M.Sc Statistics Question Paper

Loyola College M.Sc. Statistics April 2008 Statistics For Competitive Examinations Question Paper PDF Download

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NO 57

 

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – APRIL 2008

ST 4804 – STATISTICS FOR COMPETITIVE EXAMINATIONS

 

 

 

Date : 03-05-08                  Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION A

Answer ALL the Questions                                                                  (40 X 1 = 40 Marks)

 

  1. Let  A, B, and C  be independent and exhaustive events. If P (A) = ½ and P(B) = ⅓, fuid P(C).

(A) 1/6            (B) ⅔           (C) ½                (D) 1.

  1. If P (AUB) =5/6 and P (A) = ½, then P(B/AC) is

(A) 2/3            (B) 3/5         (C) 1/3              (D) ½

  1. For what value of λ, the random variable, whose distribution function is

F(x) =      0                 if x < 0

1-λe –x/3         if x ≥ 0

is continuous?

(A) 1               (B) 1 /√e      (C) ⅓                (D) √e

  1. A random variable X takes the values -1, 0, 2, and 4 with respective probabilities 1/6, ⅓, ⅓, 1/6. What is the expected value of (X-2)?

(A) 1/18          (B) 1/9         (C) -5/6            (D) -1/36

  1. If X1 and X2 are independent and identically distributed Geometric random variables with parameter 1/3, then the distribution of Y= min (X1, X2) is Geometric with the parameter

(A) 5/9            (B) 1/3         (C) 1/9              (D) 4/9

  1. A box contains 10 balls of which 4 are red and the rest are black. If 3 marbles are drawn from the box at random without replacement, what is the probability that 2 marbles are red?

(A) 1/10          (B) 1/2       (C) 3/10             (D) 1/12

  1. A random variable X distributed uniformly is such that P(X < 5) =1/2 and

P(X > 7) = 1/6. What is the support of X)?

(A) (5,7)            (B) (0,8)         (C) (1,9)                (D) (2,8)

  1. If (X, Y) has standard bivariate normal distribution with correlation coefficient ρ, what should be the value of λ in order that (λX-Y) and Y are independently distributed?

(A) –              (B)             (C)                (D)

  1. If X1 and X2 are independent random variables each having the distribution function G, then the distribution function of max(X1, X2) is

(A) G              (B) G 2           (C) G (2-G 2 )     (D) G (2-G)

  1. If X is an exponential random variable with E (eX) = 3, then E(X) is

(A) 1/2            (B) 1             (C) 2                 (D) .

  1. A random variable X has the probability density function

f (x) =  (e-x x m )/m!         if x > 0,m>0

=    0     otherwise

The variance of X is

(A)  (m+1)2    (B) m+1        (C) 2m              (D)

  1. If R1.23 =1, then the value of R2.13 is

(A) 0              (B)-1            (C) ½                (D) 1

  1. If (0.1, 0.2) is the strength of a SPRT, its approximate stopping bounds are

(A) (2/9, 8)     (B) (1/8, 9/2)  (C) (1/8,8)         (D) (2/9, 9/2)

  1. Choose the correct statement in connection with a standard LP problem.

(A)  constraints can involve non-linear term.

(B)  All constraints should be less than or equal to type (or) greater than or                        equal to type

(C) The objective function can involve non-linear terms.

(D) All variables must be nonnegative

  1. When a LPP has feasible solution, at then end of Phase-I in two-phase method,

the objective function’s value will be

(A) >0       (B) <0              (C) 0                (D) infinity

 

  1. The number of basic cells in any IBF solution for a TP is

(A) m+n+1            (B) m+n-1       (C) m-n+1       (D) n-m+1

  1. Consider the following LPP:

Max Z = X1 +X2

Subject to  X1 3

X1, 2X24

X1,X2, 0

The solution is

(A) (0,2)          (B) (3;0)          (C) (3, )        (D) (4,0)

  1. An LPP has 6 variables and 4 constraints. How many sets of basic variables are

possible ?

(A) 10        (B) 6                (C) 20              (D) 15

  • The power function associated with the UMPT for testing against the alternative in is always

(A) strictly increasing in      (B) strictly decreasing in

(C) periodic in                      (D) constant.

  1. Which of the following is the form of UMPT for testing  against the alternative in

(A)                        (B)

(C)                        (D)

  1. A UMPUT can be found for a testing problem on finding the UMPT in the class

of all

(A) Unbiased tests      (B) Similar tests

(C) Invariant tests       (D) All the three mentioned in (A), (B) and (C)

  1. Stratified sampling is advantageous over SRS

(A) If the within stratum variances more than the population variance.

(B) If the within stratum variances are less than the population variance.

(C) Without any condition.

(D) If the averages within all the strata are equal.

  1. The variance of a random vairable in terms of conditional expectations and

conditional

variance is given by

(A) V(Y) = V –

(B) V(Y) = V +

(C) V(Y) = V +

(D) V(Y) = V +

  1. If ₣ is the trivial sigma field, than E(X l ₣) is

(A) Constant   (B) E (X)         (C) X   (D) none of the above

  1. The inclusion probability of any two specified units in a SRSWOR is

(A)

(B)

(C)

(D)

  1. Choose the correct statement,

(A) Systematic sampling is a particular case of cluster sampling

(B) Cluster sampling is a particular case of systematic sampling

(C) While forming strata, we should ensure that within-stratum variability is more

(D) Proportional allocation is better then optimum allocation

 

 

 

 

  1. Which of the following is a tree wrt a network consisting of 5 nodes?

 

 

 

(A)                                                          (B)

 

 

 

 

 

 

(C)                                                                   (D)

 

 

 

 

  • The function f(x) = | x – 3| is NOT differentiable at

(A) 0                (B) +3              (C) –3              (D) f is differentiable everywhere

29.A tosses a fair coin twice and  B throws a fair die twice. Let

a = Probability of getting an odd number of heads

b = Probability that the sum of the numbers that show up is at least 7.

Then

(A) a > b        (B)  b               (C) a = b         (D) a + b < 1

  • The system of equations

x – y +3z = 3

4x – 3y +2z = 7

(3m – 1)x – y – 4 = 2 m2 – 1

has infinitely many solutions if m equals

(A) 0                (B)1                 (C)2                 (D)3

  • The mean and variance of 6 items are 10 and 5 respectively. If an observation 10 is deleted from this data set, the variance of the remaining 5 items is

(A)5                 (B)6                 (C)7                 (D)8

  • Which of the following forms a basis for R3 along with (1, 2, – 1) and (2, – 2, 4)?

(A) ( 0, 0, 0)     (B) (2, 1, 1)      (C) (3, 0, 3)      (D) (1, 4, – 2)

  • In a bivariate dataset {(Xi, Yi), i =1, 2, …,n}, X assumed only two values namely 0 and – 1 and the correlation coefficient was found to be –0.6. Then , the correlation coefficient for the transformed data {(Ui, Vi), i =1, 2, …,n}, where Ui = 4 + 2 Xi3 and Vi = 3Yi + 5, is

(A) 0,6             (B) – 0.6          (C)0                 (D) cannot be determined

  • Which one of the following could be the 1st and 2nd raw moments for a Poisson distribution?

(A) 2, 8                        (B) 4, 12          (C) 5, 30          (D) 6, 40

  • Let X and Y be random variables with identical means and variances. and if X and Y are un correlated then ca (X+Y, X-Y) is

(A)  (X)                (B)  0               (C)  2V(Y)      (D)  1

  • If X1, X2, …, Xn is a random sample form N (q ,1), – µ < q < µ, which of the following is UMVUE of ?

(A) S Xi2         (B)S (Xi – )2           (C) (SXi , SXi2)         (d) None of these

  • Let be an unbiased estimator of a parameter. The theorem which provides an improved unbiased estimator is

(A) Factorization theorem (B) Basis theorem (C) Rao-Blackwell theorem

(D) Gauss-Markor theorem.

  • The upper control limit of a c- chart is 48. The lower control limit is

(A) 0                (B) 8                (C) 4                (D) Cannot be determined

  • The linear model appropriate for two-way classification is

(A) Yij = ai + bj + eij                                        (B) Yij = m + ai + eij

(C) Yij = m + bj + eij                                                   (D) Yij = m + ai + bj + eij

  • In a 24 factorial experiment with 5 blocks, the degrees of freedom for Error Sum of Squares is

(A)20               (B)40               (C)60               (D)70

SECTION B

Answer any SIX Questions                                                                   (6 X 10 = 60 Marks)

 

  1. Let X1 and X2 be independent Poisson random variables. Show that the conditional distribution of X1 given X1+ X2= n is binomial.
  2. State Liapunov form of central limit theorem. Give an application.
  3. State and establish any two properties of a Poisson process.
  4. Explain how will you solve the following game theory problem using linear
    programming technique (Complete solution needed)

 

B1       B2       B3

 

A1       3          -1         -3

 

A2       -2         4          -1

 

A3       -5         -6         2

  1. Derive UMP level test for testing  based on a random

sample from

  1. Given a random sample from , find the moment estimators of

.

