Loyola College M.Sc. Statistics April 2004 Reliability Theory Question Paper PDF Download

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)

 

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

 

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

 

  1. Let the minimal path sets of f be P1, P2, …, Pp and the minimal cut sets be K1, K2,…, K. Show that f (.

 

  1. Show that the minimal path sets for f are the minimal cut sets of fD, where fD represents the dual of f.

 

  1. 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.

 

 

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

 

  1. If T1, T2,…, Tn are associated random variables not necessarily binary, show that

P ( T1 £ t1, T2 £ t2, …, Tn £ t) ≥

  1. Examine whether the Gamma distribution is IFR.

 

SECTION – C

 

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

 

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

 

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

increasing in each pi whenever 0 < pi < 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.

 

  1. 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.

 

  1. 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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