M.Sc Statistics Question Paper

Loyola College M.Sc. Statistics April 2004 Design Of Experiments Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – APRIL 2004

ST 4800/S 1015 – DESIGN OF EXPERIMENTS

01.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

 

SECTION – A

           

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

 

  1. Define mutually orthogonal contrasts with an example.
  2. Briefly explain the term “Replication”.
  3. Define D – optimalilty.
  4. State the MINIMAL FUNCTION for 32 factorial design.
  5. Give sum of squares for 3 factors interaction for 33 factorial design.
  6. Define a ‘SIMPLE LATTICE SQUARE’.
  7. State any two applications of ‘FINITE FIELD’ in experimental design.
  8. Distinguish between an RBD and BIBD.
  9. Give an example for ‘SYMMETRIC BIBD’ which is NOT ‘RESOLVABLE’.
  10. Briefly explain the term CRITICAL DIFFERENCE.

 

SECTION – B

 

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

 

  1. Develop the Analysis of variance for a 32 Factorial design.
  2. Distinguish between ‘COMPLETE’ and ‘PARTIAL’ Confounding in LSD.
  3. Develop the Analysis of variance for a 24 Factorial Design, when the highest order interaction is confounded.
  4. State clearly the model used in YOUDEN SQUARE and describe the analysis of variance.
  5. Construct a BIBD with V = 3 b = 6    r = 4     k = 2    and = 2.
  6. Explain the term Repeated Latin Square design with suitable illustration.
  7. Distinguish between ‘FIXED EFFECT MODEL’ and ‘RANDOM EFFECT MODEL’ with an example.
  8. State and prove all the parametric conditions of a BIBD.

 

SECTION – C

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

 

  1. a) Briefly explain the term MUTUALLY ORTHOGONAL LATIN SQUARES

(M.O.L.S)                                                                                                                    (5)

  1. b) Construct M.O.L.S. when the number of treatments V = 7. (7)
  2. c) Hence construct a SERIES OF BIBDS using the above MOLS, when the block size

k = 4,  k = 5 and k = 6.                                                                                                             (8)

  1. a) Distinguish between ‘Confounding’ and ‘FRACTIONAL REPLICATION’. (8)
  2. b) Develop the analysis of variance for a 24 FRACTIONAL FACTORIAL design stating all the hypothesis, effects and conclusions.            (12)

 

  1. a) Discuss in detail confounding in ‘MORE THAN TWO BLOCKS’ using finite field.

(10)

  1. b) Develop the analysis of variance for a BIBD. (10)

 

  1. Write shot notes on the following:
  2. SPLIT – PLOT design
  3. PBIBD
  4. RESPONSE SURFACE DESIGN
  5. CONCOMITANT VARIABLE.             (4 ´ 5 = 20 marks)

 

 

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

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2004

ST 1800/S 715 – ANALYSIS

03.04.2004                                                                                                           Max:100 marks

9.00 – 12.00

 

SECTION – A

 

Answer ALL questions                                                                                (10 ´ 2 = 20 marks)

 

  1. Define a bijective function.
  2. Define a metric.
  3. Is the set (0,1) complete? How?
  4. Define the symbols Big O and small o.
  5. Let f(x) = 1   if    x is rational

 

0   if    x is irrational,   0 £  x £ 1

Is the function Riemann integrable over  [0,1]?

  1. Define lim inf and lim sup of a sequence xn.
  2. Define the linear derivative of a function f: X Rn;  where X  Rm.
  3. Find the double limit of xmn = and .
  4. Define uniform convergence of a sequence of functions.
  5. Let f(x,y) = be defined on R2 – {(0,0)}.  Show that  f(x,y) does not exist.

 

SECTION – B

 

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

 

  1. State and prove Cauchy’s Inequality.
  2. Show that R’ is complete.
  3. Show that any collection of open sets is open and any collection of closed sets is closed.
  4. State and prove Banach’s fixed point theorem.
  5. Let {fn} be a sequence of real functions integrable over the finite interval [a, b]. If fn ® f uniformly on [a, b] then show that i) f is integrable over [a, b] and  ii) .
  6. State and prove Weierstrass M-Test.
  7. Show that A is the upper limit of the sequence {xn} if and only if, given Î > 0

xn <    for all sufficiently large n

xn >    for infinitely many n

  1. Show that if f Î R [ g; a, b] then Î R [g; a,b] and .

If is R.S integrable, can you say f R.S. integrable?  Justify.                                (3+3+2)

 

 

SECTION – C

 

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

 

  1. a) State and prove Cauchy’s root test.
  2. b) Discuss the convergence of the infinite series whose nth terms are
  3. i)                                                                          (8+6+6)

 

  1. a) Define a compact metric space. Show that a compact set in a metric space is also

complete.                                                                                                                       (5)

  1. b) State and prove Heine – Borel theorem. (15)

 

  1. a) State and prove a necessary and sufficient condition that the function f is Riemann –

Stieltjes interable.

  1. b) If f is continuous then show that f Î R [g; a,b]
  2. c) If f1, f2 Î R [g; a,b] then show that f1 f2 Î R [g; a,b]                           (6+6+8]

 

  1. a) Let (X, r) and Y, s) be metric spaces. Show that the following condition is necessary

and sufficient for the function f: X ® Y to be continuous on X: whenever G is open in

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

  1. b) Let V,W be normed vector spaces. If the function f: V ® W is linear, then show that

the following three statements are equivalent.

  1. f is continuous on V
  2. There is a point at which f is continuous.
  • is bounded for x V – {0}.

 

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Loyola College M.Sc. Statistics April 2004 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 2004

ST 4951/S 1052 – ADVANCED OPERATIONS RESEARCH

12.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

SECTION – A

 

Answer ALL questions                                                                                (10 ´ 2 = 20 marks)

 

  1. What is the need for an integer programming problem?
  2. Define a covex function.
  3. Is the following quadratic form negative definite?

j (x1, x2) = –

  1. Is the function f(x) = x1 separable? x = (x1, x2).
  2. Define a quadratic programming problem.
  3. Explain the Markovian property of dynamic programming.
  4. When do you say the Khun-Tucker necessary conditions are also sufficient for a maximization problem?
  5. Explain the need for Goal programming.
  6. What is zero-one programming?
  7. When do we need Geometric programming problem?

 

SECTION – B

 

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

 

  1. Solve the following LPP by dynamic programming.

max z = 3x1 + 4x2

Subject to

2x1+x2 £ 40

2x1+5x2 £ 180

x1+x2 ≥ 0

  1. State and prove the necessary condition for a function of n variables to have a minimum. Also prove the sufficient condition.
  2. Derive the Gomery’s constraint for a mixed algorithm.
  3. Solve by Beale’s method

max Z = 2x1 +3x2 –

Subject to

x1 + 2x2 £ 4, x1, x2 ≥ 0.

  1. Solve by using Khun – Tucker conditions

max Z = 10x1 + 4x2 – 2

Subject to: 2x1 + x2 £ 5, x1, x2 ≥ 0

 

 

 

 

  1. Reduce the following separable programming problem to an approximate linear programming problem.

f(x1, x2) = 2x1 + 3

Subject to 4x1 + 2£ 16, x1, x2 ≥ 0

  1. Consider the chance constrained problem

max Z = 5x1 + 6x2 + 3x3

Subject to

Pr [a11 x1 + a12 x2 + a13 x3 £ 8] ≥ .95

Pr [5x1 + x2 + 6x3 £ b2] ≥ .1, xj ≥ 0 “j = 1,2,3

~ N

b2 ~ N (7,9).  Reduce this problem to a deterministic model.

  1. Solve

minimize f(x1, x2) = 3 + 2

Subject to

x1 + x2 = 7

x1, x2≥ 0

 

SECTION – C

 

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

 

  1. a) Solve the following all Integer programming problem

max  Z = x1 + 2x2

Subject to

x1 + x2 £ 7

2x1 £ 11

2x2 £ 7     x1, x2 ≥ 0, x1, x2 integers.

  1. b) Explain branch and bound method with an example.                                            (12+8)
  2. Solve by Wolfe’s method

max Z =  2x1 + 3

Subject to

x1 + 4 x2  £  4

x1 + x2 £ 2

x­1, x2 ≥ 0

  1. a) A student has to take examination in 3 courses A,B,C. He has 3 days available for the study.  He feels it would be best to devote a whole day to the study of the same course, so that he may study a course for one day, two days or three days or not at all.  His estimates of the grades he may get by study are as follows:-

Course  A      B    C

Days

0                  0      1     0

1                  1      1     1

2                  1      3     3

3                  3      4     3

How should he study so that he maximizes the sum of his grades?  Solve by Dynamic

Progrmming.

