LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 40

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 p_i, whenever 0<p_i<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 λ_i, 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, …, X_n are associated binary random variables, show that

(1-X₁), (1-X₂),…,(1-X_n) 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)

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