M.Sc Statistics Question Paper

Loyola College M.Sc. Statistics April 2003 Testing Of Hypothesis Question Paper PDF Download

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LOYOLA  COLLEGE (AUTONOMOUS), CHENNAI-600 034.

M.Sc. DEGREE EXAMINATION – STATISTICS

second SEMESTER – APRIL 2003

ST   2802/  S  817   testing of hypothesis

24.04.2003

1.00 – 4.00                                                                                          Max: 100 Marks

 

 

SECTION – A                      (10 ´ 2 = 20 Marks)

Answer ALL questions.  Each carries TWO marks.

 

  1. Let X be an observation from

Find the power of the test

 

 

for testing H: q =1 against K: q = 2

  1. Give the test function

 

for testing H : q £    vs K: q >  based on an observation drawn from B (3,q),

find the  probability of rejecting H when q =.

  1. When do you say that a family of density functions has MLR property?
  2. What is a similar test?
  3. Define: Confidence set.
  4. Examine the validity of the statement “A MPT is always unbiased”.
  5. Give an example of a family having MLR property but not a member of one

parameter exponential family.

  1. Define UMPUT.
  2. Suppose a test function is of the form

for a family having MLR property in T(x).  Can such a test function satisfy

the condition    bf (q1) = bf (q2) (q1¹q2) ?

  1. Define: Maximal Invariance

 

 

 

 

 

 

 

Section – B                       (5 ´ 8 = 40 Marks)

Answer any FIVE.  Each carries EIGHT marks

  1. Let = (X1 , X2,….., xn) where Xi’s  are  i i d  with pdf  pq(x) = e-(x–q),x >q, q >0 .

Show that the family of densities pq() has MLR property and hence derive

the UMPT of level a for testing H :  q £ q0 Vs K: q >q0 .

  1. For each q0 ÎW, let A (q0) be the acceptance region of a level – a test for testing.

H(q0): q = q0 and for each sample point x let S(x) denote the set  of parameter

values S(x) = {q|x ÎA(q), qÎW}

  • Show that s(x) is a family of confidence sets for q at confidence level 1-a.
  • If A(q0) is UMP for testing H (q0) at level a against K(q0), then for each q0ÎW,

Show that S(x) minimizes pq(q0ÎS(x)) ” q ÎK(q0) among all level (1-a) families of confidence sets for q .

  1. Solve the problem of minimizing ò f fm+1dm subject to ò f fi d m = ci , i  = 1,2,….,m ,

where f1,f2, …..,fm , fm+1     are (m+1), m integrable functions.

  1. Let the  distribution of X be given by

 

X  0       1            2                  3
Pq (X = x)  q      2q       0.9.-2q           0.1-q

 

Where 0 < q < 0.1.    For testing H: q =0.05 against K: q > 0.05 at

level a =0.05, determine  which  of the following tests (if any) is UMP.

(i)         f (0) = 1, f (1) = f(2) = f(3) = 0

  • f (1) = 5, f(0) = f(2) = f(3) = 0
  • f (3) = 1, f(0) = f(1) = f(2) = 0
  1. Let X be an observation drawn from a population with pdf

pq (x) = q e–qx, x >0, q >0

Derive the UMPT of level a = 0.05 for testing H : q £ 1,  q ³ 2  Vs K : 1< q < 2.

  1. State and prove a necessary condition for all similar tests to have Neyman structure.
  2. Let X and Y be independent Poisson varictes with means l and m

H : l £ m Vs K:  l >m.

  1. Write a descriptive note on invariant tests.

SECTION – C                      (2 ´ 20 = 20 Marks)

Answer any TWO.  Each carries twenty marks.

  1. State and prove Neyman – Pearson lemma.
  2. Derive the UMPUT for testing H: q =q0 Vs K: q ≠ q0 in

pq()  = c(q) eqT(x)      h(x)

  1. Let (X1,X2, ….,Xm) and (Y1,Y2,…,Yn) be samples of sizes m and n respectively

from N(x, s2) and   N (ך,s).  Derive the UMPUT

(unconditional) for testing     (i)  H :ך £ x  Vs  K: ך > x   and

(ii)  H:  ך = x  Vs K:  ך ¹ x

  1. Illustrate, with an example, the steps involved in developing unconditional

UMPUT’s for one-sided testing problems in the multi-parameter

exponential setup

 

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Loyola College M.Sc. Statistics April 2003 Sampling Theory Question Paper PDF Download

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LOYOLA  COLLEGE (AUTONOMOUS), CHENNAI-600 034.

M.Sc. DEGREE  EXAMINATION  – STATISTICS

FIRST SEMESTER  – APRIL 2003

ST  1802/ S  717   SAMPLING  THEORY

08.04.2003

1.00 – 4.00                                                                                             Max: 100 Marks

SECTION – A                                (10 ´ 2 = 20 Marks)

Answer ALL the questions.  Each carries two marks

  1. Let the sampling design be

 

               If N=3 then what is the value of p68?

 

  1. Given a fixed size sampling design yielding sample of size 5, what is the value

of ?

  1. Under what condition the mean square error of an estimator becomes its variance?
  2. List all possible balanced systematic samples of size 4 when N = 12.
  3. Under usual notations order VSRS, VSYS,VSTR assuming the presence of linear trend.
  4. Name any two methods of PPS selection.
  5. When N=16 and n = 4, what will be your choice for random group sizes in random group method? Give reason.
  6. Define ratio estimator for the population total.
  7. Name any two randomised response techniques.
  8. Explain the term: Optimum allocation.

 

SECTION  – B                                              (5 ´ 8 = 40 Marks)

Answer any FIVE.  Each carries eight marks.

  1. Show that under SRS,

 

where

  1. Explain any one method of PPS selection in detail with a supportive example.
  2. Show that under balanced systematic sampling, the expansion estimator coincides with the population total in the presence of linear trend.
  3. Derive the mean square error of and obtain the condition under which is more

efficient than .

 

 

 

 

 

  1. Explain the usefulness of two phase sampling in pps sampling.
  2. Describe in detail any one method of Randomised Response technique.
  3. Derive under Neyman allocation.
  4. Verify the following relations with an example

 

(Proof should not be given)

 

Section – C                                    (2 ´ 20 =20 Marks)

Answer any TWO questions.  Each carries twenty marks.

 

  1. Describe random group method. Suggest an unbiased estimator for the

population total and derive its variance.

  1. Derive the first and second order inclusion probabilities in Midzeno sampling and

show that the Yates-Grundy estimator is nonnegative

  1. Develop Yates-corrected estimator.
  2. (a)  Describe how double sampling is employed in ratio estimation              (10)

 

(b)  Write a descriptive note on two stage sampling.                                     (10)

 

 

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Loyola College M.Sc. Statistics April 2003 Programming Languages Question Paper PDF Download

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LOYOLA  COLLEGE (AUTONOMOUS), CHENNAI-600 034.

