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)
- Show that a parallel system is coherent.
- Derive MTBF when the system failure time follows Weibull distribution.
- Show that independent random variables are associated.
- What is the conditional probability of a unit of age t to fail during the interval (t, t+x)?
- Define a) System Reliability b) point availability
- With usual notation show that MTBF = R* (0), where R* (0) is the Laplace Transform of R (t) at s = 0.
- Show that a device with exponential failure time, has a constant failure rate.
- Obtain the Reliability of a (k,n) system with independent and identically distributed failure times.
- State lack of memory property.
- Define a minimal path set and illustrate with an example.
SECTION – B
Answer any FIVE questions (5 ´ 8 = 40 marks)
- Define hazard rate and express the system reliability in terms of hazard rate.
- For a parallel system of order 2 with constant failure rates l1 and l2 for the components, show that MTBF = .
- Let the minimal path sets of f be P1, P2, …, Pp and the minimal cut sets be K1, K2,…, Kk. Show that f (.
- Show that the minimal path sets for f are the minimal cut sets of fD, where fD represents the dual of f.
- 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.
- Obtain the reliability of (i) parallel system and (ii) series system.
- If T1, T2,…, Tn are associated random variables not necessarily binary, show that
P ( T1 £ t1, T2 £ t2, …, Tn £ tn) ≥
- Examine whether the Gamma distribution is IFR.
SECTION – C
Answer any TWO questions (2 ´ 20 = 40 marks)
- Derive the MTBF of a standby system of order n with parallel repair and obtain the same when n = 3 and r = 2.
- 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.
- 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.
- a) If two sets of associated random variables are independent, show that their union is a
set of associated random variables.
- b) Let the probability density function of X exist. Show that F is DFR if r (t) is
decreasing.
- a) State and establish a characterization of exponential distribution based on lack of
memory property.
- b) State and prove IFRA closure theorem.
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