  1. b) Solve the following using dynamic programming

min Z =

Subject to

u1+u2+u3 ≥ 10,  u1, u2, u3 ≥ 0                                                       (15+5)

  1. a) Solve the following Geometric programming problem

f(X) = .

  1. b) Explain how will you solve if there is a constraint.        (15+5)

 

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Loyola College M.Sc. Statistics Nov 2004 Sampling Theory Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – NOVEMBER 2004

ST 1805/1802 – SAMPLING THEORY

25.10.2004                                                                                                           Max:100 marks

9.00 – 12.00 Noon

 

SECTION – A

 

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

 

  1. Define probability Sampling Design and explain the meaning of probability Sampling.
  2. Distinguish between varying size sampling design and fixed size sampling design. Give an example for each design.
  3. Define the following:
  4. a) Inclusion indicator
  5. First and Second order inclusion probabilities.
  6. Prove the following:
  7. Ep [Ii (s)] = pi ; i = 1, 2, …, N
  8. Ep [Ii (s) Ij (s)] = pij ; i, j = 1,2, …, N ; i ¹
  9. Show that an unbiased estimator for the population total can be found if an only if the first order inclusion probabilities are positive for all N units in the population.
  10. Derive the formula for pi and pij under Simple Random Sampling Design.
  11. Describe the Linear Systematic Sampling Scheme and write its probability sampling design.
  12. Derive the approximate bias of the ratio estimator for the population total Y.
  13. In cluster sampling, suggest an unbiased estimator for the population total. Write the variance of the unbiased estimator.
  14. Explain Multistage Sampling.

 

SECTION – B

 

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

 

  1. Show that an estimator can be unbiased under one design but biased under another design.

 

  1. For any sampling design, prove the following:

 

  1. Suggest an unit drawing mechanism for simple random sampling design and prove that the unit drawing mechanism implements the simple random sampling design.

 

  1. Explain Lahiri’s method of PPS sampling. Show that Lahiri’s method of selection is a PPS selection method.

 

  1. Write the reason for using Desraj ordered estimator instead of Horwitz – Thompson estimator under PPSWOR sampling scheme. Also prove that the Desraj ordered estimator is unbiased for the population total.
  2. Describe the Random Group Method of Sampling. Find an unbiased estimator of population total under this method and derive its variance.
  3. Compare V () and V () assuming the population values Yi satisfy

Yi = a + bi, i = 1, 2, …N.

[

  1. Explain Warner’s randomized response technique for estimating the proportion pA of the persons belonging to group A in a population.

 

SECTION – C

 

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

 

  1. a) Under any design P(×), derive the variance of Horwitz – Thompson estimator and find

its estimated variance.                                                                                                (16)

  1. b) Define Midzuno Sampling Design and state the unit drawing mechanism for this

design.                                                                                                                          (4)

 

  1. After the decision to take a simple random sample has been made, it was realized that the value of unit with level 1 would be unusually low and the value of unit with label N would be unusually high. In such cases, it is decided to use the estimator.

 

 

 

 

 

where C is a pre-determined constant.  Show that

  • is unbiased for  for any C.
  • Derive V ()
  • Find the value of C for which is more efficient than .

 

  1. a) Show that Linear Regression estimator is more efficient than Ratio estimator

unless b = R.                                                                                                                (4)

 

  1. b) Assuming samples are drawn using SRS in both the phases of double sampling,

suggest , and  when

  • the second phase sample is a subsample of the first phase sample.
  • the second phase sample is independent of the first phase sample. (16)

 

  1. a) In stratified sampling, deduct , V() and () assuming
  • SRS is used in all strata
  • PPSWR sampling is used in all strata.            (12)

 

  1. b) Obtain the variance of the following:
  • Hansen – Horwitz estimator in Double Sampling.
  • Estimator in Two – stage Sampling.                                            (8)

 

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Loyola College M.Sc. Statistics Nov 2004 Measure Theory Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – NOVEMBER 2004

ST 1902 – MEASURE THEORY

03.11.2004                                                                                                           Max:100 marks

9.00 – 12.00 Noon

 

SECTION – A

 

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

 

  1. Let {An, n ≥ 1} be a sequence of subsets of a set W. Show that lim inf An C lim sup An.
  2. Define minimal s – field.
  3. What is a set function.?
  4. Give an example of a counting measure.
  5. Show that any interval is a Borel set but Borel set need not be an interval.
  6. Define an Outer measure.
  7. Define Lebesgue – Stieltjes measure.
  8. Show that a composition of measurable functions is measurable.
  9. Define a simple function with an example.
  10. State Borel-Cantelli lemma.

 

SECTION – B

 

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

 

  1. If {Ai, i ≥ 1} is a sequence of subsets of a set W then show that

(Ai ).

 

  1. If D is a class of subsets of W and A C W, we denote D A the class {B A½B Î D}.  If the minimal s – field over D is    W (D), Show that    A (D  A) =     W  (D)

 

  1. Let 0 be a field of subsets of W.  Let P be a probability measure on    0.  Let {An, n ≥ 1} and {Bn, n ≥ 1} be two increasing sequences of sets such that .        Then show that

 

  1. State and establish monotone class theorem.

 

  1. If h and g are IB – measurable functions, then show that max {f, g} and min {f, g} are also IB – measurable functions.

 

  1. If m is a measure on (W, ) and A1, A2,… is a sequence of sets in    , Use Fatou’s lemma to show that
  2. m
  3. If m is finite, then show that m .

 

  1. Define absolute continuity of measures. Show that l < < m if and only if  < < m.

 

  1. State Radon – Nikodym theorem. Mention any two applications of this theorem to probability / statistics.

 

SECTION – C

 

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

 

  1. a) Let {xn} be a sequence of real numbers, and let An = (-¥, xn). What is the connection

between  sup xn and   Similarly what is the connection between

inf  xn and  inf An.

 

  1. Show that every s – field is a field. Is the converse true?                        (8+12)

 

  1. a) Let W be countably infinite set and let consist of all subsets of W.  Define

0       if A is finite

m (A) =     ¥     if A is infinite.

 

  1. Show that m is finitely additive but not countably additive.
  2. Show that W is the limit of an increasing sequence of sets An with

m (An) = 0 “n but m (W) = .

 

  1. b) Show that a s – field s is a monotone class but the converse is not true.            (7+7+6)

 

  1. a) State and establish Caratheodory extension theorem.

 

  1. b) If exists and C Î IR then show that = .                           (12+8)

 

  1. a) State and establish extended monotone convergence theorem.

 

  1. b) State and establish Jordan – Hahn Decomposition theorem. (10+10)

 

 

 

 

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Loyola College M.Sc. Statistics Nov 2004 Mathematical Statistics – I Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – NOVEMBER 2004

ST 3951 – MATHEMATICAL STATISTICS – I

02.11.2004                                                                                                           Max:100 marks

1.00 – 4.00 p.m.

 

SECTION – A

 

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

  1. Find C such that f (x) = C satisfies the conditions of being a pdf.
  2. Let a distribution function be given by

0                 x < 0

F(x) =        0 £ x < 1

1            x ≥  1

 

Find     i) Pr               ii) P [X = 0].

  1. Find the MGF of a random variable whose pdf is f (x) = , -1 < x < 2, zero elsewhere.
  2. If the MGF of a random variable is find  Pr [X = 2].
  3. Define convergence in probability.
  4. Find the mode of a distribution of a random variable with pdf

f (x) = 12x2 (1 – x), 0 < x < 1.

  1. Define a measure of skewness and kurtosis using the moments.
  2. If A and B are independent events, show that AC and BC are independent.
  3. Show that E (X) = for a random variable with values 0, 1, 2, 3…
  4. Define partial correlation.

 

 

SECTION – B

 

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

 

  1. Show that the distribution function is non-decreasing and right continuous.

 

  1. ‘n’ different letters are placed at random in ‘n’ different envelopes. Find the probability that none of the letters occupies the envelope corresponding to it.

 

  1. Show that correlation coefficient lies between -1 and 1. Also show that p2 = 1 is a necessary and sufficient condition for P [Y = a + bx] = 1 to hold.