M.Sc. DEGREE EXAMINATION – STATISTICS

second SEMESTER – APRIL 2003

ST   2803/  S  818   programming languages

28.04.2003

1.00 – 4.00                                                                                          Max: 100 Marks

 

PART – A                               (10 ´ 2 = 20 Marks)

Answer ALL the questions

  1. What is the output of the following program?

#  include < iostream. h>

void main (  ).

{  int x, y, z;

x =10;

y = x++;

z = ++y;

cont <<y <<z; }

  1. Explain the switch structure.
  2. Write a program to get the output as

 

1      2        3       4        5

 

2     3       4        5

 

3      4       5

 

  • 5

 

5

  1. Write a recursive program to get the factorial of a number.
  2. What is prototyping?
  3. What is encapsulation?
  4. Explain destructor.
  5. What is a pointer variable?
  6. What is the use of the dot operator and scope resolution operator while using

a class?

  1. Explain strcpy ( ), strcat ( )

 

PART – B                    (5 ´ 8 = 40 Marks)

 Answer any FIVE questions.

  1. Explain constructors with examples.
  2. Explain structures with examples.
  3. Write a program to find the largest element in a matrix.

 

 

 

 

  1. Illustrate how runtime polymorphism is achieved with the help of virtual functions.
  2. Explain the following
  3. i) ios : : in,    ii) ios :  : out       iii) ios :  : ate   iv) ios :  : no create
  4. v) ios: : bin vi) ios : : no replace.                                                              (6 ´ 1 ½)
  5. Illustrate the use of default arguments.
  6. Illustrate with an example the use of friend functions.
  7. Explain with an example passing objects as arguments.

PART – C                                       (2 ´ 20 = 40 Marks)

Answer any TWO questions

 

  1. a) Using Trapezoidal rule, write a program to evaluate

 

  1. Write a program to find without using a separate function for factorials

Sin x  =                                    (10 +10)

with an error  .0001 between the consecutive terms.

 

  1. Explain the concept of function overloading and operator overloading

with examples.                                                                                        (5 +15)

 

  1. Explain the concept of simple inheritance, multiple inheritance and multilevel

inheritance with examples.                                                                    (4+8+8)

 

  1. Write an interactive menu driven program that will create a data file containing

the  telephone numbers and names of the customers and implement the

following tasks.

  1. Determine the telephone number of a given person.
  2. Determine the name if a telephone number is known.
  • Update the telephone number whenever there is a change. (20)

 

 

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Loyola College M.Sc. Statistics April 2003 Mathematical Statistics II Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.

M.Sc. DEGREE EXAMINATION – STATISTICS

FourTH SEMESTER – APRIL 2003

St 4953 / s 1072  –  mathematical statistics -ii

 

26.04.2003

1.00 – 4.00                                                                                                      Max : 100 Marks

 

                                                                section – A                                (10´ 2=20 marks)

      Answer ALL the questions. Each carries two marks.

 

  1. Define empirical distribution function.
  2. For the sample central moment compute E (b2).
  3. Let X1, X2, …, Xn be a sample from a Compute the .
  4. Define t and F statistics.
  5. Let X1, X2, X3, X4 be iid random variables with pdf f. Find the marginal pdf of the order statistic X(2) .
  6. Let X1, X2, … be iid N random variables. Show that is an unbiased and consistent estimator of  .
  7. Let X ~ N (0,q). Check if the family of pdf’s is complete.
  8. State Neyman-Pearson fundamental lemma.
  9. Define a likelihood ratio test.
  10. Define a process with independent increments.

 

 

 

                                                         section – B                                         (5´ 8=40 marks)

      Answer any FIVE questions. Each carries EIGHT marks.

 

  1. Let X(1), X(2), X(3) be the order statistics of iid random variables X1, X2, X3 with common pdf

Let Y1 = X(3) – X(2) and Y2 = X(2) . Show that Y1 and Y2 are independent.

  1. State and prove factorization criterion for determining sufficient statistics.
  2. State and prove Rao-Blackwell theorem.
  3. Let X1, X2, …,Xn be iid random variables, where both n and p are unknown. Find the estimates of p and n by the method of moments.
  4. Let X1, X2, …,Xn be a sample from U [q -, q + ]. Show that the statistic T(X1,X2,…,Xn) such that max Xi – T (X1, X2, …, Xn) min Xi +

 

is an MLE of q .

 

  1. Let X1, X2,…,Xn ~ U[0, q], q>0. Show that the family of uniform densities on [0,q] has
    an MLR in max

 

  1. (a)  Explain Chi-square test of goodness of fit.
  • Explain normal test for single proportion.
  1. (a) Classify the stochastic processes with respect to time and state space.
  • State the characteristics of the Brownian motion process.

 

 

 

                                                        section – C                                         (2´20=40 marks)

Answer any TWO questions. Each carries TWENTY marks.

 

  1. (a) Derive the distribution of   in sampling from a normal population.
  • Derive the pdf of Chi-square distribution.
  1. (a) State and prove Cramer-Rao inequality.
  • Let X ~ P (). Find the UMVUE of based on a sample of size one.
  1. (a)  Let X ~ N . Obtain a 100 confidence interval for based on a
    random sample of size n.
  • Let X ~ N (0, 1) under H0 and X ~ C (1, 0) under H1. Find an MP size  test of H0 against H1, and obtain the power of the test.
  • Explain F – test for the equality of population variances.
  1. (a) Establish Chapman – Kolmogorov equation and hence show that the   is
    the power of 1-step tpm.
  • State the postulates for Poisson process.
  • Show that if {N (t)} is a Poisson process, then for , the conditional distribution of N (s) given N (t) = n is binomial .

 

 

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Loyola College M.Sc. Statistics April 2003 Industrial Statistics Question Paper PDF Download

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LOYOLA  COLLEGE (AUTONOMOUS), CHENNAI-600 034.

M.Sc. DEGREE  EXAMINATION  – STATISTICS

FourTh SEMESTER  – APRIL 2003

ST  4801/ S  1016   indusTrial STATISTICS

16.04.2003

1.00 – 4.00                                                                                             Max: 100 Marks

PART – A                                       (10 ´ 2 = 20 Marks)

Answer ALL the questions.

 

  1. Discuss the logic and statistical basis underlying the general use of 3s limits on control charts.
  2. Mention the theoretical basis of p-chart and setup its control limits.
  3. How is lack of control of a process determined using control chart techniques?
  4. What is process capability ratio (PCR)?
  5. Briefly explain the process -capability analysis.
  6. What is Average run length (ARL)?
  7. Explain an attribute single sampling plan.
  8. What purpose does an OC curve serve?
  9. Define AOQ and give an expression for AOQ for a single sampling plan.
  10. Explain six-sigma quality.

 

PART – B                                          (5 ´ 8 = 40 Marks)

Answer any FIVE questions.

 

  1. Define the terms : i) rational subgroups  ii) Specification limits

iii) natural tolerances iv) chance and assignable causes.

  1. For the following case, state:
  2. Which control charts(s) would be appropriate to use for ongoing

statistical process control;

  1. Why you suggest the chart(s) given in a); ?
  2. What assumptions are you making in suggesting the charts given in a)?