  1. (a) Let X1, X2, X3 have the multinomial distribution with parameters q1,q2,q3 and

q4, where q4 = 1 – (q1 + q2 + q3) and n = 30. If the observed values of

the random variables are X1 = 9, X2 = 8, X3 =9, find the MLEs of the

parameters.

(b) Obtain the MLE of q based on a random sample of size from the double

exponential distribution with p.d.f  f(x,q­) = exp (– |x – q | )/2, – µ < x , q < µ .                (7 + 3)

48 (a)If X1,….,Xn is a random sample from U(0,), find the distribution of the

range.

(b)Given a random sample from , find Bhattacharyya lower bound of order 2 for estimating .                                                                                                    (5 + 5)

49.(a)Let ‘p’ be the probability that the mean of a sample of size ‘n’ falls outside

the control limits of a control chart. Derive an expression for the following:

P(r out of k samples give a point out of control limits).

(b) If the mean is normally distributed and the control limits are 3-sigma limits, find the above probability with r=2, k=4.                      (4 + 6)

 

 

 

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Loyola College M.Sc. Statistics April 2008 Statistical Process Control Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

NO 51

M.Sc. DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – APRIL 2008

ST 4806 – STATISTICAL PROCESS CONTROL

 

 

 

Date : 21/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION – A

Answer ALL the  questions                                                                                    10×2 =20

 

  • Discuss the statistical basis underlying the general use of 3 – sigma limits on control charts.
  • What are chance and assignable causes of variation?
  • Describe the rational subgroup concept
  • Discuss the relationship between a control chart and statistical hypothesis testing.
  • Why is the np chart not appropriate with variable sample size?
  • What is process capability ratio (PCR) when only the lower specification is known ?.
  • Define a) Specification limit. b) Natural tolerance limit.
  • A Chain SP– I plan has n = 4 , c = 0 and  i = 3 .Draw the OC curve for this plan.
  • Explain sequential sampling plan.
  1. Write a note on modified control chart.

 

SECTION- B

Answer any FIVE questions                                                                                            5 x 8= 40

 

11.A quality characteristic is monitored by a control chart designed so that the probability that a

certain out of control condition will be detected on the first sample following the shift to that

      is 1 – b. Find the following:

a). The probability that the out of control condition will be detected on the second sample

following the shift.

b). The expected number of subgroups analyzed before the shift is detected.

12.You consistently arrive at your office about one-half hour late than you would like. Develop

a cause  and effect diagram that identifies and outlines the possible causes of this event.

13.Explain the method of constructing control limits for  and R charts when the sample sizes

are different for various subgroups.

14.The manufacturer wishes to set up a control chart at the final inspection  station for a gas

water heater. Defects in workmanship and visual quality features are checked in this

inspection. For the last 22 working days , 176 water heaters were inspected and a total of 924

nonconformities reported .

 

  1. a) What type of control chart would you recommend here and how would you use it?.
  2. b) Using four water heaters as the inspection unit ,calculate the center line and control

limits that are consistent with the past 22 days of inspection data.

15.Estimate process capability usingand R charts for the power supply voltage data . If

specifications are at 350  5 V, calculate PCR, PCRk and PCRkm. Interpret these capability

ratios.

Sample # 1 2 3 4 5 6 7 8 9 10
X1 6 10 7 8 9 12 16 7 9 15
X2 9 4 8 9 10 11 10 5 7 16
X3 10 6 10 6 7 10 8 10 8 10
X4 15 11 5 13 13 10 9 4 12 13

16.Explain  the V- Mask procedure with an illustration.

17.Describe Six- sigma quality .

18.Consider the single – sampling plan for which p1 = 0.01 , a = 0.05 , p2 = 0.10 and b = 0.10 .

Suppose that lots of N = 2000 are submitted. Draw the AOQ curve and find the AOQL.

 

SECTION- C

Answer any two questions                                                                                        2 X 20 = 40

 

  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) Find 0.900 and 0.100 probability limits for a c-chart when the process average is

equal to 16 non- conformities.                                                                                  (14+6)

  1. a)Suppose the process was in control with mean m0 and something happened today and mean

shifted to m0  + as , s is known. Using X – chart and – chart, find out which chart has a

greater chance of detecting the shift.

 

  1. b) Discuss Multivariate Quality Control                                                                    (10+10)
  2. a)Explain the Tabular CUSUM for monitoring the process mean with an illustration.
  3. b) What is Exponentially Weighted Moving Average control chart ? (10+10)
  4. a) Describe AQL and LTPD concepts.

b).Write short notes on:-

  1. Skip – Lot sampling plan with an illustration
  2. Continuous sampling plans with an illustration                                                   ( 8 + 12 )

 

 

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Loyola College M.Sc. Statistics April 2008 Statistical Computing – III Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

NO 54

M.Sc. DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – APRIL 2008

    ST 4808 – STATISTICAL COMPUTING – III

 

 

 

Date : 25/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

Answer ALL the Questions. Each question carries 33 marks

  1. (a.)

i.) Draw the OC curve of a single sampling with n = 100 , c =2.Also draw the AOQ

and ATI  curves.

ii.) Draw the tabular CUSUM for the following data.∆ = 0.5, α = 0.005, β = 0.10, σ = 1, n = 5

 

The sample mean values are given below.

34.5, 34.2, 31.6, 31.5, 35, 34.1, 32.6, 33.8, 34.8, 33.6, 31.9, 39.6, 35.4, 34, 37.1, 34.9, 33.5, 31.7, 34, 35.1

 

                                                     OR

(b)

(i) The data below represents the results of inspecting all units of a personal computer produced for the last 10 days.

Obtain the control limits .

 

Day                      1     2          3          4          5          6          7          8         9          10       

 

Units inspected    80   110      90        75        130      120      70        125      105      95

 

No of defectives   4    7          5          8          6          6          4          5          8          7

 

(ii.) The following fraction non confirming control chart with n = 100 is used to control a process.

UCL = 0.075    CL   = 0.04    LCL = 0.005

  • Find probability of type I error.
  • Find probability of type II error when p = 0.06.
  • Draw the OC curve for the control chart.
  • Find the ARL when p= 0.06.

 

  1. ( a)

The following data were collected from a 25 factorial experiment in two replicates with blocks of size 8 by completely confounding the effects ABC, ADE and BCDE. Analyse the data and identify the significant effects.

 

 

 

 

 

 

Treatment

Combination

 Yields (Rep I) Yields (Rep II)
 

00000

01100

10110

11010

11001

10101

00011

01111

 Block 1

56

68

70

73

71

81

69

86

 Block5

60

48

77

81

55

51

43

56
 

11000

10100

00010

01110

00001

01101

11011

10111

 Block 2

82

68

59

83

72

88

84

76

 Block 6

81

76

56

40

70

56

46

72
 

10000

11100

01010

01001

00101

10011

11111

00110

 Block 3

81

61

56

57

75

72

72

84

 Block7

57

37

77

52

51

64

62

70
 

00111

01011

11101

10001

11110

10010

00100

01000

 Block 4

74

69

60

49

46

74

54

72

 Block 8

68

46

59

89

50

42

98

62

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(OR)

(b) The yield of a chemical process was believed to be dependent mainly on standing time of the process. However, other factors also come into play. The chemical engineers who wanted to compare the effects of various standing times planned to

account for three other factors. So, they conducted an experiment using five types of

 

raw materials, five acid concentrations, five standing times (A, B ,C, D, E) and five catalyst concentrations ( α,β,γ,δ,ε). The following Graeco-Latin square design was

used. Analyse the data and draw your conclusions. Would you recommend a particular standing time over the others to maximize the yield? If so, which standing time is that?

 

Acid Concentrations

Raw

Material type          1                      2                       3                     4                     5

1                (Aα)16             (Bβ) 6                (Cγ) 9           (Dδ) 6             (Eε) 3

 

2                (Bγ) 8              (Cδ) 11               (Dε) 8           (Eα) 1             (Aβ) 11

 

3                (Cε) 10             (Dα) 2                (Eβ) 6           (Aγ) 15            (Bδ) 3

 

4                (Dβ) 5               (Eγ) 5                (Aδ) 12         (Bε) 4               (Cα) 7

 

5                (Eδ) 1               (Aε) 14               (Bα) 7          (Cβ) 7               (Dγ) 4

 

 

(3) (a)

(i) A business man is engaged in buying and selling identical items. He operates from a warehouse having a capacity of 500 items. Each month he can sell any quantity that he chooses up to the stock at the beginning of the month. Each month, he can buy as much as he wishes for delivery at the end of the month so long as his stock does not exceed 500 items. For the next four months he ahs the following error-free forecasts of cost and sales price:

 

Month:                  1       2       3       4

Cost Cn:               27     24     26     28

Sales Price pn:     28     25     25     27

 

If he has a stock of 220 unit, what quantities should he sell and buy in the next four months. Find the solution using dynamic programming.