 

  1. Derive the MGF of gamma distribution and obtain its mean and variance.

 

  1. Let f (x, y) = 2 0 < x < y < 1 the pdf of X and Y.  Obtain E [X | Y] and E [Y | X].  Also obtain the correlation coefficient between X and Y.

 

  1. Show that Binomial distribution tends to Poisson distribution under some conditions.

 

  1. State Chebyshw’s inequality. Prove Bernoulli’s weak law of large numbers.

 

  1. 4 distinct integers are chosen at random and without replacement from the first 10 positive integers. Let the random variable X be the next to the smallest of these 4 numbers.  Find the pdf of X.

 

 

SECTION – C

 

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

 

  1. a) Let {An} be a decreasing sequence of events. Show that

P .  Deduce the result for increasing sequence.

 

  1. b) A box contains M white and N – M red balls. A sample of size n is drawn from the

box.  Obtain the probability distribution of the number of white balls if the sampling is

done     i) with replacement     ii) without replacement.                                       (10+10)

 

  1. a) State any five properties of Normal distribution.

 

  1. In a distribution exactly Normal 7% are under 35 and 89% are under 63. What are the mean and standard deviation of the distribution?
  2. If X1 and X2 are independent N and Nrespectively, obtain the distribution of a1 X 1 + a2 X2.                                                                                  (5+10+5)

 

  1. a) Show that M (t1, t2) = M (t1, 0) M (0, t2) “, t1, t2 is a necessary and sufficient condition

for the independence of X1 and X2.

 

  1. b) Let X1 and X2 be independent r.v’s with

f1 (x1) =  ,  0 < x1 < ¥

f2 (x2) =  ,  0 < x2 < ¥

Obtain the joint pdf of Y1 = X1 + X2 and Y2 =

Also obtain the marginal distribution of Y1 and Y2

 

  1. c) Suppose E (XY) = E (X) E (Y).  Does it imply X and Y are independent.

(6+10+4)

 

  1. a) State and prove Lindberg-Levy central limit theorem.

 

  1. b) Let Fn (x) be distribution function of the r.v Xn, n = 1,2,3… Show that the

sequence{Xn} is convergent in probability to O if and only if the sequence Fn (x)

satisfies

 

=   0     x < 0

1    x ≥ 0

 

  1. c) Let Xn, n = 1, 2, … be independent Poisson random variables. Let y100 = X1 + X2 + …+

X100.  Find  Pr [190 £ Y100 £ 210].                                                                        (8+8+4)

 

 

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Loyola College M.Sc. Statistics Nov 2004 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 2004

ST 3803 – COMPUTATIONAL STATISTICS – III

02.11.2004                                                                                                           Max:100 marks

1.00 – 4.00 p.m.

 

SECTION – A

 

Answer any THREE questions without omitting any section .                   (3 ´ 34 = 102 marks)

 

  1. a) Use two phase method to solve

Max. z = 5x  – 2y + 3z

Subject .to

2x + 2y – z   ≥  2

3x – 4y         £  3

y + 3z <  5

x, y, z ≥ 0                                                        (17 marks)

 

  1. b) An airline that operates seven days a week between Delhi and Jaipur has the time-table

as shown below.  Crews must have a minimum layover of 5 hours between flights.

Obtain the pairing of flights that minimizes layover time away from home.  Note that

crews flying from A to B and back can be based either at A or at B.  For any given

pairing, he crew will be based at the city that results in smaller layover:

 

Flight No. Departure Arrival Flight No. Departure Arrival
1 7.00 a.m. 8.00 a.m 101 8.00 a.m. 9.15 a.m
2 8.00 a.m. 9.00 a.m 102 8.30 a.m. 9.45 a.m
3 1.30 p.m. 2.30 p.m 103 12.00 noon 1.15 p.m
4 6.30 p.m 7.30 p.m 104 5.30 p.m 6.45 p.m

 

(17 marks)

  1. a) Solve the following unbalanced transportation problem:

 

To

1      2      3   Supply

From

Demand          75    20   50                                                                  (17 marks)

 

  1. b)  Consider the inventory problem with three items.  The parameters of the problem are

shown in the table.

 

Item Ki bI hi ai
1 Rs.500/- 2 units Rs.150/- 1 ft2
2 Rs.250/- 4 units Rs.  50/- 1 ft2
3 Rs.750/- 4 units Rs.100/- 1 ft2

 

Assume that the total available storage area is given by A = 20ft2.  Find the economic

order quantities for each item and determine the optimal inventory cost.      (17 marks)

 

SECTION – B

 

 

  1. a) Suppose the one step transition probability matrix is as given below:

Find i) p00(2)         ii) f00(n)          iii) f13(n)          and      iv) f33(n).

 

 

.

(17 marks)

 

  1. For a three state Markov chain with states {0,1,2} and transition probability matrix

 

Find the mean recurrence times of states 0, 1, 2.                                (17 marks)

 

  1. a) An infinite Markov chain on the set of non-negative integers has the transition function

as follows:

pk0 = (k+1) /(k+2)        and  pk,k+1 1/(k+2)

 

  1. Find whether the chain is positive recurrent, null recurrent or transient.
  2. Find the stationary distribution, incase its exists. (17 marks)

 

  1. b) Consider a birth and death process three states 0, 1 and 2, birth and death rates such

that m2 = l0.  Using the forward equation, find p0y (t), y = 0,1,2.                   (17 marks)

 

SECTION – C

 

 

  1. a) From the following data test whether the number of cycles to failure of batteries is

significantly related to the charge rate and the depth of discharge using multiple

correlation coefficient at 5% level of significance.

 

X1

No. of cycles to failure

 X2

Charge rate in (amps)

 X3

Depth of discharge
101 0.375 60.0
141 1.000 76.8
  96 1.000 60.0
125 1.000 43.2
  43 1.625 60.0
  16 1.625 76.8
188 1.00 100.0
  10 0.375 76.8
386 1.00 43.2
160 1.625 76.8
216 1.00 70.0
170 0.375 60.0

(20 marks)

 

  1. For the above data given in 5a Test for the significance population partial correlation

coefficient between X1 and X2.                                                                          (14 marks)

 

  1. The stiffness and bending strengths of two grades of Lumber are given below:

 

                 I grade                II grade
    Stiffness  Bending strength   Stiffness Bending strength
1,232 4,175 1,712 7,749
1,115 6,652 1,932 6,818
2,205 7,612 1,820 9,307
1,897 10,914 1,900 6,457
1,932 10,850 2,426 10,102
1,612 7,625 1,558 7,414
1,598 6,954 1,470 7,556
1,804 8,365 1,858 7,833
1,752 9,469 1,587 8,309
2,067 6,410 2,208 9,559

 

Test whether there is significant difference between the two grades at 5% level of

significance, by testing the equality of mean vectors.  State your assumptions.

(34 marks)

 

 

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Loyola College M.Sc. Statistics April 2006 Testing Statistical Hypothesis Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 34

SECOND SEMESTER – APRIL 2006

                                         ST 2809 – TESTING STATISTICAL HYPOTHESIS

(Also equivalent to ST 2807/2802)

 

 

Date & Time : 21-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

SECTION  A                           Answer all the questions                            10 x 2 = 20

  1. Define test function and randomized test function.
  2. Let X be B(1, q), q = 0.2,0.4,0.5. For testing H: q = 0.2,0.4 Vs K: q = 0.5, a test is given by

f(x)   =  0.3,    x = 0

=  0.6,    x =  1.

Find the size of the test.

  1. Show that a UMP level a test is unbiased.
  2. Define MLR property and give an example.
  3. Show that a test with Neyman structure is similar.
  4. Describe Type I and Type II right censoring.
  5. Give two examples for multiparameter exponential family.
  6. Define location family and give an example.
  7. Describe likelihood ratio test.
  8. Explain UMA and UMAU confidence sets.

 

SECTION B                            Answer any five questions                           5 x 8 = 40

  1. Let X be DU{1,2,…, q }, q = 1,2. For testing H: q = 1 Vs K: q = 2,  find MP level a test using LP technique.
  2. Give an example of a testing problem for which UMP test does not exist.
  3. Given a random sample of size n from E(0, q ), q > 0, derive UMP level a test for testing H: q £ q 0 Vs K: q > q 0.Examine whether the test is consistent.
  4. If the power function of an unbiased test is continuous, show that the test is similar.

15.Given a random sample of size n from P( q ), q > 0, derive UMPU level a test for testing H: q = q 0 Vs K: q ¹ q 0.

16.Show that a statistic is invariant if and only if it is a function of a maximal invariant statistic.

