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)

 

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 l₁ and l₂ for the components, show that MTBF = .

 

13. 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 (.

 

14. Show that the minimal path sets for f are the minimal cut sets of f^D, where f^D represents the dual of f.

 

15. 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.

 

 

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

 

17. If T₁, T₂,…, T_n are associated random variables not necessarily binary, show that

P ( T₁ £ t₁, T₂ £ t₂, …, T_n £ t_n­) ≥

18. Examine whether the Gamma distribution is IFR.

 

SECTION – C

 

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

 

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 (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.

 

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.

 

21. 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.

 

22. 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 (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)

 

11. 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)

 

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

f(x) = .

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

.

15. Define convergence in quadratic mean and convergence in probability. Show that the former implies the latter.
16. Establish the following:
17. If X_n ® X with probability one, show that X_n ® X in probability.
18. Show that X_n ® X almost surely iff for every > 0,  is zero.
19. {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.

18. State and prove Kolmogorov zero – one law.

 

SECTION – C

 

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

 

19. 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}.

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.

20. a) State and prove Borel – Zero one law.
21. 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

.

21. a) State and prove Levy continuity theorem for a sequence of characteristic functions.
22. 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).

22. 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.

1. 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.

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

.

 

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 31

SECOND SEMESTER – APRIL 2006

                                                       ST 2805 – PROBABILITY THEORY

(Also equivalent to ST 2800)

 

 

Date & Time : 24-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

Section-A

Answer ALL questions                                                                   (10 ´ 2 = 20 marks)

1. With reference to tossing a regular die once and noting the outcome, identify completely all the elements of the probability space (Ω, A, P).
2. Show that the limit of any convergent sequences of events is an event.
3. Let F(x) = P [X < x], x Є R. Prove that F (.) is continuous to the left.
4. Write down any two properties of the distribution function of a random vector (X, Y).
5. If X is a random variable with P[X = (-1) ^k₂^k] = 1/2^k, k = 1, 2, 3…, examine whether E[X] exists.
6. If X² and Y² are independent, are X and Y independent?
7. Define almost sure convergence and convergence in probability for a sequence of random variables.
8. If Φ is the characteristic function (CF) of a random variable X, find the CF of (2X+3).
9. Let {Xn, n = 1, 2, …} be a sequence of independent and identically distributed (iid) n

N (μ, σ²) random variables. Define Yn = 1/n   Σ X² _k, n = 1, 2, 3,… Examine                                                                                  K=1

whether Kolmogorov strong law of large numbers (SLLN) holds for
{Y_n, n = 1, 2, 3…}

1. State Lindeberg – Feller Central limit theorem.

 

Section – B

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

1. Define the distribution function F(x) of a random variable X. State and establish its defining properties.
2. State and prove Minkowski’s inequality
3. State and prove Borel Zero- one law.
4. Find var(Y), if the conditional characteristic function of Y given X=x is

[1+ (t² /x)]^-1 and X has frequency function f(x) = 1/x²   for x ≥ 1

=   0      otherwise

1. 1. Show that convergence in probability implies convergence in distribution.

 

1. Define convergence in quadratic mean for a sequence of random variables.

X is a random variable, which takes on positive integer values.

Define X_n =   (n+1) if X=n

=    n     if X = (n+1)

=    X    otherwise

Show that X_n converges to X in quadratic mean.

17. Show that X_n → X in probability if and only if every subsequence of {X_n} contains a further subsequence, which converges almost surely.
18. Let {X_n, n ≥ 1} be a sequence of independent random variables such that X_n has uniform distribution on (-n, n). Examine whether the central limit theorem holds for the sequence {X_n, n ≥ 1}.