 

(ii) Use the Kuhn-Tucker conditions to solve the following non-linear programming problem:

 

Minimize  Z = 2 x12 + 12 x1 x2 – 7 x22

Subject to the constraints

2 x1 + 5 x2 ≤ 98,       x1, x2 ≥ 0

(OR)

(b)

(i) Use integer programming to solve the LPP

 

Maximize Z = x1 – 2x2

Subject to the constraints

4 x1 + 2 x2 ≤ 15,    x1, x2 ≥ 0 and integers

 

(ii) Use Wolfe’s Method to  solve the QPP

 

Maximize Z = 2 x1 + 3 x2 – 2 x12

Subject to the constraints

x1 + 4 x2 ≤ 4

x1 + x2 ≤ 2 ,    x1, x2 ≥ 0

 

 

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Loyola College M.Sc. Statistics April 2008 Statistical Computing – I Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

NO 35

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2008

    ST 1812 – STATISTICAL COMPUTING – I

 

 

 

Date : 06/05/2008            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

Answer the following questions. Each question carries 33 marks

  1. (a) Write the quadratic form associated with the matrix

A =

       Verify whether it is positive definite.

(b) Obtain the characteristic roots and vectors of the following matrix:

A =

        Obtain the matrix U such that UTAU = L.

 

(OR)

 

        Find the inverse of the following matrix by partitioning method or

sweep out process.

A =

 

Sale Price (in lakh Rs) No. of Rooms Age of building
25.9 7 42
27.9 6 40
44 6 44
28.9 7 32
31.5 5 30
30.9 6 32
36.9 8 50
40.5 5 17
37.5 5 40
44.5 7 45
  1. The data on sale prices of houses are given below with information on the number of rooms and age of the building:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Build a model with an intercept. Test for overall significance and the significance of the individual regressors. Comment on the adequacy of the model.

(OR)

(a) A model with a maximum of four regressors is to be built using a sample of

size 30. Carry out ‘Stepwise Building Process’ to decide the significant

regressors given the following information:

SST = 1810.50,   SSRes(X1) = 843.79,   SSRes(X2) = 604.22,   SSRes(X3) = 1292.93,                   SSRes(X4) = 589.24, SSRes(X1, X2) = 38.60, SSRes(X1, X3­) = 818.05,                                           SSRes(X1, X4) = 49.84, SSRes(X2,X3) = 276.96, SSRes(X2, X4) = 579.25,                                SSRes(X3,X4) = 117.16, SSRes(X1, X2, X3) = 32.07, SSRes(X1, X2, X4) = 31.98,                           SSRes(X1,X3, X4) =33.89, SSRes(X2, X3, X4) =49.21, SSRes(X1, X2, X3,X4­) = 31.91

 

(b) The following are observed and predicted values of the dependent variable for a model with an intercept and two regressors.

 

Y Y^
16.68 21.7
12.03 12.07
13.75 12.19
8 7.55
17.83 16.67
21.5 21.6
21 18.84
19.75 21.6
29 29.67
19 16.65

 

 

 

 

 

 

 

 

 

 

 

 

Compute the standardized residuals and find if there are any outliers.

 

  1. The number of accidents taking place in a high way is believed to have mixture

of two Poisson distributions with mixing proportion 2/7 and 5/7. Fit the

distribution for the following data corresponding to one such distribution.

 

Marks Number of days
0 98
1 78
2 56
3 73
4 40
5 8
6 2
7 1
>8 0

 

 

(OR)

  • Generate Five observations from a Normal distribution with mean 20 and variance 36 truncated at zero
  • Generate a sample of size 2 from a mixture of two Cauchy variates one of them has scale parameter 1 and location Parameter 1 and the other has Cauchy distribution with scale parameter 1 and location parameter 0.

 

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Loyola College M.Sc. Statistics April 2008 Sampling Theory Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

NO 37

M.Sc. DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – APRIL 2008

    ST 2810 – SAMPLING THEORY

 

 

 

Date : 24/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

SECTION – A

 

Answer  ALL  questions.  Each  carries TWO  marks.     (10 x 2 =  20 marks)

 

  1. Define Population. I i (s)   What are the assumptions made about population size?
  2. Distinguish between parameter and statistic.  Give an example for both.
  3. Find the following:

(i)   E p­ [ I i (s) ] ,  i  =  1, 2, …, N,

(ii)   E p­ [ I i (s) E p­ [ I i (s) I j (s)] ;   i , j  =  1, 2, …, N ;   i  ≠ j .

  1. Show that an unbiased estimator for the population total can be found iff the first order inclusion probabilities are positive for all units in the population.
  2. In  SRSWOR ,  show that  E p­ [ s xy  ]  =  S xy .
  3. Define Midzuno Sampling Design and show that this method is a PPS selection method.
  4. Write the estimator for population total Y under Random Group Method and show that this estimator is unbiased for Y.
  5. Show  that the expansion estimator is equal to the population total under Balanced Systematic Sampling, in the presence of  linear trend.
  6. Derive the approximate bias of the ratio estimator for population total.
  7. Show that  LR  is more efficient  than   R  unless  β = R .

 

SECTION – B

 

Answer  any FIVE  questions.  Each  carries EIGHT  marks.     (5 x 8 =  40 marks)

 

  1. Give an example to show that an estimator can be unbiased under one design but

biased under another design.

  1. Under any design P( . ),  derive the variance of  Hurwitz – Thompson estimator for population total.
  2. Describe the unit drawing mechanism for simple random sampling design and prove that the mechanism implements the design.
  3. If  T( s, s′ ) is a statistic based on the sets s and s′ which are samples drawn in the first phase  of randomization and the second phase of randomization respectively, then prove that

V( T( s, s′ ) )  =  E1 V2 ( T( s, s′ ) )  +  V1 E2 ( T( s, s′ ) ) ,

where E2 is the expectation taken after fixing the subset s and E1 is the

expectation with respect to the randomization involved in the first phase.

 

  1. Show that the estimated variance  v( HT ) is  non-negative under Midzuno        Sampling Design for all s receiving positive probabilities.
  2. Show that LSS is more efficient  than SRS for population with linear trend.
  3. Obtain Yate’s corrected estimator under LSS in the presence of linear trend to estimate population total without error.
  4. Describe Simmon’s unrelated randomized  response model and obtain the estimate of  ΠA  when ΠY is unknown.

 

SECTION – C

 

Answer  any TWO  questions.  Each  carries TWENTY  marks.     ( 2 x 20 =  40 marks)

 

19 ( a ) After the decision to take a SRS has been made, it was realized that  Y1 the value of unit with

label 1 would be unusually low and YN the value of unit with label N would be unusually high.  In

such cases it is decided to use the estimator

 

if  1    s,    N     s

*   =         if   1 s,    N  s

otherwise,

where c is a pre-determined constant.  Show that  ( i )  *   is unbiased for   for any c.

( ii ) Derive  V(*  ).  ( iii ) Find the value of c for which *  is more efficient than  .      .

( 14 )

19 ( b )  State the unit drawing mechanism for Midzuno Sampling Design and show that

the mechanism implements the design.                                                                              ( 6 )

20 ( a ) Derive the estimated variance of   DR .                                                                     ( 10 )

20 ( b ) Show that the expansion estimator is equal to the population total under

Balanced Systematic Sampling in the presence of linear trend .                                         ( 10 )

  1. Derive the expressions for the approximate bias and MSE of the estimator R

and deduce their expressions under ( i ) SRSWOR,  (ii) PPSWOR, and ( iii ) Midzuno Sampling.

( 20 )

22 ( a ) Show that Hansen-Hurwitz estimator dhh  under double sampling is unbiased

for Y and derive its variance.                                                                                               ( 12 )

22 ( b ) Explain Stratified Sampling.  Deduce the expressions for   St ,   V (St )   and

v (St ) when samples are drawn independently from different strata using

( i )  SRSWOR,  and  ( ii ) PPSWR.                                                                                       ( 8 )

 

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Loyola College M.Sc. Statistics April 2008 Probability Theory Question Paper PDF Download

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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( ∩  An  ) =1 if  P(An ) =1, n=1,2,3,…

n=1

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

distribution function.

  1. 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.

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

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

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

 

Section-B (5×8=40)

 

Answer any FIVE questions.

 

  1. 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?

  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 )= 2/ x3 ,    x ≥1.

  1. In the usual notation, prove that

∞                                                           ∞

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

n=1                                                      n=1

 

  1. Define convergence in probability. If Xn → X in probability, show that

Xn2 + Xn  → X2+ X in probability.

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

variables.

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

n

lim   e-n   Σ        nk   = 1/2.

n→∞        k=0       k!

  1. 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

 

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

properties.                                           (8 marks)

(b) Show that the vector X =(X1, X2,…, Xp ) is a random vector if and only if Xj,

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 {Xn , 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 {Xn} be a sequence of normal variables with E (Xn) = 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 Xn 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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NO 38

 

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – APRIL 2008

         ST 2902 – PROBABILITY THEORY AND STOCHASTIC PROCESSES

 

 

 

Date : 29-04-08                  Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

Part-A

             Answer all the questions:                                                                 10 X 2 = 20

 

1.) Find the constant c such that p(x) = c (2/3)x   x=1,2,3,… zero elsewhere, satisfies the

conditions of a probability distribution.