 

Case: A factory manufactures power sockets for use in the refining oil.  These

sockets look like those use by automobile mechanics, except that they are much

larger than and the surface in covered with block oxide instead of stainless steel

to indicate that they are specially designed for power tools.   The hexagonal

inside of  the socket is  prepared by an electrolytic method, and the resulting

diameter of the hexagon is very critical.

Rather than measure every possible diagonal in the socket, one diagonal is

selected at random. from each socket and recorded.  A sample of five

consecutively  produced sockets is taken each day, and the measures are made.

The process will be to collect data for a month, which consists of 22 observations,

and  develop trial  control limits.

 

 

 

 

 

 

 

 

 

 

 

  1. A control chart indicates that the current process fraction non-conforming is

0.02. If 50 items are inspected each day, what is the probability of detecting a

shift  in the fraction  non-conforming to 0.04 on the first day after the shift?

By the end of the third day following the shift?

  1. Write a detailed note on the moving average control chart.
  2. Consider a modified control chart with center line at m= 0 and s =1.0. If n = 5, the tolerable fraction non-conforming is s = 0.00135 and the control limits

are at 3, sketch the OC curve for the chart.

  1. Viscosity measurements on polymer are made every 10 minutes by an on-line viscometer. Eighteen observation are shown below.  The target viscosity for this process is mo = 3200.

 

3169 3173 3162 3154 3139 3145 3160 3172 3175
3205 3203 3209 3208 3211 3214 3215 3209 3203

 

Construct a tabular cusum for this process using standardized values

of h = 8.01  and k = 0.25.

  1. For the sampling plan N =120, n1= n2 =13, c1 = 0 and c2 =1, obtain Pa, ASN

(full inspection), AOQ (defective found replaced) and ATI when the submitted

lot has the fraction non-conforming at p = 0.18

  1. What are acceptance and rejection lines of a sequential sampling plan for attributes? How are the OC and ASN values obtained for this plan?

 

PART – C                                       (2 ´ 20 = 40 Marks)

Answer any TWO questions.

 

  1. a) Distinguish between c and u charts. Explain the situations where c and u

charts are applicable and how are the limits obtained for these charts.

  1. b) Explain the procedure of obtaining the OC curve for a p-chart

with an illustration.                                                                                (10+10)

  1. a) What are modified control charts? Explain the method of obtaining control

limits for modified control charts.

  1. b) Write a detailed note on control charts based  on extreme values.          (12+8)

 

  1. a) What purpose does a cumulative sum chart serve?
  2. b) Outline the procedure of constructing V-                                      (5+15)

 

  1. a) Explain with an illustration the methods of obtaining the probability of

acceptance   for a triple sampling plan.

  1. What are continuous sampling plans and mention a few situations where these

plans are applied.                                                                                   (10 +10)

 

 

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Loyola College M.Sc. Statistics April 2003 Computational Statistics  II Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.

M.Sc. DEGREE EXAMINATION – sTATISTICS

SECOND SEMESTER – APRIL 2003

ST 2804 / s 819  –  computational statistics – ii

 

30.04.2003

1.00 – 4.00                                                                                                     Max : 100 Marks

 

Answer any three questions.

 

  1. a) In a genetical experiment the frequencies observed in 4 different classes are 1997,
    906, 904,32. Theory predicts that these should be in the proportion
    and   respectively. Find the maximum likelihood estimate of the parameter q and
    also obtain the estimate of its variance.

 

  1. b) The following table gives the number of flower heads each having exactly ‘x’ gall cells.

 

No.of gall cells in a flower head 1 2 3 4 5 6 7 8 9 10
No.of flower

heads

 287 272 196 79 29 20 2 0 1 0

Estimate the parameter ‘q’ by the method of maximum likelihood and also obtain its
asymptotic variance by assuming the frequency distribution of gall cells to be of truncated
Poisson type.                                                                                                               (14+20)

 

  1. a) A survey of 200 families with four children each revealed the following data :
No.of boys 0 1 2 3 4
No.of families 8 48 76 54 14

Assume that X ~ B (4, p) where ‘p’ is the probability of a child being a boy.

  • Obtain the MLE of

(ii) Obtain the MLE of the probability that a randomly selected family has atleast
3 boys.

 

  1. b) The following data relates to the results of a genetical experiment:
Gene type Relative frequencies Observed frequencies
I 508
II 432
III 397
IV 518

 

Estimate the parameter q by the method of modified minimum Chi-Square.    (17+17)

 

  1. a) Let X be the concentration in parts per billion of chromium in the blood of healthy
    person and Y be the same measurement done on a person with some disease.  8
    healthy persons and 10 persons with disease were taken up for studying and the
    following observations were obtained.
X 15 23 12 18 9 28 11 10
Y 25 20 35 15 40 16 10 22 18 33

It is believed that X and Y have normal distributions with variance s x2 and s y2 respectively.  Obtain 98% Confidence interval for the ratio.

 

  1. Obtain a 95% confidence interval for the parameter q of Poisson distribution based on the following data:                                                                                           (17+17)
No.of blood corpuscles 0 1 2 3 4 5
No.of Cells 142 156 96 27 5 1

 

  1. a) Let X1, X2, …,X5 be a random sample of SAT mathematical scores assumed to be
    N (m1,s2) and let Y1, Y2, …,Y8 be an independent random sample of SAT verbal
    scores assumed to be N(m2,s2). If the following data are observed find 90% confidence
    interval for m1 – m2 .

X1 = 644;         X2 = 493;         X3 = 532;         X4 = 462;         X5 = 565.

Y1 = 623;         Y2 = 472;         Y3 = 492;         Y4  = 661;        Y5 = 540;         Y6 = 502;         Y7 = 549;         Y8 = 518.

 

  1. b) Test for randomness of the following 14 observations at 5% level:
    4,     76.3,    85.6,    76.4,    88.4,    80.2,    85.6,    84.6,    78.3,    82.8,    88.1,    85.4,    87.7,    86.6.                                                                                        (20+14)

 

  1. a) Let Y ~ B (200, p). To test H0: p = 0.75 against H1: p > 0.75. we observe Y and reject
    H0 if Y150. Use the normal approximation to compute the level and power function
    of the test for values of  p starting from 0.75 at intervals of 0.02 up to 0.85.

 

  1. b) Let X1, X2, …,Xn be a random sample from a normal distribution with mean ‘m’ and
    variance 64. Show that C = {(x1, x2, …, xn):}  is a best critical region for testing
    H0: m = 80 against H1: m = 76. Find ‘n’ and ‘c’ so that  = 0.05 and b = 0.05
    (17+17)

 

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Loyola College M.Sc. Statistics April 2003 Advanced Operations Research Question Paper PDF Download

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LOYOLA  COLLEGE (AUTONOMOUS), CHENNAI-600 034.

M.Sc. DEGREE  EXAMINATION  – STATISTICS

FourTh SEMESTER  – APRIL 2003

ST  4951/ S  1052   advanced operations  reSEaRCH

26.04.2003

1.00 – 4.00                                                                                             Max: 100 Marks

 

 

SECTION – A                      (10 ´ 2 = 20 Marks)

Answer ALL the questions.

