Loyola College M.Sc. Statistics April 2004 Reliability Theory Question Paper PDF Download

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)

 

  1. Show that a parallel system is coherent.
  2. Derive MTBF when the system failure time follows Weibull distribution.
  3. Show that independent random variables are associated.
  4. What is the conditional probability of a unit of age t to fail during the interval (t, t+x)?
  5. Define a) System Reliability b) point availability
  6. 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

 

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

 

  1. Define hazard rate and express the system reliability in terms of hazard rate.

 

  1. For a parallel system of order 2 with constant failure rates l1 and l2 for the components, show that MTBF = .

 

  1. Let the minimal path sets of f be P1, P2, …, Pp and the minimal cut sets be K1, K2,…, K. Show that f (.

 

  1. Show that the minimal path sets for f are the minimal cut sets of fD, where fD represents the dual of f.

 

  1. 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.

 

 

  1. Obtain the reliability of (i) parallel system and (ii) series system.

 

  1. If T1, T2,…, Tn are associated random variables not necessarily binary, show that

P ( T1 £ t1, T2 £ t2, …, Tn £ t) ≥

  1. Examine whether the Gamma distribution is IFR.

 

SECTION – C

 

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

 

  1. Derive the MTBF of a standby system of order n with parallel repair and obtain the same when n = 3 and r = 2.

 

  1. 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.

 

  1. 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.

 

  1. a) If two sets of associated random variables are independent, show that their union is a

set of associated random variables.

 

  1. b) Let the probability density function of X exist. Show that F is DFR if r (t) is

decreasing.

 

  1. a) State and establish a characterization of exponential distribution based on lack of

memory property.

 

  1. b) State and prove IFRA closure theorem.

 

 

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Loyola College M.Sc. Statistics April 2004 Probability Theory Question Paper PDF Download

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)

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

 

SECTION – B

 

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

 

  1. 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)

 

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

f(x) = .

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

.

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

 

 

0      if     x <  1

F(x) =

1-  if   1 £  x

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

  1. State and prove Kolmogorov zero – one law.

 

SECTION – C

 

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

 

  1. a) Let F be the range of X. If  and B FC imply that PX (B) = 0,  Show that P can

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

PX (B) = P {X Î B}.

  1. 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.

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

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

.

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

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

distributed over (-n, n).

  1. a) Let {Xn} 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.

  1. b) If Xn and Yn are independent for each n, if Xn ® X, Yn ® Y, both in distribution, prove

that ® (X2 + Y2) in distribution.

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

.

 

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