Section-C

Answer any TWO questions.                                                          (2 × 20 = 40 marks)

19. a. Define the probability distribution of a random variable. Show that the probability distribution of a random variable is determined by its distribution function.(8 marks)
20. Show that the vector X = (X₁, X₂, …, X_p) is a random vector if and only if X_j,

j = 1, 2,… , p is a real random variable.                                                (8 marks)

1. If X is a random variable with continuous distribution function F, obtain the probability distribution of F(X).                                        (4 marks)

20.a. Show that convergence in quadratic mean implies convergence in probability. Illustrate by an example that the converse is not true.                               (8 marks)

1. State and prove Kolmogorov zero-one law.                                         (12 marks)

21.a.  State and prove Kolmogorov three series criterian for almost sure convergence of the series    ∞

Σ X_n of independent random variables.         (12 marks)

n=1

1. Let {X_n} be a sequence of normal variables with E (X_n) = 2 + (1/n) and

var (X_n) = 2 + 1/n², n= 1, 2, 3 … Examine whether the sequence converges in distribution.                                                                              (8 marks)

22.a. State and prove the continuity theorem for a sequence of characteristic functions.

(12 marks)

1. Let {X_k} be a sequence of independent random variables with

P [X_k = k^λ]  = P [X_k = -k^λ] = 1/2, k = 1, 2, 3… Show that central limit theorem holds for

λ ≥ -1/2.                                                                                                  (8 marks)

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – APRIL 2007

ST 2805 / 2800 – PROBABILITY THEORY

 

 

 

Date & Time: 27/04/2007 / 1:00 – 4:00 Dept. No.                                            Max. : 100 Marks

 

 

Section – A

Answer all the Questions                                                         10 x 2 = 20

 

1. With reference to tossing a regular coin once and noting the outcome, identify completely all the elements of the probability space. (W,       , P).
2. Show that the limit of any convergent sequence of events is an event.
3. Define a random variable and its probability distribution.
4. If X is a random variable with continuous distribution functions F, obtain the probability of distribution of F(X).
5. Write down any two properties of the distribution function of a random vector (X,Y).
6. If X² and Y² are independent, are X and Y independent?
7. Define (i) convergence in quadratic mean and   (ii) convergence in distribution for a sequence of random variables.
8. If  f  is the characteristic function (CF) of a random variable X, find the CF of (3X+2).
9. State Kolmogorov’s strong law of large numbers(SLLN).
10. State Linde berg – Feller central limit theorem.

 

SECTION – B

Answer any FIVE questions.                                                  5 x 8 = 40

 

1. If X  and Y are independent, show that the characteristic function of X+Y is the product of their characteristic functions.  Is the converse true?  Justify.
2. State and prove Minkowski’s inequality.
3. Show that convergence in probability implies convergence in distribution.
4. State and prove Borel zero – one law.
5. Find the variance of Y, if the conditional characteristic function of Y given X=x is      and X has frequency function

for x  ³ 1

f (x) =

0,  otherwise

 

1. Show that X_n  ® X in probability if and only if every subsequence of {Xn} contains a further subsequence, with convergence almost surely.
2. Using the central limit theorem for suitable Poison random variables, prove that

=

 

 

1. Deduce Liapounov theorem from Lindeberg – Feller theorem.

 

 

Section – C

Answer any TWO questions                                                   2 x 20 = 40

 

19. a) Show that the probability distribution of a random variable is determined by its

distribution function.  Is the converse true?                                        (8  marks)

1. b) Show that the vector X = (X_1, X₂, …, X_p)  is a random vector if and only X_i,  i=1,2,…p is a real

random variable.                                                                                                          (8 marks)

1. c) The distribution function of a random variable X is given by

 

0       if  x < 0

F(x) =          if 0 £ x < 1

1       if  1 £ x < ¥

 

Obtain E(X).                                                                                       (4 marks)

 

20. a) State and prove Kolmogorov zero – one law for a sequence of independent
    random variables. (10 marks)
21. b) If {X_n , n ³ 1} is a sequence of independent and identically distributed random
    variables with common frequency function e^-x,  x  > 0,

prove that P[lim sup   ]=1

(10 marks)