  1. State the Bellman’s principle of optimality.
  2. Define a general non-linear programming problem.
  3. What is posynomial and where it is used?
  4. Define the mathematical formulation of an quadratic programming problem.
  5. Define separable programming problem with an example.
  6. Convert the chance constraint into equivalent deterministic constraint for the following problem.

Min  Z = 3x1+ 4x2

St   Pr (3x1-2x2 £ b1) ³

 

x1, x2 ³ 0, where b1, b2, b3 are independent random variables uniformly distributed

in the intervals (-2,2), (0,2) and (0,4) respectively.

  1. Explain the concept of integer programming problem.
  2. Explain the mathematical model of a stochastic linear program.
  3. Explain a scenario where the goal programming concepts are applied.
  4. State the Kuhn-Tucker conditions to solve an NLPP program.

 

SECTION – B                                   (5 ´ 8 = 40 Marks)

Answer any FIVE questions

  1. Explain Wolfe’s Algorithm in solving a non linear programming problem.
  2. Solve the following geometric programming problem

 

  1. Explain clearly the piece wise linear approximation.
  2. Solve the following Integer programming problem

Max Z = x1+ x2

s.t     3x1+ 2x2 £ 5

x2 £ 2

x1, x2 ³ 0 and integers.

  1. Derive the Gomary’s constraint in solving an integer programming problem
  2. Solve the non linear programming problem using Lagrangian multipliers.

Maximize

s.t       x1+ x2 + x3 =15

2x1 -x2 +2x3 = 20

x1,  x2, x3 ³ 0

 

 

 

  1. Explain the concepts in solving the stochastic programming problem for

the following scenario  assuming the usual notations

(i) Aircraft Allocation problem    (ii) Two stage programming

  1. Solve the following cargo loading problem assuming the usual notations when

there is no volume restriction  (i.e., Q = ¥) , W = 5 and N = 3 with the

numerical  data given below

 

item (n) Weight (Wn) Value (Vn)
1 2 7
2 3 10
3 1 3

 

SECTION – C                                  (2 ´ 20 = 40 Marks)

Answer any TWO questions.

  1. Explain Beale’s Algorithm in solving a non linear programming problem.
  2. Solve the following non linear programming problem using Beale’s Algorithm

 

s.t   x1 +2x2 £ 10

x1 + x2 £ 9

x1 ,x2 ³ 0

  1. a) Explain Branch and Bound Technique in solving an Integer programming

problem

  1. Derive the geometric -arithmetic mean inequality in solving a geometric

programming problem

21  a)   Explain the dynamic programming problem concepts in solving the Cargo

loading problem, assuming the usual notations.

  1. Solve the following linear programming problem through Dynamic programming problem

Max  Z = 3x1 + 4x2

s.t         2x1 + x2 £ 40

2x1 + 5x2 £ 180

x1, x2 ³ 0

  1. a) Approximate the following NLPP to LPP using separable convex

programming and piece wise linear combination concepts

 

Max f(x) = 3x1 + 2x2

                                                               

  1. Explain the scenario of n component system in series in Reliability and

provide a solution to solve using a Dynamic programming problem.

 

 

 

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Loyola College M.Sc. Statistics Nov 2003 Stochastic Processes Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2003

ST-3800/S915 – STOCHASTIC PROCESSES

03.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

Answer ALL the questions.                                                                              (10×2=20 marks)

 

  1. Define a stochastic process clearly explaining the time and state space.
  2. Examine if a sequence of independent random variables possesses independent increment property.
  3. Define a Markov chain. Give an example.
  4. Let the transition probability matrix of a Markov chain with the state space S= {0,1,2,3} be P = . Find the periodicities of the states.
  5. Define i) recurrence and ii) mean recurrence time of state i.
  6. Describe a Poisson process.
  7. Define current life and excess life associated with a renewal process.
  8. Find the distribution of excess life if N(t) ~ P (l t).
  9. If {Xn} is martingale with respect to {Yn} , Show that E [Xn+k½Y0 Y1­­ ….Yn] = Xn for all k .
  10. Define Branching Process.

 

SECTION-B

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

 

  1. State and establish Chapman – Kolmogorov equations satisfied by a discrete time Marhov – chain.
  2. Show that d(i) = g cd {n ³ 1 | f> 0}.
  3. Describe Yule process. Show that the marginal distirbution of the process is negative binomial with p = e–bt if the initial size is greater than 1.
  4. Describe a Birth – Death process and derive kolmogorov forward differential equation.
  5. Show that the renewal function corresponding to the life time density

 

 

l2 x e–lx  ,   x > 0

f (x) =

0              ,    elsewhere

 

is given by      M (t) =

  1. Define a renewal process. Show that renewal function M(t) satisfies the renewal equation.
  2. Let Y0 = 0, and Y1, Y2….be iid random variables with E (Yk) = 0 and

E (Y) = s2, k = 1, 2, ….

Let X0 = 0 and Xn =  .

Show that {Xn}  is a martingale w.r.t {Yn}.

  1. Suppose the probability generating function of the off-spring distribution for a Branching process is f (s) = 0.1 + 0.4 s + 0.5 S2. Obtain the extinction probability.

 

SECTION-C

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

 

  1. i) Show that if i j, then d(i) = d(j). (8)
  2. ii) Show that state i is recurrent if and only if . (12)
  3. i) Prove that the three dimensional symmetric random walk on the set of integers is a transient Markov chain.          (15)
  4. ii) Let {Xn, n ³ 0} be an irreducible FMC with doubly stochastic tpm. Show that the stationary probabilities are equal. (5)
  5. i) Show that the stationary distribution for a single server queueing model is geometric, and also show that the distribution of waiting time is an exponential.                         (7+8)
  6. ii) Derive Kolnogorov forward differential equations for Telephone trunking model. (5)
  7. i) Let the renewal counting process be Poisson. Find the joint distribution of  and .  Deduce that the two random variables are independent.                                                        (10)
  8. ii) State and prove elementary renewal theorem. (10)

 

 

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Loyola College M.Sc. Statistics Nov 2003 Sampling Theory Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – NOVEMBER 2003

ST-1802/S717 – SAMPLING THEORY

08.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

 

SECTION-A

 

Answer ALL  the questions.                                                                             (10×2=20 marks)

 

  1. Explain probability sampling design.
  2. Given N = 5, n = 3, X2 = 2, X3 = 3, X = 25. Compute p23 under Midzuno Sampling design.
  3. Distinguish between inclusion probabilities and inclusion indicators.
  4. List all possible Balanced Systematic Samples when N = 30 and n = 6.
  5. Define Des-Raj ordered estimator.
  6. Define Horvitz-Thompson estimator.
  7. Write a short note on Yates corrected estimator under Linear systematic sampling.
  8. Describe two phase sampling.
  9. When is stratified sampling used?
  10. Define proportional allocation.