2.) If the events A and B are independent show that A and B are independent.

3.) The MGF of a random variable X is (1-2t)-n/2  .Find the E(X).

4.) Define the period of a state. When do you say that a Markov chain is aperiodic?

5.) Define a process with independent increments. When do you say that it is stationary?

6.) Define a Markov process.

7.) Define covariance between two random variables X and Y. What happens to the Covariance when

they are independent?

8.) Given the MGF of a random variable X as (1/3 + 2/3 et)5  .Find P(X=0)

9.) What is a Renewal process?

10.) State the additive property of Poisson distribution.

 

Part-B

       Answer any 5 questions:                                                                        5 X 8 = 40

11.) The transition probability matrix of a Markov chain with three states 0,1,2 is

 

P  =

 

and the initial distribution is given by   P(X0= i) = 1/3  i=0,1,2

Find  i.)P[X2 = 2]     ii.)P[X3 =1 ; X2=2 ; X1=1 ; X0=2]

12.) Show that discrete queue forms a Markov chain.

13.) Show that the distribution function F(x) is non decreasing and right continuous.

14.) Let X have a pdf  f(x)=4x3 , 0<x<1, zero elsewhere .Find the distribution function and

pdf of Y= -2 ln  X4

15.) Derive the MGF of Normal distribution.Hence obtain the mean and variance.

16.) Given the joint pdf of the random variables X and Y as

f (x, y) = 8xy,   0<x<y<1, zero elsewhere.

Obtain i.) Marginal pdf of x1

ii.)conditional pdf of Y given X=x

iii.)E[Y/x]

iv.)Var [Y/x]

17) In a Markov chain if  i  ↔ j then show that

i.) d(i) = d(j)

ii.) If i is recurrent then j is also recurrent .

18.) Derive the expression for Pn (t) in Yule-Furry process

 

                             Part-C

     Answer any 2 question:                                                               2 X 20 = 40

 

19.)a.)Let An be increasing sequence of events. show that  P(lim An) =   lim P( An) .

Deduce the result for decreasing sequence.                              (8+4)

b.)Five numbers are drawn without replacement from the first 10 positive integers.

Let X represent the next to the smallest of the numbers drawn .Find the probability

distribution of X.Also obtain F(x).                                           (5+3)

20.)a.)State and prove addition theorem for n-events.

b.)Bowl I contains 3 red chips an7 blue chips.Bowl II contains 6 red chips and 4 blue

chips .A bowl is selected at random and then 1 chip is drawn from  the bowl.

i.)Compute the probability that this chip is red.

ii.)Given that the drawn chip is red,find the conditional probability that it is drawn

from bowl II.
c.)Show that Binomial distribution tends to Poisson distribution under some

conditions.

21.)a.)State the postulates of a Poisson process and obtain the expression for P(t).

b.)If the Process {Xt} has stationary independent increments with a finite mean, show  that

E[Xt}=m0+m1 t       where m0 = E (X)

m1=  E  (X) – m0

22.)a.)Given the Markov chain with transition probability matrix with states 1,2,3,4

and

 

P  =

 

 

 

i.)Show that the chain is irreducible, aperiodic and non-null recurrent.

ii.)Obtain the probability of reaching the various states as n → ∞.   (8+6)

b.)A gambler has Rs.2.He bets Rs.1 at a time and wins Rs.1 with probability 1/2.He

stops the playing if he loses Rs.2 or wins Rs.4.

What is the transition probability matrix of the Markov chain?             (6)

 

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Loyola College M.Sc. Statistics April 2008 Multivariate Analysis Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

NO 42

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – APRIL 2008

    ST 3808 / 3801 – MULTIVARIATE ANALYSIS

 

 

 

Date : 26/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

Answer all the questions.                                                                         (10 X 2 = 20)

 

  1. If X and Y are two independent standard normal variables then obtain the distribution of two times of the mean of these two variables.
  • If X = ( X1, X2 )’~N2 then write the c.f. of the marginal distribution

of X2.

  1. Mention any two properties of multivariate normal distribution.

4 . What is meant by residual plot?.

  1. Explain the use of partial and multiple correlation coefficients.
  2. Define Hotelling’s T2 – statistics.
  3. Define Fisher’s Z-transformation
  4. Write a short on discriminant analysis.
  5. Explain how canonical correlation is used in multivariate data analysis
  6. Explain classification problem into two classes.

 

 

PART B

Answer any FIVE questions.                                                                     (5 X 8 = 40)

 

11.Find the multiple correlation coefficient between X1 and X 2 , X3, …     , X p.

Prove that the conditional variance of X1 given the rest of the variables can not be

greater than unconditional variance of X1.

  1. Derive the c.f. of multivariate normal distribution.
  2. Let Y ~ Np ( 0 , S ). Show that Y’S -1 Y   has     distribution.
  3. Obtain a linear function to allocate an object of unknown origin to one of the two

normal populations.

  1. Giving a suitable example describe how objects are grouped by complete linkage

method.

  1. Let X ~ Np (m, S). If X(1) and X(2) are two subvectors of X, obtain the conditional

distribution of X(1) given X(2).

  1. Prove that the extraction of principal components from a dispersion matrix is the

study of characteristic roots and vectors of the same matrix  .

  1. Explain step-wise regression.

 

 

 

 

 

PART C

Answer any two questions.                                                                 (2 X 20 = 40)

 

 

               19 .  Derive the MLE of  m and S when the sample is from N ( m ,S ).

 

  1. a) Derive the procedure to test the equality of mean vectors of two p-variate

normal populations when the dispersion matrices are equal.

 

  1. b) Test at level 0.05 ,whether µ = ( 0 0  )’ in a bivariate normal population with

σ11 = σ22= 10 and  σ12= -4 , by using the sample mean vector= (7  -3)‘

based on a sample size 10.

(15 + 5)

 

  1. a) Define i) Common factor ii) Communality  iii) Total variation
  2. b) Explain the principal component ( principal factor )method of estimating L

in the factor analysis method.

  1. c) Discuss the effect of an orthogonal transformation in factor analysis method.

( 6 + 6 + 8 )

  1. What are canonical correlations and canonical variables? .Describe the extraction

of canonical correlations and their variables from the dispersion matrix. Also

show that there will be p canonical correlations if the dispersion matrix is of size p.

( 2+10+8 )

 

 

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Loyola College M.Sc. Statistics April 2008 Measure And Probability Theory Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

NO 32

 

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2008

ST 1809 – MEASURE AND PROBABILITY THEORY

 

 

 

Date : 30-04-08                  Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

SECTION A

 

Answer ALL questions                                                                     2 *10 = 20

 

  1. Show that  =
  2. Define : Sigma field
  3. Mention any two properties of set functions.
  4. If   exists, then show that
  5. Define Singular measure.
  6. State the theorem of total probability.
  7. If E (Xk) is finite, k > 0, then show that E (Xj) is finite for 0 < j < k.
  8. Let X1 and X2­ be two iid random variables with pdf  . Find V(X1 + X2).
  9. Show that E ( E ( Y | ġ ) ) = E (Y).
  10. Define Convergence in rth  mean of a sequence of random variables.

 

SECTION B

Answer any FIVE questions                                                                                    5 * 8 = 40.

 

  1. Show that finite additivity of a set function need not imply countable additivity.
  2. Consider the following distribution function.

If μ is a Lebesgue measure corresponding to F, compute the measure of

a.)          b.)

 

  1. State and prove the Order Preservation Property of integral of Borel measurable functions.
  2. Let μ be a measure and λ be a singed measure defined on the σ field  of subsets of Ω. Show that λ << μ   if and only if | λ | << μ.
  3. State and prove Borel – Cantelli lemma.
  4. Derive the defining equation of conditional expectation of a random variable given a σ field.

 

 

  1. Let Y1,Y2,…,Yn be n iid random variables from U(0,θ). Define

Xn = max (Y1,Y2,…,Yn). Show that

 

  1. State and prove the Weak law of large numbers.

 

SECTION C

Answer any TWO questions.                                                                         2 * 20 = 40

 

  1.  a.) Show that every finite measure is a σ finite measure but the converse need not

be true.

b.) Let h be a Borel measurable function defined on. If   exists,

then show that  = , v  c € R.                                              (8+12)

  1. a.) State and prove the extended monotone convergence theorem for a sequence

of Borel measurable functions.

b.) If X = (X1,X2,…,Xn) has a density f(.) and each Xi has a density fi, i= 1,2,…,n ,

then show that X1,X2,…,Xn are independent if and only if

a.e. [λ] except possibly on the Borel set of Rn with Lebesgue measure zero.

(12+8)

 

  1. a.) If Z is ġ measurable and Y and YZ are integrable, then show that

E ( YZ | ġ ) = Z E (Y | ġ) a.e. [P].

b.) Show that  implies  but the converse is not true.