17.Derive likelihood ratio test for testing H: q = q 0 Vs K: q > q 0 based on a random sample from E(0,q), q >0.

18.Explain shortest length confidence interval and illustrate with an example.

 

 

 

SECTION C                           Answer any two questions                         2 x 20 = 40

19 a).   State and establish the sufficient part of Neyman-Pearson lemma.

  1. b) Let X1,X2,…Xn denote a random sample of size n from E(q ,1), q e Examine if there exists UMP level a test for testing H: q = q 0 Vs K: q ¹ q 0.

20 a)  In the case of one-parameter  exponential family show that there exists UMP level a  test for testing one-sided hypothesis against one-sided alternative. State your assumptions.

  1. b) Derive UMPU level a test for testing H:  q1 £ q £ q2 Vs K: q < q1  or q > q                               based on a random sample from N(q , 1), q e R. Explain the determination of the constants.Is the test unique?

21 a)   Discuss the relation between similar tests and tests with Neyman structure.

  1. b) Let X1,X2,…Xn  be a random sample from P( ) and Y1,Y2,…Ym be a random sample from an independent  Poisson population P( ).Derive UMPU level a test for testing H:l £ m  Vs K:l > m. Determine the constants when  n = 2 and m = 1, X1 = 1, X2 = 2 and Y1 = 3.

22 a)   State and establish the asymptotic null distribution of the likelihood ratio statistic.

  1. b) For testing H:(X1 , X2 ) is BVN(q, q ,1,1, 0.5) Vs K: (X1 , X2 ) is BVN(q, q,1, 4, 0.5), derive UMPI level a test with respect to location transformations.

 

 

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Loyola College M.Sc. Statistics April 2006 Stochastic Processes Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – APRIL 2006

                                                   ST 3806 – STOCHASTIC PROCESSES

 

 

Date & Time : 17-04-2006/AFTERNOON   Dept. No.                                                       Max. : 100 Marks

 

 

SECTION  A                           Answer all the questions                           10 ´ 2 = 20

 

  1. Define stationary independent increment process.
  2. Show that the square of a stochastic matrix is stochastic.
  3. Suppose the one-step tpm is an identitiy matrix, show that the states are all recurrent.
  4. Find a stationary disribution of an MC with one-step tpm

 

P =       0.3    0.7

0.7    0.3

  1. For a Poisson process, find the covariance function.
  2. Describe Pure birth process.
  3. Define excess life and current life of a Renewal process.
  4. For a martingale { Xn , n = 0.1.2,…}, show that E(Xn) = E(Xn +1), n = 0,1,2,…
  5. Describe a branching process.
  6. Define a covariance stationary process and give an example.

 

SECTION  B                               Answer any five questions                       5 ´ 8 = 40

 

  1. For a stationary independent increment process, show that the variance of the marginal distribution is linear in the time parameter.
  2. Define periodicity and show that it is a class property.
  3. If the one step tpm of an irreducible finite state Markov Chain is symmetric, show that the stationary distribution is uniform.
  4. Describe a Poisson Process and derive its marginal distribution.
  5. For a linear growth process with immigration, find the average size of the population if the initial population is i units.
  6. Derive the generating function relations satisfied by a Branching process.
  7. If { X(t), t ³ 0} is a Brownian motion process, show that the distribution of (X(t1),X(t2)) is bivariate normal.
  8. If the interoccurrence distribution of a Renewal process is exponential, find the distributions of i ) current life and ii ) excess life.

 

 

 

 

 

 

SECTION  C                              Answer any two questions                      2 ´ 20 = 40

 

19 a). State and establish Chapman-Kolmogorov equations satisfied by a Markov Chain.

b).   Illustrate Basic limit theorem with an example.

20 a). Describe Birth-Death process. Derive Kolmogorov backward equations satisfied by the Birth-Death process.

  1. b) Describe telephone trunking model and find its stationary distribution.

 

21 .a) State and prove Elementary renewal theorem in Renewal theory.

  1. b) Find the renewal function associated with a renewal process having the interoccurrence distribution with pdf

 

f(x) = l2 x exp(-l x), x > 0, l > 0.

 

22 a) Let { Xn , n = 0.1.2,…}be a Branching process with X0 = 1. Find the mean and variance of  Xn in terms of those of the offspring distribution.

  1. b) Let { Xn , n = 0.1.2,…}be a covariance stationary process with zero mean and the covariance function Rx(v) . Find the best predictor of Xn+1 of the form aXn, where a is a real constant.

 

 

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Loyola College M.Sc. Statistics April 2006 Statistics For Competitive Examinations Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 49

FOURTH SEMESTER – APRIL 2006

                               ST 4804 – STATISTICS FOR COMPETITIVE EXAMINATIONS

 

 

Date & Time : 25-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

SECTION A

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

 

  1. If the difference between two numbers is 1.2, then the variance of them is

(A) 0.18          (B) 1.44        (C) 0.72       (D) 0.36

  1. To test the hypothesis that the variance of a normal distribution is 2, the test procedure used is

(A) Normal test        (B) Chi-square test    (C) F-test   (D) t-test

  1. To test which one of the following hypothesis, F-test is used?

(A) Goodness of fit (B) equality of means of two normal populations

(C)  Significance of correlation coefficient (D) Equality of Variances of two normal populations

  1. If X has Poisson distribution with 3P[X = 2] = 2P[X ≤ 1], then the expected value of X is

(A) 3               (B) -2/3           (C) 2           (D) 3/2

  1. X1, X2 and X3 are independent observations on a normal random variable with

mean μ and variance σ2.What is the efficiency of   (3X1+2X2+X3 ) / 6 as an estimator of  μ ?

(A) 6/7             (B) 1           (C) 1/3           (D) 1/12

  1. If E(Y/X) = α X + β and X has standard normal distribution, then E(Y) is

(A) 0               (B) 1                (C) β          (D) α

∞

  1. If P ( An) = 1, n=1, 2, 3… then the value of P (An) is

n=1

(A) 0               (B) 1                (C) 1/2       (D) ¼

  1. A random variable X has characteristic function

Φ (t) = (sin t)/t,    t ≠0

1          otherwise

Then, Var(X) is equal to

(A) 1        (B) 0       (C) 1/6        (D) 1/3

  1. If X1 and X2 are independent and identically distributed random variables with p(x) = qx.p, x = 0, 1, 2, 3… and (p+q) = 1, then the distribution of (X1+X2) is

(A) Geometric        (B) degenerate          (C) Negative Binomial    (D) Hyper- geometric

 

  1. In a random sample of size n from the distribution

dF (x) = e-x.dx,   0<x<∞,

the mean of the smallest sample value is

(A) 1/n      (B) 1/n2      (C) 0       (D) 1

  1. If the degrees of freedom for error in the analysis of Variance for a Latin square design is 30, the number of treatments is

(A) 5               (B) 6           (C) 7           (D) not possible to determine

  1. If a population consists of 10 units and the population Variance is 20, the Variance of the sample mean of a simple random sample pf size 4 without replacement is

(A) 5               (B) 2           (C) 20           (D) 3

  1. The number of simple random samples of size 4 that can be drawn without replacement from a population with 12 units is

(A) 124           (B) 495        (C) 11,880        (D) 48

  1. The standard deviation of a symmetric distribution is 4. The value of the fourth moment about the mean in order that the distribution be leptokurtic is

(A) greater than 768              (B) equal to 768           (C) equal to 256       (D) less than 48

  1. Given Maximize subject to

 

 

 

For what values of the above problem will have several optimum solutions?

(A) 2    (B) 3    (C) 6    (D) 1

 

  1. The objective function in the phase-I (when we use two phase simplex method) is formed by
  • summing all the variables
  • summing all the artificial variables

(C)   taking the product of artificial variables

  • subtracting the sum of artificial variables from the sum of other variables

 

  1. The following set of constraints require x artificial variables

 

 

where x is

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

 

  1. Given the following simplex table (associated with a maximization problem)

 

Basic   z          x1        x2        x3        x4        Solution

 

z          1          -4         -2         0          0          8

 

x3        0          4          3          1          0          1

 

x4        0          -1         1          0          1          2

 

The leaving and entering variables are

(A) x1, x3        (B) x1, x4        (C) x2,x3         (D) x2,x4

 

  1. An LPP has 4 variables and 2 constraints. How many sets of basic variables are possible?

(A) 10        (B) 6                (C) 3                (D) 20

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

(A) Strictly increasing in      (B) Strictly decreasing

(C) Periodic in                                 (D) can’t say

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

alternative in

(A)               (B)

(C)                (D)

  1. Choose the correct statement

(A) Power functions of UMPTs are always monotone

(B) A UMPT is always UMPUT

(C) MPT’s are not unique

(É)All similar tests will have Neyman structure

  1. Choose the correct statement

(A) RR methods are not associated with sensitive attributes

(B)Yates Grundy estimator is non-negative under Midzuno scheme

(C) HTE can not be used under PPSWOR

(D)Balanced systematic sampling is not recommended for populations with linear

trend.

