LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS
|
SECOND SEMESTER – April 2009
ST 2957 / ST 2955 – RELIABILITY THEORY
Date & Time: 27/04/2009 / 1:00 – 4:00 Dept. No. Max. : 100 Marks
SECTION –A (10 x 2 = 20)
Answer any TEN questions. Each question carries TWO marks
- Define the following: (i) Mean time before failure(MTBF)
(ii) Steady state availabilty
- If the hazard function r(t)=3t2, t>0, obtain the corresponding probability distribution of
time to failure
- Obtain the reliability of a parallel system consisting of n components, when the
reliability of each component is known. Assume that the components are non-repairable.
- Explain in detail an n unit standby system.
- What is meant by reliability allocation?
- Define a coherent structure and give two examples.
- If x is a path vector and y ≥ x, show that y is also a path vector.
- Write down the structure function for a two out of three system.
- 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, i=1,2,3,…,n .
- Give an example of a distribution, which is IFR as well as DFR.
SECTION-B (5×8 =40 marks)
Answer any FIVE questions. Each question carries EIGHT marks
- Obtain the reliability function, hazard rate and the system MTBF for the following
failure time density function
f(t) = 12 exp(-4 t3)t2, t>0.
- What is a series system? Obtain the system failure time density function for a series system with
n independent components. Suppose each of the n independent components has an exponential
failure time distribution with the parameter λi, i= 1,2,…,n. Find the system reliability.
- Find the system MTBF for a (k,n) system, when the lifetime distribution is
exponential with the parameter λ. Assume that the components are non-repairable.
- Assuming that the components are non-repairable and the components have identical
constant failure rate λ, obtain the MTBF of the series-parallel system.
- Let Φ be a coherent structure. Show that
Φ(x .y ) ≤ Φ(x ) Φ(y )
Show that the equality holds for all x and y if and only if the structure is series.
- Let h be the reliability function of a coherent system. Show that
h( p Ц p‘) ≥ h( p ) Ц h( p‘) for all 0 ≤ p , p‘ ≤ 1.
Also, show that the equality holds if and only if the system is parallel.
- If two sets of associated random variables are independent, show that their union is
the set of associated random variables.
- Show that Wiebull distribution is a DFR distribution.
SECTION-C (2X20=40 marks)
Answer any two questions. Each question carries TWENTY marks
- a) Obtain the reliability function, hazard rate and the system MTBF for exponential
failure time distribution with the parameter λ. (8 marks)
- b) Obtain the system failure time density function for a (m, n) system. Assume that
the components are non-repairable. (12 marks)
20.a) Define the terms: (i) Hazard rate and (ii) Interval reliability (4 marks)
- b) For a simple 1 out of 2 system with constant failure rate λ and constant repair rate
μ, obtain the system of differential-difference equations. Also, obtain
the system reliability and system MTBF. (16 marks)
21.a) Define: (i) Dual of a structure (ii) Minimal path vector and (iii) Minimal cut vector
(6 marks)
- b) Let h be the reliability function of a coherent system. Show that
h( p . p‘) ≤ h( p ) . h( p‘) for all 0 ≤ p , p‘ ≤ 1. (10 marks)
Also show that the equality holds if and only if the system is series.
- c) If X1, X2, …, Xn are associated binary random variables, show that
(1-X1), (1-X2),…,(1-Xn) are also associated binary random variables.(4 marks)
22.a) If the probability density function of F exists, show that F is an IFR
distribution iff r(t)↑t. (10 marks)
- b) Examine whether Gamma distribution G(λ, α) is IFR or DFR. (10 marks)
Latest Govt Job & Exam Updates: