=0.89. Determine system reliability

4. Define Parallel-Series system
5. Define MTBF
6. Define a (k,n) system
7. Define Standby system
8. R(t) = e^{– 0.2t} determine the warranty period for a reliability of 0.9
9. An equipment has a hazard function r(t) = 6×10^-8t². The equipment is required to operate a 100

hours. What is the reliability at 100 hours?

10. Define a) DFRD b)IFRD

 

SECTION-B

Answer any five questions                                                                                                          (5X8=40)

 

11. Obtain the system reliability function R(t) and hazard function r(t) when the system failure time distribution follows Weibull distribution
12. Establish the following (3+3+2)
13. i) ii) If R^*(s) = LT{R(t)} then MTBF = R^*(0) iii)If T~Exponential distribution then MTBF=1/λ
14. Obtain system failure time density function for a (k,n) system
15. Define Series-Parallel system. Obtain system hazard function r(t) and MTBF for a

Series-Parallel System

15. Consider a series system consisting of two components with first component following a

exponential failure time distribution with λ=1/10,000 and second component following a

weibull with parameters β=6 and η=10,000. i)Obtain system reliability ii)Obtain system’s cdf

and pdf  iii) Given that the system has performed 500 hrs what is the reliability of the system

for an additional 1000hr mission  iv)Obtain the system failure rate v)What should be the

warranty period for a system reliability of 90%

16. Explain the methods of obtaining the reliability of a Complex system

17 Establish the following

1. F is IFR ó on [0,∞)
2. F is IFR ó
3. i) Establish: r(t) is a conditional probability function but not a conditional pdf
4. ii) Establish: r(t)↓t ó F is DFRD

 

SECTION-C

 

Answer any two questions                                                                                                         (2X20=40)

 

19. Obtain the reliability function R(t) and hazard function r(t) for the following failure time

distributions  i) Exponential   ii) Gamma

20. Obtain MTBF for the case when failure time(T) of a system is distributed as i) Exponential

ii)Weibull  iii) Gamma

21. Consider a Standby system of order 3 with T_i ~ Exponential(λ_i), i=1,2,3 . obtain the system

failure time density function and hence obtain the reliability function R(t) for the case when

λ₁= λ₂= λ₃ and λ₁≠ λ₂≠ λ₃(20)

22. Obtain system mean time between failure (MTBF) for a (k,n) system

 

 

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