  1. Lahiri’s method

(A) is a PPS selection method involving a given number of attempts

(B) is a PPS selection method involving unknown number of attempts

(C) is an equal probability selection method involving a given number of attempts

(D) is an equal probability selection method involving unknown number of

attempts

 

  1. Ratio estimator is

(A) a particular case of regression estimator

(B) an unbiased estimator

(C)more suitable when y and x have high negative correlation

(D) more suitable when y and x have no correlation

 

  1. Random group method is due to

(A) Desraj       (B) Murthy      (C) Hartley-Ross         (D) Rao-Hartley-Cochran

 

  1. Randomised response methods are meant for

(A) homogeneous data

(B) heterogeneous data

(C) sensitive data

(D) stratified populations

 

  1. Which name is associated with shortest route problems

(A) Kuhn-Tucker

(B) Floyd

(C) Charnes

(D) Karmakar

 

  1. Which of the following functions is NOT continuous at 0?

(A) |x|              (B) ex               (C) x – [x]                   (D)Sin x

 

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

a = Probability of getting at least two heads

b = Probability that the sum of the numbers that show up is less than 6

Then

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

 

31.The system of equations

2x + 4y – z = 3

x + 2y +2z = 2

x + (m2+1) y + 7z = 4m – 1

has infinitely many solutions if m equals

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

 

  1. The mean and variance of 8 items are 10 and 100 respectively. An observation 3

is deleted from the data. The variance of the remaining 7 observations is

(A)100             (B)106             (C)112             (D)120

 

  1. Let T: R3 ® R2 be defined as T(x, y, z) = (3x + y – z , x + 5z). The matrix

corresponding to this linear transformation is

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

 

  1. Which of the following is NOT true of a normal variable with mean 0?

(A) E(X2) = 1, E(X3) = 0                        (B) E(X2) = 1, E(X4) = 2

(C) E(X2) = 2, E(X4) = 12                      (D) E(X2) = 1/2 , E(X4) = 3/4

 

  1. If X and Y are uncorrelated random variables with equal means and variances,

then

(A) X + Y and X – Y are identically distributed

(B) X + Y and X – Y are independent

(C) X + Y and X – Y are negatively correlated

(D) X + Y and X – Y have equal variance

 

  1. 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 –0.3. Then , the correlation coefficient for the transformed data {(Ui, Vi), i =1, 2, …,n}, where Ui = 3 – 5 Xi2 and Vi = 2Yi –3  is

(A) –0.3           (B) 0.3             (C) 0                (D) cannot be determined

 

  1. If X1, X2,…, Xn is a random sample from U( 0, q), which of the following is a biased estimator of q?

(A) 2           (B) X(n)                        (C)X1 + Xn      (D) (n+1)X(n) / n

 

  1. The Cramer-Rao lower bound for estimating the parameter l of a Poisson distribution based on a random sample of size n is

(A) l               (B) nl              (C) l / n           (D) l1/2

 

  1. The lower control limit of a c-chart is 4. The upper control limit is

(A)16               (B)20               (C)24               (D) none of these

 

  1. In a 24 factorial experiment with 4 blocks the degree of freedom for Error Sum of Squares is

(A) 25              (B)35               (C)45               (D)55

SECTION  B

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

 

  1. Explain the procedure for solving a game theory problem graphically

 

  1. Show that family of Uniform densities binomial densities has

MLR in

  1. A sample has two strata with relative sizes and . He believes

that . For a given cost , show that (assuming stratum

sizes are large)

 

 

 

  1. The exponent of a bivariate normal density is given below:

– ⅔(x2+9y2-13x-3xy+60y+103)

Find μ1, μ2, σ1, σ2 and ρ.

 

  1. The number of accidents in a town follows a Poisson process with the mean of 2

accidents per day and the number of people involved in ith accident has the

distribution

P[X1=k] = 1/ 2k, k≥1.

Find the mean and variance of the number of people involved in accidents per

week.

 

  1. If Φ is a characteristic function, show that e λ (Φ -1) is a characteristic function for

all λ>0.

 

  1. (a)Let X­1, …,Xn, Xn+1 be a random sample from N(m, s2). Let M be the average of the first ‘n’ observations and S2 be the unbiased estimator of the population variance based on the first ‘n’ observations. Find the constant ‘k’ so that the statistic k( M – Xn+1) /S follows a t- distribution.

 

(b) Let X have a Poisson distribution with parameter q. Assume that the

unknown q is a value of a random variable which follows Gamma distribution

with parameters a = a / ( 1- a) and  p = r, where ‘r’ is a positive integer. Show

that the marginal distribution of X is Negative Binomial.                          (5 + 5)

 

  1. (a) Let X1,…,Xn be a random sample from Poisson distribution with parameter l. Starting with the initial estimator X12 – X1­ for l2, use Rao-Blackwellization to get an improved estimator by conditioning on the sufficient statistic S Xi. State whether the resulting estimator is UMVUE and justify.

(b) Let X1, …, Xn be a random sample from N(m, s2). Obtain an unbiased and

consistent estimator of s4.                                                                           (6 + 4)

 

  1. (a) Derive an expression for E(Mean Treatment Sum of Squares) in LSD.

(b) Consider four quantities T1, …,T4 and let T1 – 2T2 + T3 be a contrast. Find

two other contrasts so that all the three are mutually orthogonal.              (7 + 3)

 

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Loyola College M.Sc. Statistics April 2006 Statistical Process Control Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 48

FOURTH SEMESTER – APRIL 2006

                                           ST 4803 – STATISTICAL PROCESS CONTROL

 

 

Date & Time : 22-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

SECTION A  

Answer all the questions                                                                             10 x 2 = 20

  1. Discuss the statistical basis underlying the general use of 3 – sigma limits on control charts.
  2. Define rational subgroup concept.
  3. How is lack of control of a process determined by using control chart techniques?
  4. What is process capability ratio (PCR)?
  5. Why is the np chart not appropriate with variable sample size?
  6. Explain an attribute single sampling plan.
  7. What purpose does an OC curve serve?
  8. Define AOQ.
  9. Define a). Specification limit. b). Natural tolerance limit.
  10. Explain the concept of TQM.

SECTION B  

Answer any five questions                                                              5 x 8= 40

  1. What are the dimensions of quality? Explain.
  2. 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.

  1. A control chart for the fraction non-conforming is to be established using a CL of p = 0.10. What sample size is required if we wish to detect a shift in the process fraction non-conforming to 0.20 with probability 0.50?
  2. Explain the method of constructing control limits for X – bar and R charts when the sample sizes are different for various subgroups.
  3. In designing a fraction non-conforming chart with CL at p =0.20 and 3-sigma control limits, what is the simple size required to yield a positive LCL? What is the value of n necessary to give a probability of .50 of detecting a shift in the process to 0.26?
  4. Estimate process capability using X – bar and 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
  1. Find a single sampling plan for which p1 = 0.05, a = 0.05 p2 = 0.15 and b = 0.10.
  2. What are chain sampling and skip-lot sampling plans?

 

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 are the limits obtained for these charts.
  2. 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)
  3. a) Write a detailed note on the moving average control chart.
  4. b) What are modified control charts?. Explain the method of obtaining control limits for modified control charts.                                                                                                         (8+12)
  5. a) Outline the procedure of constructing V-mask.
  6. b) What is Exponentially Weighted Moving Average control chart?.           (15+5)
  7. a) Write a detailed note on six-sigma quality.
  8. b) Explain with an illustration the method of obtaining the probability of acceptance for a triple sampling plan.                                                                                                               (10 + 10)

 

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Loyola College M.Sc. Statistics April 2006 Statistical Computing – I Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 29

FIRST SEMESTER – APRIL 2006

                                                 ST 1812 – STATISTICAL COMPUTING – I

 

 

Date & Time : 22-04-2006/1.00-4.00 P.M.   Dept. No.                                                       Max. : 100 Marks

 

 

 

Answer any THREE questions

 

  1. a) Find a G- inverse of the matrix

A =

 

  1. b) Check whether the following vectors are linearly dependent:
  2. i) X¢  = (1, -1, 2),       Y¢ = (2, 0, -1),             Z¢ = (0, -2, -5).
  3. ii) X¢ = (1, 0,0), Y¢ = (0, 1, 0), Z¢ = (0, 0, 1).                            (20+14)

 

  1. a) The following data relates to the results of an experiment the relative frequencies for 4 different types of genes are expected to be and
    where 0 < < 1.

