LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – APRIL 2004

ST 4950 – RELIABILITY THEORY

06.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

 

SECTION – A

 

Answer ALL questions                                                                                (10 ´ 2 = 20 marks)

 

1. Show that a parallel system is coherent.
2. Derive MTBF when the system failure time follows Weibull distribution.
3. Show that independent random variables are associated.
4. What is the conditional probability of a unit of age t to fail during the interval (t, t+x)?
5. Define a) System Reliability b) point availability
6. With usual notation show that MTBF = R^* (0), where R^* (0) is the Laplace Transform of R (t) at s = 0.
7. Show that a device with exponential failure time, has a constant failure rate.
8. Obtain the Reliability of a (k,n) system with independent and identically distributed failure times.
9. State lack of memory property.
10. Define a minimal path set and illustrate with an example.

 

SECTION – B

 

Answer any FIVE questions                                                                        (5 ´ 8 = 40 marks)

 

11. Define hazard rate and express the system reliability in terms of hazard rate.

 

12. For a parallel system of order 2 with constant failure rates l₁ and l₂ for the components, show that MTBF = .

 

13. Let the minimal path sets of f be P₁, P₂, …, P_p and the minimal cut sets be K₁, K₂,…, K_k­. Show that f (.

 

14. Show that the minimal path sets for f are the minimal cut sets of f^D, where f^D represents the dual of f.

 

15. Explain the relative importance of the components. For a system of order 3 with structure function f (x₁ x₂ x₃) = x₁ (x₂ x₃), compute the relative importance of the components.

 

 

16. Obtain the reliability of (i) parallel system and (ii) series system.

 

17. If T₁, T₂,…, T_n are associated random variables not necessarily binary, show that

P ( T₁ £ t₁, T₂ £ t₂, …, T_n £ t_n­) ≥

18. Examine whether the Gamma distribution is IFR.

 

SECTION – C

 

Answer any TWO questions                                                                        (2 ´ 20 = 40 marks)

 

19. Derive the MTBF of a standby system of order n with parallel repair and obtain the same when n = 3 and r = 2.

 

20. a) Let h (be the system reliability of a coherent structure.  Show that h ( is strictly

increasing in each p_i whenever 0 < p_i < 1 and i = 1,2,3,…,n.

 

1. b) Let h be the reliability function of a coherent system. Show that

h (    ‘) ≥ h ()     h () ” 0 £ , ‘ £ 1.

Also show that equality holds  when the system is parallel.

 

21. a) If two sets of associated random variables are independent, show that their union is a

set of associated random variables.

 

1. b) Let the probability density function of X exist. Show that F is DFR if r (t) is

decreasing.

 

22. a) State and establish a characterization of exponential distribution based on lack of

memory property.

 

1. b) State and prove IFRA closure theorem.

 

 

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