21. a) State and prove Kolmogorov three series theorem for almost sure convergence
    of the series S X_n of independent random variables.                         (12 marks)
22. b) Show that convergence in quadratic mean implies convergence in probability.
    Illustrate by an example that the converse is not true.                         (8 marks)

 

22. a) State and prove Levy continuity theorem for a sequence of characteristic
    functions.           (10 marks)

 

1. b) State and prove Inversion theorem.           (10 marks)

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

NO 58

SECOND SEMESTER – APRIL 2008

ST 2805/ 2800 – PROBABILITY THEORY

 

 

 

Date : 06-05-08                  Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

Section-A (10 × 2=20)

 

Answer ALL the questions.

∞

1.Show  that P( ∩  A_n  ) =1 if  P(A_n ) =1, n=1,2,3,…

n=1

2. Show that the limit of any convergent sequence of events is an event.
3. Define a random variable X and the σ- field induced by X.
4. Show that F(x) = P [X≤ x], x є R is continuous to the right.
5. Show that the probability distribution of a random variable is determined by its

distribution function.

6. Calculate E(X), if X has a distribution function F(x), where

F(x)  =   0         if x<0

x/2      if 0≤ x<1

1         if x ≥ 1.

7. If X₁ and X₂ are independent random variables and g₁ and g₂ are Borel functions, show

that g₁(X₁) and g₂(X₂ )are independent.

8. If Φ is the characteristic function (CF) of a random variable X, find the CF of (3X+4).
9. State Glivenko-Cantelli theorem.
10. State Lindeberg-Feller central limit theorem.

 

Section-B (5×8=40)

 

Answer any FIVE questions.

 

11. Explain the independence of two random variables X and Y. Is it true that if X and

Y are independent, X² and Y² are independent? What about the converse?

12. If X is a non-negative random variable, show that E(X) <∞ implies that
13. P(X > n) →0 as n→∞. Verify this result given that

f(x )= 2/ x³ ,    x ≥1.

13. In the usual notation, prove that

∞                                                           ∞

Σ   P [׀X׀ ≥ n]  ≤ E׀X׀ ≤    1 + Σ   P [׀X׀ ≥ n].

n=1                                                      n=1

 

14. Define convergence in probability. If X_n → X in probability, show that

X_n² + X_n  → X²+ X in probability.

15. If X_n → X in probability, show that X_n → X in distribution.
16. State and prove Kolmogorov zero-one law for a sequence of independent random

variables.

17. Using the central limit theorem for suitable Poisson random variables, prove that

n

lim   e^-n   Σ        n^k   = 1/2.

n→∞        k=0       k!

18. Find Var(Y), if the conditional characteristic function of Y given X=x is (1+(t²/x))^-1

and X has frequency function

f (x)  =1/x²,   for x ≥1

0,      otherwise.

 

Section-C (2×20= 40 marks)

 

Answer any TWO questions

 

19. (a) Define the distribution function of a random vector. Establish its

properties.                                           (8 marks)

(b) Show that the vector X =(X₁, X_2,…, X_p ) is a random vector if and only if X_j,

j =1, 2, 3… p is a real random variable.                 (8 marks)

(c)  If X is a random variable with continuous distribution function F, obtain the

probability distribution of F(X).                              (4 marks)

20 (a)  State and prove Borel zero –one law.

• 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

P [lim (X _n / (log n)) >1] =1.                                        (12+8)

21  (a) State and prove Levy continuity theorem for a sequence of characteristic

functions.

• Let {X_n} be a sequence of normal variables with E (X_n) = 2 + (1/n) and

var(X) = 2 + (1/n²), n=1, 2, 3…Examine whether the sequence converges in distribution.                                                    (12+8 marks)

22 (a)  State and prove Kolmogorov three series theorem for almost sure convergence of

the series Σ X_n of independent random variables.                                    (12)

(b)  Show that convergence in quadratic mean implies convergence in probability.