(10+10)

  1.  a.) State and prove Levy inversion theorem.

b.) Using Central limit theorem for suitable Poisson variable, prove that

.                                                                           (12+8)

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Loyola College M.Sc. Statistics April 2008 Fuzzy Theory And Applications Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

NO 45

 

THIRD SEMESTER – APRIL 2008

ST 3875 – FUZZY THEORY AND APPLICATIONS

 

 

 

Date : 30-04-08                  Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

 

SECTION – A

Answer ALL the Questions                                                                    (10 x 2 = 20 marks)

 

  1. Define a fuzzy t-conorm.
  2. Give an example of a discontinuous t-norm. Justify?
  3. Define pseudo inverse of a increasing continuous function on [0, 1].
  4. What are the two methods for defining fuzzy arithmetic?
  5. State the law of excluded middle and the law of contradiction.
  6. Define α-cut and strong α-cut of a fuzzy set.
  7. Give a rough graphical depiction of the membership function of a convex fuzzy set.
  8. State the Axiomatic Skeleton for fuzzy complements.
  9. Give an example for a parametric class of membership functions.
  10. Define an ‘Artificial Neural Network’.

 

 

SECTION – B

Answer any FIVE Questions                                                                    (5 x 8 = 40 marks)

 

  1. (a) Prove that the standard fuzzy union is the only idempotent t-conorm.

(b). Prove that , for all a,b [0, 1].                   (4+4)

  1. Given

Determine

 

  1. Prove that u(a,b)=c(i(c(a), c(b))) is a t-conorm for all a,b[0, 1], where c is the involutive fuzzy complement.
  2. Under what conditions distributive law hold good for fuzzy numbers? Justify your answer with suitable examples.
  3. If c: [0,1] → [0,1] is involutive and monotonically decreasing, show that it is continuous and that c(0) = 1 and c(1) = 0.
  4. Explain the ‘indirect method with one expert’ for constructing membership functions.
  5. What is an activation function? Explain the three basic types of activation functions.
  6. Discuss the problem of fuzzy clustering with an example.

 

 

 

 

 

 

 

SECTION -C

Answer any TWO Questions                                                                 (2 x 20 = 40 marks)

  1. (a) Let A and B be two fuzzy numbers. If

 

and

Determine the product fuzzy number (A . B) and the division (A/B).

 

(b) Explain the basic arithmetic operations on the intervals.                            (16+4)

 

  1. (a) Prove the characterization theorem for fuzzy numbers.

(b) Let A and B be fuzzy numbers.  Prove that   is

also a fuzzy number, where * is one of the basic arithmetic operations.      (10+10)

 

  1. (a) Explain the ‘direct method with multiple experts’ for constructing membership

functions.

(b) Let X ={x1, ..,x5} be a universal set and suppose three experts E1, E2, E3 have

specified the valuations of these five as elements of two fuzzy sets A and B as

given in the following table:

Membership in A                       Membership in B

Element E1 E2 E3
x1

x2

x3

x4

x5

 0

1

1

0

1

 1

0

0

1

1

 1

0

1

0

1
Element E1 E2 E3
x1

x2

x3

x4

x5

 1

0

1

0

0

 1

0

1

0

1

 1

0

0

1

0

 

 

 

 

 

 

 

Assuming that for set A, the evaluations by the three experts have to be given

weights as c1 = 1/4 , c2 = 1/2, c3 =1/4 and for set B as equal weights, find the

degree of membership of the five elements in A and in B. Also, evaluate the

degree of membership in A∩B using the Algebraic prroduct operator and in

AUB using the Drastic union operator.                                                         (6 +14)

 

  1. (a) Briefly explain the three fundamental problems of ‘Pattern Recognition’.

(b) Describe the single-layer and multi-layer feed forward and recurrent neural

network architectures.                                                                                    (6 +14)

 

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Loyola College M.Sc. Statistics April 2008 Estimation Theory Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

NO 36

SECOND SEMESTER – APRIL 2008

ST 2808 – ESTIMATION THEORY

 

 

 

Date : 17/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

SECTION – A                            Answer all the questions                                   (10 x 2 = 20)

  1. Give an example of a parametric function for which unbiased estimator does not exist.
  2. Define a loss function for simultaneous estimation problem and give an example.
  3. If δ is a UMVUE, then show that 5δ is also a UMVUE.
  4. Find the Fisher information in the Bernoulli distribution with the parameter θ.
  5. Define completeness and bounded completeness.
  6. Given a random sample of size 2 from N(0, σ2), σ>0, suggest two ancillary statistics.
  7. Give two examples for location equivariant estimator.
  8. Let X follow E( θ,1), θ = 0.1,0.2. Find the MLE of θ .
  9. Define a consistent estimator and give an example.
  10. Explain prior distribution and Conjugate family.

 

SECTION – B                                Answer any  five questions                      (5 x 8 = 40)

‌11. Let X follow DU{1,2,…N}, N = 2,3,4,… Find the class of unbiased estimators of  N .

  1. State and prove Cramer-Rao inequality for the multiparameter case.
  2. Discuss the importance of Bhattacharyya inequality with a suitable example.
  3. Let X1,X2,…,Xn be a random sample from N(θ, θ2), θ >0. Find a minimal sufficient statistic and

examine whether it is complete.

  1. Using Basu’s theorem show that the sample mean and the sample variance are independent in the

case of  N( θ, 1), θ ε R.

16.Given a random sample from E(0, τ), τ > 0, find MREE of τ and τ2 with respect to  standardized

squared error loss.

17.Give an example in which MREE of a location parameter exists with respect to squared error loss but

UMVUE does not exist.

  1. Let X1,X2,…,Xn be a random sample from B(1, θ), 0<θ<1. If the prior distribution is U(0,1), find the

Bayes estimator of θ with respect to the squared error loss.

 

SECTION – C                   Answer any two questions                                       (2 x 20 = 40)

19 a) State and establish Bhattacharya inequality

  1. b) Let X follow DU{1,2,…,N}, N = 3,4,…Find the UMVUE of N using Calculus approach.

20 a) Show that an estimator δ is QA – optimal if and only if each component of δ is a UMVUE.

  1. b) Given a random sample from N(μ,σ2), μ ε R, σ > 0, find UMRUE of (μ, μ/σ) with

respect to any loss function, convex in the second argument.

21 a) Discuss the problem of equivariant estimation of the scale parameter.

  1. b) Given a random sample of size n from U(ξ, ξ+1), ξ ε R,find the MREE of ξ with respect to

standardized squared error loss.

22 a) Give an example for an MLE which is consistent but not CAN.

  1. b) Stating the regularity conditions, show that the likelihood equation estimator is CAN.

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

NO 34

 

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2008

ST 1811 – APPLIED REGRESSION ANALYSIS

 

 

 

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

Time : 1:00 – 4:00

 

Section A

Answer ALL the questions   

Each question carries 2 marks                                               (10 X 2 =20 Marks)

 

1)      Define the linear regression model in the context of any applied scenario.

2)      What are the basic assumptions of a linear regression model?

3)      Mention any four major areas where categorical data analysis is used.

4)      Explain nominal and ordinal variables with examples

5)      Explain interval variable with an example.

6)      Explain the link function of a generalized linear model

7)      What is the role of binary data in the generalized linear model.

8)      What is meant by multicollinearity?

9)      Write down the sampling variance of the slope coefficients in the multiple regression model.

10)    What is the role of the variance inflation factor?

 

Section B

Answer any 5 questions        

Each question carries 8 marks                                               (5  X 8 = 40 Marks)

 

11)    Derive the estimate of the parameters of a linear regression model using the method of least squares.

12)    Explain the three components of a generalized linear model

13)    Identify the natural parameter for the binomial logit model in the context of a decision taken to purchase a particular product

14)    Identify the natural parameter for the Poisson log linear model for a count data in the context of the number of silicon wafers used in the production of a computer chip.

15)    Explain Poisson log linear model.

16)    Explain the regression model of the status of the employees on the education and income level in the context of the usage of a dummy variable.

 

17)    Explain the concept of multi collinearity in the context of regression delivery time of and item on the distance traveled and the gasoline consumed.

18)    Write short note on stepwise regression methods.

 

Section C

Answer any 2 questions        

Each question carries 20 marks                                             (2  X 20 = 40 Marks)

 

19 a) How do you write the distribution of a transformed mean of a response variable of a Poisson log linear model in the natural exponential family form.

(10 Marks)

19 b) Give an illustration for multinomial responses using baseline logit models.

(10 Marks)

20 a) Illustrate the Poisson general linear model from a study of nesting horseshoe crab. ( 10 Marks)

20 b) Explain the use of dummy variables in the logit models with an example.

(10 Marks)

21 a) Explain the regression model of the status of the employees on the education and income level in the context of the usage of a dummy variable. ( 10 Marks)

21 b) Explain the concept of multi collinearity in the context of regression delivery time of and item on the distance traveled and the gasoline consumed.

