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, …, P_p and the minimal cut sets be K₁, K₂,…, K_k. Show that f (.

Show that the minimal path sets for f are the minimal cut sets of f^D, where f^D 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,…, T_n are associated random variables not necessarily binary, show that

P ( T₁ £ t₁, T₂ £ t₂, …, T_n £ t_n) ≥

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 p_i whenever 0 < p_i < 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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – APRIL 2004

ST 2800/S 815 – PROBABILITY THEORY

02.04.2004 Max:100 marks

1.00 – 4.00

SECTION – A

Answer ALL questions (10 ´ 2 = 20 marks)

Show that if = 1, n = 1, 2, 3, …
Define a random variable and its probability distribution.
Show that the probability distribution of a random variable is determined by its distribution function.
Let F (x) = P (Prove that F (.) is continuous to the right.
If X is a random variable with continuous distribution function F, obtain the probability distribution of F (X).
If X is a random variable with P [examine whether E (X) exists.
State Glivenko – Cantelli theorem.
State Kolmogorov’s strong law of large number (SLLN).
If f(t) is the characteristic function of a random variable, examine if f(2t). f(t/2) is a haracteristic function.
Distinguish between the problem of law of large numbers and the central limit problem.

SECTION – B

Answer any FIVE questions (5 ´ 8 = 40 marks)

The distribution function F of a random variable X is

0 if x < -1

F (x) = if -1 £ x < 0

if 0 £ x < 1

1 if 1 £ x

Find Var (X)

If X is a non-negative random variable, show that E(X) < ¥ implies that
P [ X > n] ® 0 as n ® ¥. Verify this result given that

f(x) = .

State and prove Minkowski’s inequality.
In the usual notation, prove that

Define convergence in quadratic mean and convergence in probability. Show that the former implies the latter.
Establish the following:
If X_n ® X with probability one, show that X_n ® X in probability.
Show that X_n ® X almost surely iff for every > 0, is zero.
{X_n} is a sequence of independent random variables with common distribution function

0 if x < 1

F(x) =

1- if 1 £ x

Define Y_n = min (X₁, X₂ , … , X_n) . Show that Y_n converges almost surely to 1.

State and prove Kolmogorov zero – one law.

SECTION – C

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

a) Let F be the range of X. If and B F^C imply that P_X (B) = 0, Show that P can

be uniquely defined on (X), the s – field generated by X by the relation

P_X (B) = P {X Î B}.

b) Show that the random variable X is absolutely continuous, if its characteristic function f

is absolutely integrable over (- ¥, ¥ ). Find the density of X in terms of f.

a) State and prove Borel – Zero one law.
b) If {X_n, n ≥ 1} is a sequence of independent and identically distributed random

variables with common frequency function e^-x, x ≥ 0, prove that

a) State and prove Levy continuity theorem for a sequence of characteristic functions.
b) Use Levy continuity theorem to verify whether the independent sequence {X_n}

converges in distribution to a random variable, where X_n for each n, is uniformly

distributed over (-n, n).

a) Let {X_n} be a sequence of independent random variables with common frequency

function f(x) =, x ≥ 1. Show that does not coverage to zero with probability

one.

b) If X_n and Y_n are independent for each n, if X_n ® X, Y_n ® Y, both in distribution, prove

that ® (X²+ Y²) in distribution.

c) Using central limit theorem for suitable exponential random variables, prove that