 

SECTION-B

 

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

 

  1. Derive variance of Horvitz-Thompson estimator in Yates-Grundy form.
  2. Explain Lahiri’s method and show that Lahiri’s method of selection is a probability proportional to size selection method.
  3. Write a note on Warner’s model.
  4. Explain ratio estimator, also derive the approximate bias and mean square error of the estimator.
  5. Compare Linear systematic sampling and simple random sampling in the presence of a linear trend.
  6. Develop Hartly-Ross unbiased ratio type estimator.
  7. Derive variances and covariances in the two cases of two phase sampling, assuming simple random sampling is used in both the phases of sampling.
  8. Describe Two-stage sampling. Give the unbiased estimator and also derive the variance of the unbiased estimator.

 

 

 

 

 

 

 

SECTION-C

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

 

  1. a) Derive V() for n = 2.            (15)
  2. b) For any fixed size sampling design yielding samples of size n, prove that

(5)

  1. a) Describe Midzuno sampling design and show that it is a sampling design. (5)
  2. b) Derive the first and second order inclusion probabilities under Midzuno sampling

design.                                                                                                                          (15)

  1. a) Develop Yates corrected estimator under linear systematic sampling. (10)
  2. b) Suppose from a sample of n units selected with simple random sampling (SRS) a

subsample of n’ units is selected with SRS duplicated and added to the original

sample. Derive the expected value and the approximate sampling variance of , the

sample mean based on the n+n’ units.                                                                         (10)

  1. a) Write a note on proportional allocation for a given cost. Also deduct V  under it

assuming SRS is used in all strata.                                                                             (10)

  1. b) A sampler has two strata with relative sizes and . He believes that

S1, S2 can be taken as equal.  For a given cost C = C1 n1 + C2 n2,  show that (assuming

Nh is large).

(10)

 

 

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Loyola College M.Sc. Statistics Nov 2003 Operation Research Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2003

ST-3802/S917 – OPERATION RESEARCH

07.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

Answer ALL questions.                                                                                   (10×2=20marks)

 

  1. Define ‘CONVEX HULL OF A SET’.
  2. What is meant by ‘SURPLUS VARIABLE’?
  3. Distinguish between ‘PRIMAL’ and ‘DUAL’ problem.
  4. When do we go for Big-M Method?
  5. State the MAX FLOW – MIN CUT theorem.
  6. What are the disadvantages of under stockings?
  7. Give any two applications of NETWORK?
  8. What is meant by ‘MINIMALSPANNING TREE’?
  9. Briefly explain the term ‘MIXED STRATEGY’.
  10. Define i) setup cost ii) Holding cost

 

SECTION-B

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

 

  1. a) What is meant by a CONVEX SET?
  2. b) Show that the set S = is a convex set.
  3. Explain VOGEL’S method to determine the Initial Assignments for a Transportation problem.
  4. Construct a 2×3 game and hence solve using graphical method.
  5. a) What is meant by BUFFER STOCK?
  6. b) Explain how to derive an expression for Buffer Stock.
  7. Solve the steady – state equation for M/M/1 queueing model.
  8. Distinguish between ‘TOTAL FLOAT’ and ‘FREE FLOAT’ with suitable illustration.
  9. When do we go for ABC – INVENTORY SYSTEM and VED – ANALYSIS? Explain briefly with neat diagram.
  10. Distinguish between ‘FORWARD PASS’ and ‘BACKWARD PASS’ with suitable illustration.

 

SECTION-C

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

 

  1. a) Explain the BALANCED and UNBALANCED Transportation problem with an

example.

  1. Define the term DEGENERACY in transportation problem and hence construct a 3×3 degenerate transportation problem and find the optimal solution.        (6+14)
  2. a) Discuss in detail the industrial applications of QUEUENING THEORY.
  3. b) Derive the expressions Pn, Lq, Ls, Wq and Ws for (M/M/C) : (GD/¥/¥).        (8+12)
  4. a) Distinguish between ‘DETERMINISTIC’ and ‘PROBABILISTIC’ models with suitable

illustrations.

  1. b) Explain in detail the SINGLE – ITEM static model with price breaks, and derive the

optimum order quantity.                                                                                        (10+10)

  1. Write short notes on the following:
  2. TWO-PHASE METHOD
  3. ASSIGNMENT MODEL
  4. DOMINANCE PROPERTY
  5. PERT              (4X5=20)

 

 

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Loyola College M.Sc. Statistics Nov 2003 Computational Statistics – III Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2003

ST-3803/S918 – COMPUTATIONAL STATISTICS – III

10.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

Answer any THREE  questions.                                                                     

 

  1. a) Solve the following L.P.P using SIMPLEX METHOD

MAXIMIZE         Z = 12 X1 + 15X2 +14 X3

subject to                     X1 + X2 + X3  100

-0.01 X2 + 0.02 X3  0

-0.01 X1 + 0.01 X2  0

X1, X2,      X3   0

  1. b) Solve the following game using graphical method:

 

Player -B

B1    B2

Player-A

(18+15.5)

  1. a) Construct a network based on the following data:

ACTIVITY:                       A         B         C         D         E          F          G

t­o      :                              3          2          2          2         1         4          1

tm    :                      6          5          4          3         3         6          5

tP      :                             9          8          6          10        11        8         15

PREDECESSOR:               –           –           A         B         B        C,D       E

 

 

Calculate

  • The expected time and SD for each activity
  • The CRITICAL PATH
  • The probability that the project will be completed by 18 weeks.
  1. b) Consider the inventory problem with 3 items (n = 3), the parameters of the problem are

shown below

Item                 KI                    bI                            hI                     a

1                      $10                  2 units             $ 0.3                1 sq.ft

2                      $5                    4 units             $ 0.1                1 sq.ft

3                      $15                  4 units             $ 0.2                1 sq.ft

 

Assume that the total available storage space area A= 25 Sq.feet.   Determine the

optimal order quantity for the three items.

 

(20+13.5)

 

 

 

  1. a) The following correlation were obtained among the responses auditory reaction times,

audiometric hearing loss,  WAIS comprehension and WAIS digital symbol for a

sample of   N = 47 males

 

  • determine the partial correlation of reaction time and hearing loss with the two WAIS subset scores held constant. Test the hypothesis of zero partial correlation at 5% level.
  • Compute the multiple correlation of reaction time with the other three variates. Test the hypothesis of independence of the first response and the last three.

(20)

  1. b) Let X ~ N  with the mean vector .  What is the

conditional distribution of X2  (X1 = 8, X3 = 5).                                                         (13.5)

 

1     2    3    4

  1. a) Consider a Markov chain with TPM
  • Examine whether the Markov chain is irreducible.
  • Also check whether the state 4 is ergodic
  • Find the stationary distribution (3.5+4+6)
  1. b) Let A = B =   =    and

to be distributed independently according to trivariate normal population with

respective parameters.

 

 

 

What are the distribution of the following linear transformation of those variates.

 

(i) AX              (ii)        BX                  (iii) (X’ A   Y’ B’)        (iv) (X’ A’   X’ B’)

 

(4×5=20marks)

 

 

 

 

 

  1. a) Consider a Markov chain {xn, n }with the state space S = {0 , 1, 2, 3, 4, 5} and one

step TPM given by

 

P  =

Find the equivalence classes and the periodicity of the states.