(10 Marks)

22 a) Job satisfaction of the employees of a company is categorized in to 1. Not satisfied 2. A little satisfied 3. Satisfied and 4. Very much satisfied.  Construct a multinomial model for regressing job satisfaction on income and the gender.

( 10 Marks)

22 b) Explain the four methods for scaling residuals bringing out the relationship between them.          (10 Marks)

 

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Loyola College M.Sc. Statistics April 2008 Applied Experimental Design Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

NO 50

 

FOURTH SEMESTER – APRIL 2008

ST 4805 – APPLIED EXPRIMENTAL DESIGN

 

 

 

Date : 16-04-08                  Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION – A

Answer all the questions                                                          (10 x 2  = 20 marks)

 

  1. Briefly explain the term “Randomization”
  2. Define mutually orthogonal contrast with an example.
  3. Give sum of squares for 3 factors interaction for 33 Factorial Design.
  4. If G(F) = pn where p = 5 and n = 1, list the elements of the finite field.
  5. When do we go for confounding Design?.
  6. What is meant by resolvable BIBD? Give an example.
  7. Give any two advantages of fractional factorial design
  8. Define a simple Lattice Square.
  9. State any two parametric conditions of a BIBD.
  10. What is meant by optimal surfaces?

 

SECTION-B

 

Answer any Five questions                                                         (5 x 8  = 40 marks)

  1. Describe, the analysis of variance for a 33 factorial design, stating all the

hypothesis, ANOVA and conclusions

 

  1. Distinguish between FIXED EFFECT MODEL and RANDOM EFFECT MODEL with suitable

illustrations.

 

  1. Construct a mutual mutually orthogonal Latin Square when the number of treatments v = 5.

 

  1. When do go for a PBIBD? Explain briefly with an illustration.

 

  1. Discuss in detail the various RESOLUTION DESIGNS.

 

  1. Explain the term repeated Latin Square Design and hence construct 5 X 5 repeated Latin Square

Design.

 

  1. Distinguish between ANOVA and ANOCOVA.

 

  1. Explain the term SPLIT – PLOT and STRIP = PLOT with an example.

 

 

 

 

 

 

SECTION-C

 

Answer any Two questions                                                        (2 x 20  = 40 marks)

 

 

 

  1. a) Distinguish between confounding and fractional replication.

 

  1. b) Develop, the analysis of variance for a 24 fractional factorial design, stating all the

hypothesis, ANOVA and conclusions

 

(10+10-Marks)

 

  1. a) Construct a PBIBD with 2 associate classes, stating all the parametric conditions.

 

  1. b) Discuss in detail the Linear and Quadratic Response Surface Designs stating the model analysis

and conclusions.

 

(10+10-Marks)

 

  1. a) Discuss in detail confounding in MORE THAN TWO BLOCKS using finite field.
  2. b) Construct BIBDs with v = 5 with k = 2,  k = 3 and k = 4.

(10+10-Marks)

 

22 Write shorts on the following

  1. Local control
  3. Partial confounding
  4. Orthogonal Design.                                       (5+5+5+5-Marks)

 

 

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Loyola College M.Sc. Statistics April 2008 Analysis Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

NO 31

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2008

    ST 1808 – ANALYSIS

 

 

 

Date : 28/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

SECTION – A

——————-

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

  1. Define  a metric space and give an example.
  2. Let ( X , ρ) be a metric space and let Y С X  Define  σ : Y x Y → R1 as  σ(x ,y)  =  ρ(x ,y)             x ,y   Y.  Show that (Y, σ ) is a metric subspace of (X , ρ ).
  3. Let X = R2 .  Take   x n    =  (  n/ (2n+1)  ,   2n2 / (n2 – 2)  ) ; n = 1,2,….                                     Show that lim n→∞ x n    =  (  ½  ,  2 ) .
  4. Let  V  =  B [ a , b ] be the class of bounded functions defined on        [ a , b ] .  Examine whether  sup a ≤ x ≤ b  ‌‌‌| f( x) | is a norm on V .
  5. Define a linear function and give an example.
  6. Show that every convergent sequence in  (X, ρ )  is a Cauchy sequence.  Is the converse true?
  7. State any three properties of compact sets.
  8. Prove the following relations :

( i )  O ( v n )  +  o (  v n )  =   O ( v n )

(ii)   O ( v n )  .  o (  w n )  =  o ( v n w n )

 

  1. Apply Weierstrass’s  M – test to show that

p  converges uniformly on  ( -∞ , ∞ ) , whenever  p > 1.

  1.  Give an example of a function  f   not in  R( g ; a , b) whenever g is a non-constant function.

SECTION – B

——————-

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

 

  1. State and prove Cauchy – Schwartz inequality regarding inner product on a vector space.
  2. Prove that  ‛c’  is a limit point of E  iff     a sequence x n  E  э   

x n  ≠  c and  x n  → c  as n →∞.

 

  1. Prove the following:

( i  ) The union of any collection of open sets is open  .

( ii ) The intersection of any collection of closed sets is closed.

  1. Let ( X , ρ) and ( Y, ρ) be the metric spaces. Prove that a necessary

and sufficient condition for f : X → Y to be continuous at ‛ x0’ X is

that    f (x n ) → f ( x0 ) as n →∞.

 

  1. Prove that a linear function f : Rm → Rn is everywhere continuous.

 

  1. State and prove Heine – Borel theorem regarding compact sets.

 

  1. State and prove Cauchy`s root test regarding convergence of a series of complex terms.

 

  1. Let f : X → Rn ( X С Rm  ) be differentiable at  ξ  X. Then show that the linear derivative of     f at ξ  is unique.

 

SECTION – C

——————-

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

 

  1. ( a ) Let ρ   be a metric on   X.  Define σ   =  ρ / ( 1 + ρ   ) .

Show that  ( i  )   σ  is a metric

( ii )   ρ  and  σ   are equivalent                         ( 10)

( b ) State and prove  a necessary and sufficient condition for the set

F to be closed.                                                                    (10)

 

  1. ( a ) Suppose f : ( X , ρ) → ( Y , σ ) is continuous on X. Let ρ1 be a

metric on X and  σ1  be a metric on  Y э

( i  )  ρ  and ρ1 are equivalent.

( ii ). σ and σ1 are  equivalent.

Then show that f is continuous with respect to ρ1 and σ1. ( 10 )

( b ) Prove that a necessary and  sufficient  for f : ( X , ρ) → ( Y , σ )

to be continuous on X is that f -1 (G) is open in X whenever G is

open in Y.                                                                         (10)

 

  1. ( a ) Show that R1 with usual metric is complete. ( 10 )

( b ) Find all values of  x for which the series  ∑ x n  /  n x

converges.                                                                   (10)

 

  1. ( a ) State and prove Darboux theorem  regarding

Riemann – Stieltje’s  integral.                                   (10)

( b ) Let f : X  → Rn  ( X С Rm  ) be differentiable at  ξ  X.

Then show that all the partial derivatives Di fj (ξ ) ,

i = 1,2,…, m ;  j = 1,2,…,n exist and obtain the linear

derivative D f (ξ ).                                                      (10)

 

 

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Loyola College M.Sc. Statistics April 2008 Advanced Operations Research Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

NO 52

FOURTH SEMESTER – APRIL 2008

ST 4807 – ADVANCED OPERATIONS RESEARCH

 

 

 

Date : 23/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION- A

Answer all the questions.                        10 x 2 = 20 Marks

 

  1. When an LPP is said to have an unbounded solution?
  2. Write the significance of goal programming.
  3. Write a note on holding and shortage costs in an inventory system.
  4. What are the behaviours of customers in queuing analysis?
  5. Provide the Khun -Tucker conditions for a maximization problem.
  6. How Beal’s method differs from Wolfe’s method?
  7. Give an account of methods used for solving an integer programming problem.
  8. Define dynamic programming problem.
  9. How to find an optimal inventory policy for multiple item static model?

10.When an LPP is called stochastic?

 

                                                                        SECTION- B

Answer any five questions.                5 x 8 =  40 Marks

  1. Use two-phase simplex method to

Max. Z = 5x1 + 3x2

Subject to

2x1 + x2    1

x1  + 4x2  6

x1  0 and x2  0 .

  1. An item sells for $25 a unit ,but a 10 % discount is offered for lots of 150 units or more. A company uses this item at the rate of 20 units per day. The setup cost for ordering a lot is $50 and the holding cost per unit per day is $0.3. Should the company take advantage of the discount?
  2. Explain multi-item EOQ model with storage limitation.
  3. Derive the steady-state measures of performance for (M / M / 1):(FIFO/ / ).
  4. Explain generalized Poisson queuing model.
  5. Provide branch and bound algorithm for solving I P P.
  6. Use dynamic programming to

Min. Z = x12 + x22 + x32

Subject to

x1 + x2 + x3  15

x1 0 , x2  0 and  x3  0.