Illustrate by an example that the converse is not true.    (8 marks)

 

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     LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

YB 52

M.Sc. DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – April 2009

S  815 – PROBABILITY THEORY

 

 

 

Date & Time: 04/05/2009 / 1:00 – 4:00       Dept. No.                                            Max. : 100 Marks

 

 

SECTION-A (10 × 2=20)

Answer ALL the questions.

 

1. With reference to tossing a regular coin once and noting the outcome, identify

completely all the elements of the probability space (Ω, A, P).

∞

2. If P(A_n ) =1, n=1,2,3,… , evaluate P( ∩  A_n  ) .

n=1

3. Show that the limit of any convergent sequence of events is an event.
4. Define a random variable and its probability distribution.
5. Calculate E(X), if X has a distribution function F(x), where

F(x) =   0         if x<0

x/2      if o≤ x<1

• if x ≥ 1.

6. If X₁ and X₂ are independent random variables and g₁ and g₂ are Borel functions, show

that g₁(X₁) and g₂(X₂ )are independent.

7. State Glivenko-Cantelli theorem.
8. Φ is the characteristic function (CF) of a random variable X, find the CF of (2X+3).
9. State Kolmogorov’s strong law of large numbers(SLLN).
10. State Lindeberg-Feller central limit theorem.

 

SECTION-B (5 × 8 = 40)

Answer any FIVE questions.

 

11. Define the distribution function of a random variable X. State and establish its

defining properties.

12. Explain the independence of two random variables X and Y. Is it true that if X and

Y are independent, X² and Y² are independent? What about the converse?

13. State and prove Borel zero –one law.
14. State and prove Kolmogorov zero-one law for a sequence of independent random

variables.

15. Define convergence in probability. Show that convergence in probability implies

convergence in distribution.

16. a) Define “Convergence in quadratic mean” for a sequence of random variables.
17. b) X is a random variable, which takes on positive integer values. Define

X_n   = n+1   if X=n

n      if X=(n+1)

X      otherwise

Show that X_n  converges to X  in quadratic mean.      (2+6)

 

17. Establish the following:

(a) If  X_n → X with probability one, show that X_n → X in probability.

(b) Show that X_n → X almost surely if and only if for every є >0,

P [lim sup │ ‌X_n – X│> є ]= 0

18. Let { X_n ,n ≥1} be a sequence of independent random variables such that X_n has

uniform distribution on (-n, n). Examine whether the central limit theorem holds for

the sequence { X_n, n≥1}.

 

SECTION-C (2 x 20 = 40 marks)

Answer any TWO questions

 

19.a) Show that the probability distribution of a random variable is determined by its

distribution function.

1. b) Show that the vector X =(X₁, X_2,…, X_p ) is a random vector if and only if X_j,

j =1, 2, 3… p is a real random variable.

1. c) If X is a random variable with continuous distribution function F, obtain the

probability distribution of F(X).                              (6+8+6)

20.a) State and prove Kolmogorov’s inequality.        (10 marks)

1. b) State and prove Kolmogorov three series theorem for almost sure convergence of

the series Σ X_n of independent random variables.  (10 marks)

21.a) If  X_n and Y_n  are independent for each n, if  X_n →  X,  Y_n → Y, both in

distribution, prove that (X_n² + Y_n²) → (X²+Y²) in distribution.            (10 marks)

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

function f(x) = 1/x²  , x=1,2,3,… Show that X_n /n does not converge to zero with

probability one.                                                                          (10 marks)

22.a) State and prove Levy continuity theorem for a sequence of characteristic

functions.(12 marks).

b)Let {X_n} be a sequence of normal variables with E (X_n) = 2 + (1/n) and

var(X_n) = 2 + (1/n²), n=1, 2, 3…Examine whether the sequence converges in

distribution.(8 marks).

 

 

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