 

  1. b) For a Morkov chain with one step TPM 0        1       2

 

Find if the states are transient or recurrent.

 

 

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Loyola College M.Sc. Statistics Nov 2003 Econometrics Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2003

ST-3950/S919 – ECONOMETRICS

12.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

Answer ALL questions.                                                                                   (10X2=20 marks)

 

  1. What is meant by a generalized least square estimator?
  2. Explain auto regressive process.
  3. What are lagged variables?
  4. What are the properties of OLS estimators of a linear model Y =u?
  5. What is multi collinearity?
  6. Explain specification error.
  7. What is the auto correlation?
  8. What are the reasons for auto correlation disturbances?
  9. What are the sources of non spherical disturbances?
  10. Explain the homoscedasticity property.

 

SECTION-B

Answer any FIVE questions.                                                                           (5X8=40 marks)

 

  1. Consider the linear Y =u with E (u) = 0 and E(uu!) = Prove that an unbiased estimate of  is given by  where r is the residual vector.
  2. Derive the MLE of the parameters of the linear regression model Y =
  3. Derive the variance – covariance matrix of the autocorrelated disturbance terms?
  4. Explain in detail the concept of multi cotnearity.
  5. Explain the effect of excluding the relevant variables is the linear model Y =
  6. Explain clearly the concept of hetroscedasticity property.
  7. Explain the concept of structural change.
  8. Write short notes on (i) dummy variables (ii) seasonal adjustment.

 

SECTION-C

Answer any TWO questions.                                                                           (2X20=40 marks)

 

  1. For the general linear model Y =u, derive the least square estimator and

find Var  .

  1. a) State and prove gauses Markov theorem.
  2. b) Derive the test procedures to test the linear hypothesis H:Rb = S for the general linear

model.

  1. a) What are the properties of OLS estimators under Non spherical disturbances?
  2. b) Explain the Drubin – Watson test to test for auto correlation. (8+12)
  3. a) Explain cochrane – orcutt iterative estimation procedure used in the presence of

autocorrelated disturbance.

  1. b) Describe the ALMON Lag model to estimate the parameters of a distributed Lag

model.                                                                                                              (10+10)

 

 

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Loyola College M.Sc. Statistics Nov 2003 Multivariate Analysis Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2003

ST-3801/S916 – MULTIVARIATE ANALYSIS

05.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

Answer ALL the questions.                                                                              (10×2=20 marks)

 

  1. Let X ~ N . Obtain the conditional distribution of X1 given X2 = x2.
  2. Write the characteristic function of bivariate normal distribution.
  3. Explain how the collinearity problem can be solved in the multiple regression

Y = Xb + Î.

  1. If X and Y are two independent standard normal variables, obtain the distribution of two times of the mean of these two variables.
  2. Let X be trinormal with

m =  and   compute  .

  1. Define Fisher’s Z – transformation.
  2. Explain classification problem into two classes.
  3. Write down any four similarity measures used in cluster analysis.
  4. Distinguish between principal component and factor analysis.
  5. What is meant by residual plot?

 

SECTION-B

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

 

  1. Define multiple correlation coefficient between X1 and X2, …., Xp. Show that the multiple correlation coefficient between X1 and X2, …., Xp has the expression

.

  1. Let Y ~ Np ( 0, å) . Show that Y1 has – distribution.
  2. Test at level 0.05 whether m = in a bivariate normal population with s11 = s22 = 5 and s12 = -2 ,   by using the sample mean vector  based on a sample of size 10.
  3. How will you test the equality of covariance matrices of two multivariate normal distributions on the basis of independent samples drawn from two populations?
  4. Derive the characteristic function of Wishart distribution.
  5. In Principal component analysis derive the first principal component.
  6. Obtain the rule to assign an observation of unknown origin to one of two p-variate normal populations having the same dispersion matrix.
  7. Outline single linkage and complete linkage clustering procedures with an example.

 

SECTION-C

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

 

  1. a) Derive the MLE of å when the sample is from Np (m, å).
  2. b) Define Hottelling’s T2 – statistic.
  3. c) Using the likelihood ratio test procedure, show that the rejection region for testing m =

mo against m ¹ mo is given by

T2 = n( S-1 (³  T.                                                             (10+3+7)

  1. a) Prove that under some assumptions (to be stated), Variance- covariance matrix can be

written as å = LL1 +  in the factor analysis model.  Also discuss the effect of an

orthogonal transformation.

  1. b) Let X1,X2,…., Xp have covariance matrix å with eigen value vector pairs (l1, e1),…,

(lp, ep),  l1 l2 ….. ≥ lp 0,   then prove that

s11 + s22 + …..+ spp = ,

Where ,  Yi  represents the i – th principal component.

  1. c) Explain the principal component (principal factor) method of estimating L in the factor

analysis model.                                                                                                  (10+5+5)

  1. a) Explain the method of extracting canonical correlations and canonical variables. Also

explain how the theory of canonical correlation is helpful in the analysis of multivariate

data.

  1. b) State an establish the additive property of Wishart distribution. (10+10)
  2. Write shot notes on:-
  3. Roy’s Union – Intersection principle
  4. Step – Wise regression
  5. Mohalanobis Squared distance. (5+10+5)

 

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Loyola College M.Sc. Statistics Nov 2003 Measure Theory Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – NOVEMBER 2003

ST-1801/S716 – MEASURE THEORY

06.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

Answer ALL the questions.                                                                              (10×2=20 marks)

 

  1. For a sequence {An} of sets, if An A, show that .
  2. Define a monotone increasing sequence of sets and give its limt.
  3. Show that a s – field is a monotone class.
  4. Define the indicator function of a set A.
  5. Show that the set rational numbers is a Borel set.
  6. If X is a simple function, show that is a simple function.
  7. If X1 and X2 are measurable functions with respect to prove that max { X1, X2} is measurable w.r.t     .
  8. If = {1,2,3,4},      is the power set of , μ {f} = 0, μ {1} = , μ {1,2} = ,

μ {1,2,3} =  μ (W) = 1, is μ a measure on (W,     )?

  1. If μ is a measure, show that μ ≤ .
  2. If = [0,1] and μ is the Lebesgue measure, write down the value of , where C is the set of rationals, A = [0, 3/4] and B = [1/2, 1].