The frequencies observed were 508, 432, 397 and 518 respectively.  Estimate

the parameter q by the method of maximum likelihood and find the estimate of

the  standard error of the estimator.

 

  1. b) The scores of 17 students are given by the following table. Assuming that this

is a sample from normal population whose variance is s2, obtain

  1. a 95% confidence interval for s
  2. a 99% confidence interval for s

 

Scores:

(Out of 100)         45     65     68     77     95     69     56         72        75

38     68     72     65     42     66     55         62

(14+20)

 

  1. a) Below are given two random samples drawn from different normal populations:

Sample 1:    10        6          16        17     13     12     8       14     15        9

Sample 2:    7          13        22        15     12     14     18     8       21        23 10

 

Obtain a 99% confidences limits for the difference of means of the 2 populations.

 

  1. b) Fit a normal distribution to the following data

 

C.I: 60-65 65-70 70-75 75-80 80-85 85-90 90-95 95-100
Frequency 3 21 150 335 325 135 26 4

(20+14)

 

  1. a) Fit a multiple regression model of Y on X1 and X2 for the following data. Estimate Y when X1 = 1350 sq.ft and X2 = 2 years                                      (20)

 

  1. b) Also, test the significance of the population multiple correlation coefficient at

5% level of significance.                                                                                (14)

 

FLAT PRIZE IN LAKHS

(Y)

 FLAT SIZE IN SQ.FT

(X1)

 AGE OF THE FLAT IN YEARS        (X2)
12.3 1050 1
15 1200 1
14.8 1180 3
11 950 2
10.3 900 3
16.9 1300 3
18 1400 3
6 450 4
5.2 480 5
4.6 420 4
18 1450 6
9.3 850 3
12.2 1020 7

 

  1. a) Use Step-wise regression analysis to identify the most significant independent

variable(s) and comment on your finding regarding the significance of

population regression coefficients for the data given in question-4                (20)

  1. b) Compute the condition index for the data in question-4 and examine whether

the multi-co linearity problem is present in the data or not.                          (14)

 

 

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Loyola College M.Sc. Statistics April 2006 Sampling Theory Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – APRIL 2006

                                                          ST 2810 – SAMPLING THEORY

 

 

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

 

 

Section A  (10 x 2 =20)

 

Answer ALL the questions. Each carries TWO marks.

  1. Define : Midzuno sampling design
  2. State the identity which relates sample size of a sampling design with its first order inclusion probabilities
  3. Give the formula for unbiased estimator of under Warner’s RR
  4. Define balanced systematic sampling
  5. Mention the situations in which product and ratio estimators can be used instead of .
  6. When do you recommend “Two phase sampling”?
  7. Name an estimator which uses selection probabilities.
  8. Give any two limitations of “Linear Systematic Sampling “
  9. Name any one sampling-estimating strategy in which no unbiased estimator for variance of estimator can be found.
  10. Write the variance of Yates corrected estimator under LSS when there is a linear trend in the population

Section B (5 x 8 = 40)

 

Answer any Five. Each carries Eight  marks.

  1. Prove the following identities : and verify the same in the case of following sampling design

 

 

  1. From a population containing N units a sample of n units is drawn using SRS and from the drawn sample a subsample of n’ units. Suggest an unbiased estimator for the population total based on the subsample and obtain its variance
  2. Describe modified systematic sampling and show that under the model
  3. Describe Desraj ordered estimator and obtain an unbiased estimator of

 

  1. Explain proportional allocations (1) for a given cost (2) for a given sample size. Derive the variance of under the above cases assuming simple random sampling is used in all strata.

 

  1. Explain Warner’s randomized response model in detail.

 

  1. Define product estimator . Obtain an expression for its bias under simple random sampling and hence develop an unbiased estimator for the population total.
  2. Derive the approximate mean square error of estimators in the class also obtain the minimum mean square error in the class.

 

Section C  (2 x 20 =40)

 

Answer any TWO. Each carries TWENTY marks

 

  1. Define : Horvitz-Thompson estimator. Show that it is unbiased for the population total and derive its variance in Yates-Grundy form

 

  1. Derive the first and second order inclusion probabilities under Midzuno sampling scheme and show that under this design the Yates-Grundy estimator is non-negative

 

  1. Develop Yates corrected estimator under Linear Systematic Samping

 

  1. Develop Hartley-Ross ratio type unbiased estimator under simple random sampling.

 

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Loyola College M.Sc. Statistics April 2006 Reliability Theory Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 53

FOURTH SEMESTER – APRIL 2006

                                                        ST 4955 – RELIABILITY THEORY

 

 

Date & Time : 29-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

Section-A (10×2=20 marks)

Answer ALL the questions. Each question carries TWO marks.

  1. Define the terms: (a) Reliability function R(t)

(b) Hazard function r(t)

  1. In the usual notation, show that MTBF = R*(0)
  2. If n components functioning independently and having equal reliabilities are operating in parallel, find the reliability of the entire system.
  3. Comment on the following statement: Series and parallel systems are particular cases of an (m, n) system.
  4. What are (a) parallel-series and (b) series-parallel systems?
  5. Define a coherent structure and give two examples.
  6. Define (a) Minimal path vector (b) Minimal cut vector.
  7. What do you mean by (i) the number of critical path vectors of component i and (ii) relative importance of component i?
  8. Give an example of a set of random variables that are not associated.
  9. What is a cumulative damage shock model?

Section-B (5×8=40 marks)

Answer any FIVE questions

  1. Obtain the reliability function, hazard rate and the system MTBF for Weibull   distribution with the parameters λ and α.
  2. Suppose that gi(t) is the density function for Ti, the time to failure of ith component in a standby system with three independent and identical components and is given by gi(t) = λ e-λt, i = 1, 2, 3; t>0. Obtain the system failure time density function and hence find its expected value.
  3. What is a series system? Obtain the system failure time density function for a series system with n independent components. Suppose each of the n independent components has an exponential failure time distribution with constant failure rate λi, i = 1, 2, 3, …, n. Find the system reliability.
  4. Let Φ be a coherent structure. Show that

Φ(x Ц y) ≥ Φ(x) Ц Φ(y)

Further, show that the equality holds for all x and y if and only if the structure is parallel.

  1. Given the structure Φ, define the dual of the structure Φ. Also, show that the minimal path sets for Φ are the minimal cut sets for ΦD.
  2. Consider a coherent system with three components having the structure function Φ(x1, x2, x3) = x1. (x2 Ц x3)

Determine the number of critical path vectors of each component. Also determine       the relative importance of each component. Are components 2 and 3 equally important?

  1. When do you say that a set of random variables T1, T2,… , Tn are associated? Show that a set consisting of a single random variable is associated.
  2. Let the density of exist. Show that F is DFR if and only if r(t) is decreasing       in t.

Section-C (2×20 = 40 marks)

Answer any TWO questions. Each carries TWENTY marks

19.a. What is a series- parallel system of order (m, n)? Write down the system reliability and system failure rate of the same.                          (10 marks)

  1. Assuming that the components have identical constant failure rate λ, obtain MTBF of the series- parallel system. (10 marks)

20.a. Define the terms. (i) System availability.