  1. Explain the two- stage programming technique used in stochastic programming.

 

    …2

 

 

-2-

 

 

SECTION – C

Answer any two questions.                2 x 20 =  40 Marks

 

19.(a)  Solve the following LPP graphically:

Max. Z = 2x1 + 5x2

Subject to

x1 + x2  1  ,     x1 – 5x2  0  ,   5x1 – x2 0  ,  x1 – x2 -1 ,   x1 + x2  6  ,

x2  3   ,   x1  0 and x2  0 .

 

  • Use dynamic programming to solve the following LPP:

Max.Z = 3x1 + 5x2

Subject to

x1  4 ,  x2  6 ,  3x1 + 2x2   18

x1  0 and x2  0 .

 

  1. (a) Derive the probabilistic EOQ model.

(b)  Electro uses resin in its manufacturing process at the rate of 100 gallons per

month. It cost Electro $100 to place an order for a new shipment. The holding

cost per gallon per month is $2 and the shortage cost per gallon is $10.Historical

data show that the demand during lead time is uniform over the range (0, 100)

gallons. Determine  the  optimal ordering policy for Electro.

 

  1. Use Wolfe’s method to solve the following QPP:

Max. Z = 2x1 + x2 – x12

Subject to

2x1 + 3x2   6

2x1 + x2     4

x1  0 and x2  0 .

 

  1. Solve the following mixed-integer programming problem using Gomory’s cutting plane algorithm:

Max. Z = 3x1 + x2+ 3x3

Subject to

-x1  + 2x2 + x3    4

4x2 – 3x3            2

x1  – 3x2 + 2x3  3

xi   0     (i = 1,2,3) where x1 and x3 are integers.

 

 

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Loyola College M.Sc. Statistics April 2008 Advanced Distribution Theory Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AK 37

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2008

    ST 1810 – ADVANCED DISTRIBUTION THEORY

 

 

 

Date : 03/05/2008            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

SECTION – A                                                Answer all the questions                                     (10 x 2 = 20)

 

  1. Define zero truncated binomial distribution and find its mean.
  2. Show that geometric distribution obeys lack of memory property.
  3. If X is lognormal, show that X2 is lognormal.
  4. Define bivariate Poisson distribution.
  5. Let (X1, X2) have a bivariate binomial distribution. Show that the marginals are binomial.
  6. State bivariate lack of memory property.
  7. Let (X1, X2) have a bivariate normal distribution. Show that X1 – X2 has a normal distribution.
  8. Define non-central F distribution.
  9. Let X1, X2, X3, X4 be independent standard normal variables. Examine whether

(X12 + 2X22 + X32 + 2X42 – 2 X1X2 + 2X2X3 )/ 2 is distributed as chi-square.

  1. Let X have B(1,p), where p is uniform on (0,1). Show that the compound distribution is uniform.

 

SECTION – B                                               Answer any five questions                                     (5 x 8 = 40)

 

  1. Express the following distribution function as a mixture and hence find its mgf.

F(x)  =  0,                 x < 0

=  (x + 2)/8,     0 ≤ x  < 1

=   1,                x  ≥  1.

  1. State and establish the additive property satisfied by bivariate binomial distribution.
  2. Show that the regression equations are linear in the case of bivariate Poisson distribution.
  3. For lognormal distribution, show that mean > median > mode.
  4. State the mgf of Inverse Gaussian distribution. Hence find the cumulants.
  5. State and establish any two properties of Pareto distribution.
  6. Let X1, X2, X3 be independent standard normal variables. Find the mgf of

X12 + 5X22 + 4X32  – 2 X1X2 + 8X2X3 .

  1. Let X be N(,). Show that (X – )/ -1 (X – ) has a chi-square distribution.

 

SECTION – C                                            Answer any two questions                                     ( 2 x 20 = 40)

 

19 a) Let X1, X2 be independent and identically distributed non-negative integer-valued random variables.

Show that X1 given X1 + X2 is uniform if and only if X1 is geometric.

  1. b) Derive the pgf of the power series class of distributions. Hence find the pgf of binomial and Poisson

distributions.

20 a) Let X1 and X2  be independent and identically distributed random variables with finite second

moment.

Show that X1 + X2  and X1 – X2 are independent if and only if X1 is normal.

  1. b) Let X1,X2,…,Xn be independent random variables such that Xi has Inverse Gaussian distribution with

parameters i, µi , I = 1,2,…,n. Show that

i  (Xi – µi)2 / (Xi µi2)  is chi-square with n degrees of freedom.

21 a) Define bivariate exponential distribution. Show that the marginals are exponential.

  1. b) Derive the pdf of non-central chi-square distribution.

22 a) State and establish a necessary and sufficient condition for a quadratic form in N(0,1) variables to

have  a chi-square distribution.

  1. b) Using the theory of quadratic forms, show that the sample mean and the sample variance are

independent in the case of normal distribution.

 

 

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Loyola College M.Sc. Statistics Nov 2008 Stochastic Processes Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

BA 25

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – November 2008

    ST 3809 / 3800 – STOCHASTIC PROCESSES

 

 

 

Date : 05-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART-A

Answer all questions:                                                                                      (10×2=20)

 

  1. Define a Markov process.

 

  1. Define recurrent state and transient state of a Markov chain.

 

  1. Define a Martingale of the process {Xn} with respect to {Yn}.

 

  1. Obtain E[X1 + X2 +…..+ XN] where Xi, i=1, 2, 3,….. are i.i.d and independent of the

random variable N .

 

  1. Let X1, X2 be independent exponentially distributed random variables parameters λ1

and λ2 respectively. Obtain P[min(X1,X2)>t] .

 

  1. Messages arrive at the telegraph office in accordance with the laws of a Poisson

Process with mean rate of 3 messages per hour. What is the probability of getting no

message during morning hours from10 to 12?

 

  1. Obtain the pgf of a Poisson process.

 

  1. If X1 and X2 are independent random variables with distribution functions of F1 and F2 respectively. Write

an expression for the distribution function of X=X1+X2?

 

  1. Obtain P[N(t)=k] in terms of the distribution functions of the life times for a renewal

Process?

 

  1. Define a stationary process.

 

PART-B

Answer 5 questions:                                                                                        (5×8=40)

 

11) Consider the Markov chain with states 0,1,2 having the TPM

 

 

and  P[X0 = i] = 1/3,  i = 1,2,3

Obtain i) P[X2=0]

  1. ii) P[X2=0, X1=2/ X0=1]

iii) P[X2=0, X1=2, X0­=1]                                                              (4+2+2)

 

12) Verify whether the Markov chain with TPM given below is ergodic

 

 

 

 

 

13) Show that for a renewal process in the usual notation,

M(t)= F(t) + F*M(t)

 

14) Prove that if {Xn} is a super martingale with respect to {Yn} then

  1. i) E[Xn+k ç Y0,Y1,…..Yn ] ≤ Xn,
  2. ii) E[Xn­] ≤ E[Xk], 0 ≤ k ≤ n

 

15) State the postulates of birth and death process. Obtain the forward differential

equations for a birth and death process.

 

16)  Obtain the Stationary distribution of a Markov chain with TPM

 

17) Consider the times {Sk} at which the changes of Poisson process X(t) occur. If

Si = T0 + T1 + … + Ti-1, i = 1,2,3,… obtain the joint distribution function of S1,

S2,……Sn given X(t) = n.

 

18) Show the periodicity is a class property.

 

PART-C

Answer 2 questions:                                                                                        (2 x 20=40)

 

19) a) Show that i is recurrent if and only if ∑Pii n = ∞

  1. b) Show that in a one dimensional symmetric random walk state 0 is recurrent.
  2. c) if j is transient prove that for all i ∑Pij n < ∞                                              (8+7+5)

 

20) a) State the postulates of a Poisson process and obtain the expression for Pn(t).

 

  1. b) If X(t) has a Poisson process, u<t, k<n obtain P[X(u) = k çX(t) = n]             (12+8)

 

21) a) Obtain the renewal function corresponding to the lifetime density

f(x) = λ2 x e – λ x ,  x ≥ 0

  1. b) Let Y0=0, Y1, Y2,….. be i.i.d with

E[Yk] = 0    var[Yk]=σ2     k=1,2,……

E[| Yn |] < ∞   let X0=0

Show that

  1. i) X n= Yi
  2. ii) Xn = (Yi )2 – nσ2

are martingales.                                                                                                 (10+5+5)

 

22) a) Derive the p.g.f of a branching process. Hence obtain the mean and variance of Xn.

 

  1. b) Let the offspring distribution be P[ζ= i] = 1/3 , i = 0,1,2

Obtain the probability of extinction.

 

 

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Loyola College M.Sc. Statistics Nov 2008 Statistical Computing – II Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

BA 26

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – November 2008

    ST 3810 – STATISTICAL COMPUTING – II

 

 

 

Date : 07-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

Answer ALL the questions:                                                                         ( 5 x 20 = 100 )

 

1).(a) Let X ~ N4    , compute

b). Two independent samples observation are drawn from a bivariate normal distribution with common population variance matrix. Test whether the two groups have the same population mean vector.