 

SECTION-B

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

 

  1. Prove that there exists a unique and minimal s-field on a given non – empty class of sets.
  2. Define Borel s – field of subsets of real line. Show that the minimal s – field generated by the class of all open intervals is a Borel s – field.          (2+6)
  3. a) Define a finitely additive and a countably additive set functions.
  4. b) Let W = {-3, -1, 0, 1, 3} and for A W, let l (A) = with l1 = min (l, O), show that l is not even finitely additive.
  5. If l is an extended real valued s – additive set function on a ring  such that l(A) > – for every A Î Â, show that l is continuous at every set A Î Â.
  6. If X1 and X2 are measurable functions w.r.t show that (X1 + X2) is also measurable w.r.t.    prove that lim inf Xn is measurable w.r.t      .
  7. Define the Lebesgue – Stieltjes (LS) measure induced by a distribution function F on IR. If μ is the LS measure induced by

F(x) =   1 – e-x     if x > 0

 

  • if x ≤ 0,

then find (a) μ (0, 2) (b) μ [-1, +1] and (c) μ (A), where A = {0, 1, 2, 3, 4}.                 (2+6)

  1. Show that a measure on a s – field can be extended to a complete measure.
  2. State and establish Fatou’s lemma.

 

SECTION-C

 

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

 

  1. a) Distinguish between (i) a ring and a field (ii) a ring and a s – ring.
  2. b) Define the minimal s-field containing a given class of sets. Give an example.
  3. c) Show that the inverse image of a s-field is a s-field.
  4. a) Define (i) extension of a measure (ii) completion of a measure. (6)
  5. b) State and prove the Caratheodory extension theorem. (14)
  6. a) Prove that if 0 ≤ Xn X, then . (8)
  7. b) If X and Y are measurable functions on a measure space, show that

.                                                                                (12)

  1. a) If X ≥ 0 is an integrable function, prove that j (A) = A a measurable set,

defines a measure, which is absolutely continuous with respect to the measure m.  (10)

  1. b) State and prove the Lebesgue “dominated” convergence theorem. Is the

“denominated” condition necessary?  Justify your answer.                                     (10)

 

 

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Loyola College M.Sc. Statistics Nov 2003 Computational Statistics – I Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – NOVEMBER 2003

ST-1804/S719 – COMPUTATIONAL STATISTICS – I

13.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

Answer any THREE questions. 

 All questions carry equal marks                                                        

 

  1. a) Compare the exact mean square error of ratio estimator with the variance of Hartely-

Ross unbiased estimator using the following population data assuming simple random

sampling is done with sample size 3.

Booth No. Votes polled in 1991 election Votes polled in 1996 election
1

2

3

4

5

 1024

886

950

1251

684

 964

931

1014

895

768
  1. b) Consider a finite population of size 30 where the population units are arranged in

ascending order with respect to the levels,

Label : 1          2          3 …………..30

Y    : 13        21        29………….245

YI = 5+8i,   i = 1, 2, ……. 30.

Draw a linear systematic sample of size 6 and estimate the population total using Yates

corrected estimator and give your comments.                                                     (20+14)

  1. a) To estimate the volume of timer available in forest area consisting of 36 geographical

regions,  a sample of size 6 is drawn using random group method.  The following table

gives the data collected.

S.No. No.of trees Volume of timber

(in cubic units)

 Total no. of trees
1

2

3

4

5

6

 46

73

50

48

36

20

 112.6

143.6

121.2

98.7

76.3

42.3

 243

180

140

70

107

90

Estimate the total volume of timber assuming that there are 2800 trees in the forest

area and also estimate the variance of your estimator.  (Here it is assumed that the

random groups are all of equal size).

  1. b) A sample survey is conducted with the aim of estimating the total yield of paddy, the

area is divided into 5 strata and from each stratum 4 plots are selected by the method of

simple random sampling without replacement (SRSWOR).  Using the data given below

obtain the estimate of total yield along with the variance of the estimator.

Stratum No. Total No. of plots Yield of paddy for 4 plots
1

2

3

4

5

 105

87

76

98

64

 104     182     148       87

108      64       132     156

110      281     120     114

96      102     141    111

112     128      124    118

 

(17+17)

  1. a) Show that (A­-1)T = (A­T)-1 for the following matrix.
  2. b) Find g inverse for the given rectangular matrix.

 

  1. c) Find the rank of the given matrix.

 

(14+10+10)

  1. a) Find BT A-1 B without finding A-1 for given matrices A and B.
  2. b) A population containing 420 birds of same weight and age were taken for a stimulus

study.  The birds were divided into 60 equal groups.  They were then given the stimulus

to increase the growth.  The following data gives the frequency distribution of those

with significance increase in weight at the end of 6 weeks.  Fit a truncated binomial

distribution an test the goodness of fit at 5% level of significance.

No.of weeks:   1          2          3          4          5          6

No. of groups: 5          12        18        15        7          3

(14+20)

  1. a) For the following distribution, fit a negative binomial distribution and test the goodness

of fit at 5% level of significance.

X:        0          1          2          3          4          5

F:         212      128      37        18        3          2

  1. b) Fit a distribution of the type p(x) = where p1 (x) = pqx-1 , x = 1,2,…

0 ≤ p ≤ 1 and  Also test the goodness of fit at 5%

level.

X:        0          1          2          3          4          5          6          7          8

F:         72        110      119      58        28        12        4          2          2

(14+20)

 

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Loyola College M.Sc. Statistics Nov 2003 Analysis Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – NOVEMBER 2003

ST-1800/S715 – ANALYSIS

04.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

Answer ALL questions.                                                                                   (10×2=20marks)

 

  1. Let Z be the set of all integers. Construct a function form Z to Z which is not one to one and also not onto.
  2. Define a metric on a non-empty set x.
  3. The real valued function f on R2 – is defined by f (x, y) = .  Show that

lim f (x,y) does not exist as (x, y)  (0, 0) .

  1. State weirstrass’s approximation theorem.
  2. If is a convergent sequence in a metric space (X, P) then prove that it is a cauchy sequence.
  3. If Un = O (1/nk-2), for what value of k converges?
  4. Define the upper limit and lower limit of a sequence.
  5. Find and also the double limit of xmn as m,n where xmn  = .
  6. Let f: Rm.  Define the linear derivative of f at .
  7. From the infinite series where obtain the expansion for log (1+x).

 

SECTION-B

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

 

  1. Show that the space R’ is complete.
  2. State and prove cauchy’s inequality.
  3. Prove that the union of any collection of open sets is open and the intersection of any collection of closed sets is closed.
  4. a) Show that fn­ (x) = x  is not uniformly convergent
  5. b) Let and be metric spaces. Let the sequence fn : converge to f uniformly on x. If C is a point at which each fn is continuous, then show that f is continuous at C.
  6. Let V, W be normed vector spaces. If the function f : V W  is linear, then show that the following statements are equivalent.
  7. f is continuous on V
  8. there is a point at which f is continuous.
  • is bounded for
  1. Examine for convergence of if
  3. un =
  4. Let (be a metric space and let f1, f2, …..fn be functions on X to R‘.  The function

f = (f1, f2, …..fn)  : is given by f(x) = (f1(x) … fn (x).  Prove that f is continuous at x0 if and only if f1, f2,…..fn  are continuous.

  1. If f : is differentiable at then prove that the linear derivative of f at  is unique.

 

SECTION-C

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

 

  1. a) Let (and be metric spaces.  Prove that the following condition is

necessary and sufficient for the function f :  to be continuous on X:

whenever G is open in Y, then f-1 (G) is open in X.

