06.04.2004
1.00 - 4.00
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
M.Sc., DEGREE EXAMINATION - STATISTICS
FOURTH SEMESTER – APRIL 2004
ST 4950 - RELIABILITY THEORY
Max:100 marks
SECTION - A
(10  2 = 20 marks)
Answer ALL questions
1.
2.
3.
4.
5.
6.
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.
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
(5  8 = 40 marks)
Answer any FIVE questions
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 1 and 2 for the components,
1
1
1


show that MTBF =
.
1 2 1  2
13. Let the minimal path sets of  be P1, P2, ..., Pp and the minimal cut sets be K1, K2,..., Kk.
p
Show that  ( x) 
k
 xi and  ( x )  
j  1 i  Pj
j1 i  K j
xi .
14. Show that the minimal path sets for  are the minimal cut sets of D, where D represents
the dual of .
15. Explain the relative importance of the components. For a system of order 3 with structure
function  (x1 x2 x3) = x1 (x2  x3), compute the relative importance of the components.
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16. Obtain the reliability of (i) parallel system and (ii) series system.
17. If T1, T2,..., Tn are associated random variables not necessarily binary, show that
n
P ( T1  t1, T2  t2, ..., Tn  tn) ≥  P { Ti  t i }
i 1
18. Examine whether the Gamma distribution is IFR.
SECTION - C
(2  20 = 40 marks)
Answer any TWO questions
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 ( p ) be the system reliability of a coherent structure. Show that h ( p ) 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 ( p p ') ≥ h ( p ) h ( p )  0  p , p '  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.
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.
b) State and prove IFRA closure theorem.
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