(ii) Steady state availability.                                (4 marks)

  1. A system consists of a single unit, whose lifetime X and repair time Y are independent random variables with probability density functions f (.) and g (.) respectively. Assume that initially at time zero, the unit just begins to operate. Determine the reliability, availability and steady state availability of the system.                                                                                                  (4+6+6 marks)

21.a. Let Φ be a coherent structure. Show that

Φ(x .y) ≤ Φ(x) .Φ(y)

Also, show that the equality holds for all x and  y if and only if the structure is series.                                                                                      (10 marks)

  1. Let h be the reliability function of a coherent system. Show that h (p Ц p’) ≥ h (p) Ц h (p’) for all 0 ≤ p, p’ ≤ 1                                     (10 marks)

22.a.  Show that the order statistics Y1:n, Y2::n,…,Yn:n corresponding to n independent random variables are associated.                                                    (10 marks)

  1. Examine whether Gamma distribution G (λ, α) is IFR or DFR. (10 marks)

 

 

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Loyola College M.Sc. Statistics April 2006 Probability Theory Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 31

SECOND SEMESTER – APRIL 2006

                                                       ST 2805 – PROBABILITY THEORY

(Also equivalent to ST 2800)

 

 

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

 

 

Section-A

Answer ALL questions                                                                   (10 ´ 2 = 20 marks)

  1. With reference to tossing a regular die once and noting the outcome, identify completely all the elements of the probability space (Ω, A, P).
  2. Show that the limit of any convergent sequences of events is an event.
  3. Let F(x) = P [X < x], x Є R. Prove that F (.) is continuous to the left.
  4. Write down any two properties of the distribution function of a random vector (X, Y).
  5. If X is a random variable with P[X = (-1) k2k] = 1/2k, k = 1, 2, 3…, examine whether E[X] exists.
  6. If X2 and Y2 are independent, are X and Y independent?
  7. Define almost sure convergence and convergence in probability for a sequence of random variables.
  8. If Φ is the characteristic function (CF) of a random variable X, find the CF of (2X+3).
  9. Let {Xn, n = 1, 2, …} be a sequence of independent and identically distributed (iid) n

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

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

  1. State Lindeberg – Feller Central limit theorem.

 

Section – B

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

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

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

=   0      otherwise

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

 

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

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

Define Xn =   (n+1) if X=n

=    n     if X = (n+1)

=    X    otherwise

Show that Xn converges to X in quadratic mean.

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

Section-C

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

  1. a. Define the probability distribution of a random variable. Show that the probability distribution of a random variable is determined by its distribution function.(8 marks)
  2. Show that the vector X = (X1, X2, …, Xp) is a random vector if and only if Xj,

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

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

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

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

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

Σ Xn of independent random variables.         (12 marks)

n=1

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

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

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

(12 marks)

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

P [Xk = kλ]  = P [Xk = -kλ] = 1/2, k = 1, 2, 3… Show that central limit theorem holds for

λ ≥ -1/2.                                                                                                  (8 marks)

 

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Loyola College M.Sc. Statistics April 2006 Measure And Probability Theory Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 26

FIRST SEMESTER – APRIL 2006

                                       ST 1809 – MEASURE AND PROBABILITY THEORY

 

 

Date & Time : 25-04-2006/1.00-4.00 P.M.   Dept. No.                                                       Max. : 100 Marks

 

 

Part A

Answer all the questions.                                                                            10 ´ 2 = 20

 

  1. Define set of all real numbers as follows. Let An = ( -1/n, 1] if n is odd and

An = ( -1, 1/n] if n is even. Find lim sup An and lim inf An.

  1. Explain Lebesgue-Stieltjes measure with an example.
  2. Define counting measure with an example.
  3. State Borel- Cantelli Lemma.
  4. If h is B– measurable function, show that | h | is also B-measurable

function.

  1. What is induced probability space?
  2. If random variable X takes only positive integral values, show that
    E(X) = P[ X ³ n].
  3. Define convergence in r-th mean.
  4. If Xn  X and g is continuous, show that g(Xn)  g (X).
  5. State Levy’s theorem.

Part B

Answer any five questions.                                                                     5 ´ 8 = 40

 

  1. If { Ai , i ³ 1) is a sequence of subsets of a set W, show that

Ai = (A i  – A i – 1).

  1. Show that countable additivity of a set function with m(f) = 0 implies finite additivity of a set function.
  2. Prove that every finite measure is a s – finite measure. Is the converse true? Justify.
  3. Let f be B-measurable and if f = 0 a.e. [m],  show that f dm = 0.
  4. State and establish  additivity theorem of integral.

 

 

 

  1. State and establish Minkowski’s inequality.
  2. Show that Xn  X implies Xn  X. Is the converse true? Justify.
  3. If XnX, show that (Xn2 + Xn) (X2 + X).

 

Part C

Answer any two questions.                                                                   2 ´ 20 = 40

 

  1. a). State and establish extended monotone convergence theorem.

b). State and establish basic integration theorem.                                         ( 12 + 8)

  1. a). Let l (A) = dm;  A in the s – field Á, where fdm exists; thus l is a signed measure on Á. Show that l+(A) = f +dm, l– (A) = f – dm and |l|(A) = |f|dm.

b). State and establish Jordan – Hohn decomposition theorem.                    (8 + 12)

  1. a). If hdm exists and C є R, show that Chdm = Chdm.

b). Let X be a random variable defined on the space (W, A, p) and E |X|k < µ, k>0, Show that nk P[|X|>n] ® 0 as n ® µ.                                                         (10 + 10)

  1. a). Show that Xn  X implies Xn   X. Is the converse true? Justify.
    b). State and establish Lindberg Central limit theorem.                              (10 + 10)

 

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Loyola College M.Sc. Statistics April 2006 Industrial Statistics Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 46

FOURTH SEMESTER – APRIL 2006

                                                     ST 4801 – INDUSTRIAL STATISTICS

 

 

Date & Time : 25-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

PART – A

Answer all the questions.                                                                                  10 ´ 2 = 20

  1. Discuss the logic and statistical basis underling the general use of 3 – sigma limits on control charts.
  2. Mention the theoretical basis of p – chart and set up 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. Write down the control limits of a coefficient of variation chart.
  6. What is an average run length (ARL)?
  7. Explain an attribute single sampling plan.
  8. Define AOQ for a single sampling plan.
  9. Discuss the concepts of chance and assignable causes of variability.
  10. Write a short note on multivariate control chart.

 

PART – B
Answer any five questions.                                                                     5 ´ 8 = 40

 

  1. Explain the method of constructing control limits for X-bar and R charts when the sample sizes are different for various subgroups.
  2. A control chart indicates that the current process fraction non-conforming is 0.02. If 50 samples 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 shift? By the end of the third day following the shift?
  3. Write a detail note on the moving average control chart.
  4. Consider a modified control chart with CL at m = 0 and s =1. If n = 5, the tolerable fraction non-conforming d = 0.00135 and the control limits are at 3, sketch the OC curve of the chart.
  5. Design a cumulative sum control chart to detect a shift of D = 0.75 that has L(0) = 400. Is it possible to find a cumulative sum control chart for detecting this shift that has L(0.75) £ 12? What is L(0) for this chart?
  6. For the double sampling plan N = 120, n1 = n2 = 13, c1 =0 and c2 = 1, obtain Pa, ASN, AOQ and ATI when the submitted lot has the fraction non – conforming at p = 0.18.
  7. What are acceptances and rejection lines of a sequential sampling plan for attributes? How are the OC and ASN values obtained for this plan?
  8. What are modified control charts? Explain the method of obtaining control limits.

 

PART – C
Answer any two questions.                                                                   2 ´ 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.

b). Explain the procedure of obtaining the OC curve for a p – chart with an

illustration.

(10 + 10)

  1. a). Define the terms i). Rational subgroups ii). Specification limits

iii). Natural tolerance limits iv). Probability limits.

b). Explain the relevance of non-parametric method in quality control procedures with

an illustration.                                                                                              (10 + 10)

  1. a). What purpose does a cumulative sum chart serve?

b). Outline the procedure of constructing a V-mask.                                         (5 + 15)

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

plans are applied

b). Explain with an illustration the method of obtaining the probability of acceptance

for a triple sampling plan.                                                                            (10 + 10)

 

 

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Loyola College M.Sc. Statistics April 2006 Estimation Theory Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – APRIL 2006

                                                        ST 2808 – ESTIMATION THEORY

 

 

Date & Time : 19-04-2006/FORENOON     Dept. No.                                                       Max. : 100 Marks

 

 

PART – A

 

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

 

  1. If the class of unbiased estimators of  a parametric function is neither empty nor singleton, then show that the class is uncountable.
  2. Prove or disprove the uniqueness of UMVUE.
  3. If  δ  is a UMVUE and bounded, then  show that any polynomial in δ  is also a UMVUE.
  4. State Chapman – Robbin’s  inequality.
  1. Suppose δ  is sufficient for Р  and  Р0  С  Р,  then show that δ  is sufficient for Р0.
  1. Let S be a sufficient statistic. If  likelihood equivalence of x and y in the support A of the random variable X implies S(x) = S(y)   x , y  Є  A , then show that S is minimal sufficient.
  2. Let X 1  , X2   be a random sample from N ( θ ,1),   θ Є R . Verify whether or not   (X1  , X 2 ) is complete.
  3. Give two examples of a  Location-Scale family of distributions .
  4. Define Ancillary Statistic and give an example.
  5. Give an example of  M- estimator Tn  of  θ  which can be thought of as a weighted

average of the sample values with the weights depending on the data.