 

Group A

Age 55 58 59 60 62 65 68
Bp 120 125 130 100 105 120 116
Glucose 140 145 155 158 162 170 180

 

Group B

Age 59 62 58 57 56 69 65 62
Bp 100 126 95 100 105 110 115 120
Glucose 145 155 148 142 143 160 159 156

 

2). (a) Let X be normally distributed according as  N3 ( ,

with .

Find conditional distribution (X1 | X2   = 8, X3 = 5).

b). Find the maximum likelihood estimator of the 2 x  1   mean vector   and  2 x 2 covariance matrix  based on the random sample  from the bivariate normal population.

c). Income in excess of Rs. 2000 of people in a city is distributed as exponential  20 people were selected and their incomes are shown below

2200 3250 8000 8500 9500
2500 4500 6200 6000 8100
3000 7500 2100 7200 3700
2750 10000 9000 8600 97500

 

Obtain the point estimate of the expected income of a person in this city by maximum likelihood method. Obtain the estimate of its variance.

 

3) (a)  The biologist who studies the spiders was interested in comparing the lengths of female and male green, lynx spiders. Assume that the length X of the female spider is approximately distributed as       and the length Y of the male spider is approximately distributed as . Find an approximately 95 % confidence interval for () using 30 observation of X.

 

 

5.2 4.7 5.75 7.5 6.45 6.55 4.7 4.8 5.95 5.2
6.35 6.95 5.7 6.2 5.4 6.2 5.85 6.8 5.65 5.5
5.65 5.85 5.75 6.35 5.75 5.95 5.9 7 6.1 5.8

 

and the 30 observation of Y ,

 

8.25 9.95 5.9 6.55 8.45 7.55 9.8 10.9 6.6 7.55
8.1 9.1 6.1 9.3 8.75 7 7.8 8 9 6.3
8.35 8.7 8 7.5 9.5 8.3 7.05 8.3 7.95 9.6

Where measures are  in millimeters.

 

(b) .Given below are the qualities of 10 items ( in proper units) produced by two processors A and B.Test whether the variability of the quantity may be taken to be the same for the two processors

Processor A 33 37 35 36 35 34 34 35 33 33
Processor B 38 35 37 38 33 32 37 36 35 37

 

4). (a). For a 3 state Markov Chain with state {0, 1, 2,} and TPM

find the mean recurrence times.

 

  • From the following population of 10 clusters compare the following sampling designs for the estimation of the population total

(i)  Select 5 clusters by SRSWTR method

(ii) Draw an SRSWTR of 8 clusters and select a SRSWTR of size 2 from each

cluster and comment upon your results

 

C luster No.       Values    of the variates  
1 345 123 345 456
2 256 345 367 345
3 321 145 456 256
4 267 235 387 478
5 378 378 367 245
6 409 254 390 346
7 236 378 342 234
8 265 456 234 290
9 234 321 345 456
10 267 149 456 345

 

 

5) A sample survey was conducted with the aim of estimating the total yield of paddy. The area is divided into three strata and from each stratum, 4 plots are selected using SRSWTR. From the data given below,  calculate an estimate of the total yield along with an estimate of its variance.

 

Stratum No. Total No. of Plots Yield of Paddy for 4 Plots in the sample ( Kgs )
I 200 120 140 160 50
II 105 140 80 200 140
III 88 110 300 80 130

 

 

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Loyola College M.Sc. Statistics Nov 2008 Statistical Computing – I Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

BA 23

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – November 2008

    ST 1812 – STATISTICAL COMPUTING – I

 

 

 

Date : 13-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

  Answer any THREE questions. Each carries THIRTY FOUR marks.

 

  1. (a). Fit a distribution of the type

P(x) = (1/2) [P1(x) + P2(x)]

where P1(x) = (e –μ  μ x ) / x!  ; x = 0,1,2, …, μ  >0

 

and   P2(x) = (e –λ  λ x ) / x!  ; x = 0,1,2, …, λ >0

for the following data on the frequency of accidents during 106 weeks in

Chennai:

 

No. of accidents  :  0         1         2         3       4          5

 

No. of weeks       : 26       46       20        6       5          3 .

Also test the goodness of fit at 5% level of significance.

(18)

(b) In a population containing 539 live birds of same weight and age , the birds

were divided into 77 equal groups. They were then given a stimulus to increase

the growth rate . The following data gives the frequency distribution of those

with significant weight at the end or 6 weeks. Fit a truncated  binomial

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

 

No. of  birds :   1       2           3         4          5          6          7

Frequency    :               7      16         22       18         9          3          2

(16)

  1. (a). Find a g-inverse of the following matrix.

 

2         2          3          3          1

2         3          3          2          7

5         3          7          9          2

3         2          4          7          3

(17)

(b) Find the characteristic roots and vectors of the following matrix:

 

1         1        1

1        -1        2

0         1        1

( 17)

  1. (a) Generate a random sample of size 15 from Cauchy distribution with p.d.f.

f(w) = (1/π) [  λ / { λ 2 + (w – θ) 2  } ] , taking θ = 50 and  λ =5. 14)

 

 

 

 

(b)   Verify whether or not the following  matrix is positive definite:

 

12            4            -4

4          12              4

-4           4             20

(10)

(c)   Find the rank of the given matrix A by performing row operations:

 

3        2          3          1

4        3          5          2

2        1          1          0                                                                                                         (10)

 

  1. Consider the following data for a dependent variable representing repair time in hours and two independent variables representing months since last service and type of repair.

Customer             Months since              Type of Repair          Repair time

last service                                                      in hours

1                      2                                  1                                  2.9

2                      6                                  0                                  3.0

3                      8                                  1                                  4.8

4                      3                                  0                                  1.8

5                      2                                  1                                  2.9

6                      7                                  1                                  4.9

7                      9                                  0                                  4.2

8                      8                                  0                                  4.8

9                      4                                  1                                  4.4

10                    6                                  1                                  4.5

 

Using these data, develop an estimated regression equation relating repair time in hours to months

since last service and type of repair. Estimate the repair time if months since last service  = 12 and

type of repair = 1.

 

  1. The following table gives the annual return, the safety rating (0=riskiest, 10 = safest) and the annual expense ratio for 10 foreign funds (Mutual funds, March 2006).

 

Foreign Funds                   Safety Rating              Annual Expense          Annual

Ratio (%)                     Return (%)

1                                              7.1                   1.5                               49

2                                              7.2                   1.3                               52

3                                              6.8                   1.6                               89

4                                              7.1                   1.5                               58

5                                              6.2                   2.1                               131

6                                              7.4                   1.8                               59

7                                              6.5                   1.8                               99

8                                              7.0                   0.9                               53

9                                              6.9                   1.7                               77

10                                            7.7                   1.2                               61

  1. Use F-test to determine the overall significance of the relationship at 0.05 level of significance.
  2. Use t-test to determine the significance of each independent variable at 5 % level of significance.

 

 

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Loyola College M.Sc. Statistics Nov 2008 Measure And Probability Question Paper PDF Download

by

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

BA 20

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – November 2008

    ST 1809 – MEASURE AND PROBABILITY

 

 

 

Date : 06-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

SECTION A

Answer all questions.                                                                                    (10×2=20)

 

  1. Give the definition of a -field.
  2. Let. Find and.
  3. Let, be the power set of  and be a measure defined on. Define Verify whether is countably additive.
  4. Define Signed measure.
  5. Give an example for a -finite measure.
  6. Give the relation between Lebesgue – Steiltje’s measure and the distribution function and hence show that Lebesgue measure is a particular case of Lebesgue – Steiltje’s measure.
  7. Define Product measure.
  8. State Radon – Nikodym theorem.
  9. If  a.e., then show that  a.e..
  10. State Weak law of large numbers.

 

 

SECTION B

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

 

  1. Let be subsets of W. Show that for each ‘n’  .
  2. Show that every field is a -field but the converse need not be true.
  3. Explain the various ways of defining the integral of a borel measurable function.
  4. State and prove Fatou’s lemma.
  5. Show that.
  6. State and prove Holder’s inequality.
  7. Justify the following statement:

“Existence of the higher order moments implies the existence of the lower order moments but the converse need not be true”.

  1. State and prove monotone convergence theorem for conditional expectation given a -field.

 

 

 

SECTION C

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

 

  1. Let ‘h’ be a Borel measurable function such that exists. Define , . Show that is countably additive on. In particular if, then show that is a measure on.
  2. a.) Let ‘f ‘be a borel measurable function. If a.e., then show that.

b.) State and prove monotone class theorem.                                     (8+12)

  1. a.) State and prove Chebyshev’s inequality.

b.) A coin is tossed independently and indefinitely. Define the event A2n as in the 2nth toss equalization of head and tail occurs. Show that if the coin is biased, then the probability of A2n occurring infinitely often is zero and if the coin is unbiased, then the probability of A2n occurring infinitely often is one.

(8+12)

  1. a.) State and prove Kolmogorov’s strong law of large numbers.

b.) Show that almost sure convergence need not imply convergence in quadratic mean. Further, show that quadratic mean convergence need not imply almost sure convergence.                                                         (10+10)

 

 

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