  1. b) Show that if is a metric on x then so is given by  (x, y) =  and P and

are equivalent.                                                                                                   (12+8)

  1. a) State and prove Banach’s fixed point theorem.
  2. b) State and prove Heine – Borel theorem.                 (10+10)
  3. a) State and prove d’ alembert’s ratio test
  4. b) Discuss the convergence of where
  5. c) Discuss the convergence and absolute convergence of

(8+8+4)

  1. a) Show that a necessary and sufficient condition that fis that, given

there is a dissection D of [a, b] such that S (D, f, g) – s (D, f, g) < .

  1. b) If fI, f2 R [g i a, b] then prove that f1 f2 R [g i a, b]
  2. c) If f R [ g i a, b] then show that      (7+7+6)

 

 

 

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

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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,…, Kk­. 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 £ tn­) ≥

  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

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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)

 

  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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Loyola College M.Sc. Statistics April 2004 Mathematical Statistics – II Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

  FOURTH SEMESTER – APRIL 2004

ST 4953 – MATHEMATICAL STATISTICS – II

12.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

SECTION – A

 

Answer ALL questions                                                                                (10 ´ 2 = 20 marks)

 

  1. Define a consistent estimator and give an example.
  2. Show that unbiased estimators do not always exist.
  3. Let X1, X2, …, Xn be iid b (1,p) random variables.

Show that T =  is sufficient for p.

  1. State Lehmann – Scheffe theorem.
  2. State Bhattacharya Inequality.
  3. Write the test function associated with i) a non-randomized test ii) a randomized test.
  4. Define UMP test for testing a simple hypothesis against a composite hypothesis.
  5. Write any four applications of chi-square distribution.
  6. State the postulates for Poisson process.
  7. Define Markov chain and give an example.

 

SECTION – B

 

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

 

  1. Let X1, X2,…, Xn be a random sample from N (m, s2). Show that  and S2 are independent.
  2. Derive the pdf of F-distribution.
  3. Let X1, X2, …., Xn be a random sample from a distribution of continuous type with pdf f(x; q). Derive the joint pdf of jth and kth order statistics, 1 £ j < k £
  4. State and prove Factorization Criterion for determining sufficient statistics.
  5. State and prove Rao-Blackwell theorem.
  6. Let X1, X2, …, Xn be a random sample from Poisson distribution that has the mean q > 0.  Show that  is an efficient estimator of q.
  7. Let X1, X2,.., Xn be iid N (m, s2) random variables where both m and s2 are unknown. Obtain a MP test for testing Ho­: m = mo­ ; s2 =  against H1: m = m1; s2 = .
  8. Show that if {N(t)} is a Poisson process, then for s< t, the conditional distribution of N(s) given N(t) = n is binomial b (n, ).

 

 

SECTION – C

 

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

 

  1. a) State and Prove Cramer-Rao Inequality.            (10)

 

  1. b) Let X1, X2, …, Xn be iid N(m, s2) random variables. Obtain a confidence interval for m

when (i) s2 is known (ii) s2 is unknown.                                                                    (10)

 

  1. a) State and prove Neyman-Pearson lemma.            (10)

 

  1. b) Let X1, X2, …, Xn be a random sample for N(m, s2) where both m and s2 are unknown.

Derive the likelihood ratio test for testing Ho: m = mo against H1 : m ¹ mo.                  (10)

 

  1. a) Let X1, X2,…, XN be iid b (n, p) random variables, where n and p are unknown. Find

the method of moments estimator for (n,p).                                                                 (7)

 

  1. b) Let X1, X2, …,Xn be a sample for U [q – , q + ]. Show that the maximum

likelihood estimator of q is not unique.                                                                        (7)

 

  1. c) Explain normal test of significance for single mean and give an example. (6)

 

  1. a) Classify the stochastic processes with respect to time and state space. (2)

 

  1. b) State the characteristics of the Brownian motion process (4)

 

  1. c) Establish Chapman – Kolmogorov equaion and hence show that the m – step tpm is the

mth power of 1 – step tpm.                                                                                           (8)

 

  1. d) Explain chi-square test for goodness of fit and give an example. (6)

 

 

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

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – APRIL 2004

ST 2801 – ESTIMATION THEORY

05.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

 

SECTION – A

 

Answer ALL the questions                                                                          (10 ´ 2 = 20 marks)

 

  1. What is the problem of point estimation?
  2. Show that UMVUE of a given parametric function is unique almost surely.
  3. Define QA – optimality criterion.
  4. Let X ~ N ( 0, s2), s > 0. Find a minimal sufficient statistic.
  5. Classify the following as location none of the two:
  6. a) BVN (0,0, q, q, 1/2) b) BVN (q, 0, 1, 1, 0.6).
  7. State Rao-Blackwell theorem.
  8. Define Exponential family.
  9. Let X ~ B (n, p), n = 2, 3 and p = . Obtain MLE of (n, p) based on X.
  10. Define scale equivariant estimator.
  11. Explain Minimax estimation.

 

SECTION – B

 

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

 

  1. Find the Jackknified estimator of m2 in the case of f(x) = , x ≥ m; m Î

 

  1. State and establish Basu’s theorem.

 

  1. Let X1, X2, …, Xn be a random sample from N (m, s2), m Î R, s > 0. Find UMRUE of

(m, m/s) with respect to any loss function which is convex in its second argument.

 

  1. Let X1, X2,…, Xn be iid U (q – , q + ), q Π Find minimal sufficient statistic and examine whether it is boundedly complete.

 

  1. Given a random sample of size n from N (m, s2), mÎR, s > 0, find Cramer – Rao lower bound for estimating ( m/s2).  Compare it with the variance of UMVUE.

 

  1. State and establish the invariance property of CAN estimator.

 

  1. Given a random sample from a location family with the location parameter x, show that is MREE of with respect to any invariant loss function, where dO is an LEE,  = and * minimizes Eo { P (½with respect to .

 

  1. Let X ~ N (q, 1), qÎ Find Bayes estimator of q with respect to squared error loss if the prior of q is N (0, 1).

 

 

 

SECTION – C

 

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

 

  1. a) Give an example for each of the following:
  2. i) Ug is empty ii) Ug is singleton.

 

  1. b) Let X be DU {1,2,…, N}, N = 1,2,3,4,… . Find QA – optimal estimator of (N, N2).

(12+8)

  1. a) Show that a vector unbiased estimator is D – optimal if and only if each of its

components is a UMVUE.

 

  1. b) State and establish Lehmann – Scheffe theorem. (12+8)

 

  1. a) Let X1, X2, …, Xn be iid N (0, ),    Find MREE of    r with respect to

standardized squared error loss.

 

  1. b) Let (Xi­ , Yi), i = 1,2, …, n be a random sample from ACBVE distribution with pdf.

f(x,y) = {(2a+b) (a + b) / 2}  exp {-a(x+y) – b max.  (x,y)}, x, y > 0.

Find i) MLE of ( a, b)  and (ii) examine whether the MLE is consistent.           (8+8+4)

 

  1. Write short notes on:-
  2. Jackknifing method.
  3. Fisher information
  4. Location – scale family    (10+5+5)

 

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