 

PART – B

 

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

 

  1. Let X be a discrete random variable with  pdf  pθ(x)  =  θ  if  x  =  -1 and

pθ(x)  =  (1 – θ )2θx   if  x  =  0,1,2,…, where 0 < θ <  1. Find the class U0 of

unbiased estimators of ‘0’ and hence find the class Ug of unbiased estimators of

g (θ)  =  θ , 0< θ < 1.

  1. Give an example where only constant estimable parametric functions have

UMVUE.

  1. Give an example of a UMVUE whose variance is greater than

Chapman-Robbins’s  Lower Bound.

    1. Let X1 ,…,Xn   be a random sample of size n from N(θ , 1) ,  θ Є R. Using Fisher information, show that   α ixi  is sufficient iff  αi  are equal for all i .
    2. Let X1 ,… ,Xn   be a random sample of size n from U (0,θ), θ  > 0. Then show that S() = X(n)  is minimal sufficient.
    3. Show that a complete sufficient statistic is minimal sufficient if it exists.

 

  1.  Let X1 ,… ,Xn   be a random sample of size n from B (m,θ),  m known and

θ unknown.   Show that the joint distribution of  (X1 ,…,Xn  ) belongs to an

exponential family. Hence find the mgf of   Xi.

  1. Let X ~ N (θ , 1 ), θ Є R , and let the prior distribution of θ be N ( 0 , 1 ).

Find the Bayes estimator of θ when the loss function is

  • Squared error
  • Absolute error.

 

 

PART – C

 

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

 

19(a). State and prove a necessary and sufficient condition for an estimator in the class ug

to  be a  UMVUE.  (10)

19(b). Derive Chapman – Robbin’s inequality, using covariance inequality. (10)

 

20(a). Give an example of a family which is not boundedly complete.(10)

20(b). Let X1  ,….., Xn   be a random sample from N(μ ,σ2 ), μ Є R, σ  > 0. Show that the distribution of ( X1 ,…..,Xn  ) belongs to two-parameter exponential family. Hence by using Basu’s theorem, establish the independence of   and s2. (10)

21(a). Prove that  δ*  is  D-Optimal estimator of g(θ)  iff each component of δ*  is a            UMVUE.  (14)

21(b). Let X1  ,… , Xn  be a random sample from N(μ ,σ2 ), μ Є R, σ  > 0. Obtain Jackknife estimator of variance   σ2  .    (6)

 

22(a). State and prove Lehmann – Scheffe Theorem for convex loss function. (8)

22(b). Let  have the p.d.f    f ( – ξ ).  If   δ  is a location equivariant estimator ,  then

show that the bias ,  risk and variance of  δ  do not depend of  ξ  .  (6)

22( c )  Let X1  ,… ,Xn  be a random sample from N(ξ  ,1 ) ,   ξ Є R .  Find the MRE

estimator of  ξ   when the loss is squared error.  (6).

 

 

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Loyola College M.Sc. Statistics April 2006 Computational Statistics – III Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 41

THIRD SEMESTER – APRIL 2006

                                            ST 3803 – COMPUTATIONAL STATISTICS – III

 

 

Date & Time : 02-05-2006/1.00-4.00 P.M.   Dept. No.                                                       Max. : 100 Marks

 

 

Answer THREE questions choosing one from each section.

SECTION – A

  1. A Scientist studied the relationship of size and shape for painted turtles. The following table contains their measurements on 10 females and 10 male turtles.  Test for equality of the two population mean vectors.

 

FEMALE

MALE
Length (x1) Width (x2) Height (x3) Length (x1) Width (x2) Height (x3)
98 81 38 93 74 37
103 84 38 94 78 35
103 86 42 96 80 35
105 86 42 101 84 39
109 88 44 102 85 38
123 92 50 103 81 37
123 95 46 104 83 39
133 99 51 106 83 39
133 102 51 107 82 38
133 102 51 112 89 40
  1. Given the following trivariate Normal distribution with mean vector and variance covariance matrix.

 

  1. Obtain the conditional distribution of X1 and X2 given X3 = 10

(15 marks)

  1. Obtain the distribution of CX where

(6 marks)

  • Find the correlation matrix for the data of Female turtles given in question No.1. Find whether the correlations are significant.                    (13 marks)

 

SECTION – B

Answer any ONE question

  1. a) Use two-phase method to solve following linear programming problem:

Max Z =

Sub. To

 

(18.5 marks)

 

  1. b) Solve the following transportation problem:

 

9   10  11                                                                              (15 marks)

  1. a) Solve the following game graphically:

B

(18.5 marks)

  1. b) Patients arrive at a clinic according to a Poisson distribution at a rate of 30

patients per hour.  The waiting room does not accommodate more than 14

patients.  Examination time per patient is exponential with mean rate 20 per hour.

  1. Find the effective arrival rate at the clinic.
  2. What is the probability that an arriving patient will not wait? Will find a vacant seat in the room?
  • What is the expected waiting time until a patient is discharged from the clinic?                                                              (15 marks)

SECTION – C

 

Answer any ONE question

  1. a) Suppose the one step transition probability matrix (tpm) is given as below: Find Poo(2), foo(n), f13(u) and f33(u).

 

 

(17 marks)

  1. b) For a three state Markov Chain with states {0,1,2} and tpm.

P = ,   Find m0, m1 and m2.

(17 marks)

  1. a) An infinite Markov Chain on the set of non-negative integers has the transition

function as follows:

Pko =    and Pk1 k+1 =

 

  • Find whether the chain is positive recurrent, null recurrent or transient.
  • Find stationary distribution if it exists.
  1. b) For a Branching process with off-spring distribution given by p(0) = p (3) =

Find the probability of extinction, when   (i) X0 = 1    and    (ii)  X0 > 1.

(17+17 marks)

 

 

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Loyola College M.Sc. Statistics April 2006 Computational Statistics – II Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 44

THIRD SEMESTER – APRIL 2006

                                             ST 3807 – COMPUTATIONAL STATISTICS – II

 

 

Date & Time : 02-05-2006/1.00-4.00 P.M.   Dept. No.                                                       Max. : 100 Marks

 

 

Answer any three questions and each question carries 33.5 marks.

  1. The following data relates to three body measurements taken on boys and girls of the same community. Test whether boys and girls differ from each other on the basis of their measurements assuming the variance – covariance matrices of these two data sets are equal.

 

Boys (centimeter)
Serial Number Height Chest Mid Upper Arm
1

2

3

4

5

6

 78

76

92

81

81

84

 60.6

58.1

63.2

59.0

60.8

59.5

 16.5

12.5

14.5

14.0

15.5

14.0

 

 

Girls
Serial Number Height Chest Mid Upper Arm
1

2

3

4

5

6

7

8

9

 

 80

75

78

75

79

78

75

64

80

 58.4

59.2

60.3

57.4

59.5

58.1

58.0

55.5

59.2

 

 14.0

15.0

15.0

13.0

14.0

14.5

12.5

11.0

12.5

 

  1. a) For the girls data given in question number 1 obtain

i). sample correlation matrix and test for the significance of the correlations.

ii). Partial correlation r12..3

iii). Multiple correlation R1.23 and test its significance.

  1. b)   Draw a Q-Q plot for the girls data on chest measurement in question number 1 to find whether the data is from a normal distribution.

 

  1. a). The following frequencies with the corresponding probabilities observed in a genetical experiment are given below :

 

Cell Number 1 2 3 4
Probabilities (1+q) / 4 (1 – q) / 4 (1 – q) / 4 (1 + q) / 4
Frequencies 1997 906 904 32

Obtain the Maximum Likelihood Estimator of q and the variance of this estimator.

 

b). Obtain the estimator by the method of modified minimum chi-square for the data in question 3. a).

c). Estimate the parameter q assuming a truncated Poisson distribution truncated at 0 for the given data

X 1 2 3 4 5 6 7 8
Frequency 60 50 40 25 10 8 4 3

 

  1. a) Classify the states of the Markov chain having the following transition probability    matrix (tpm)

 

Find Pn and   lim  Pn

n→∞

  1. Consider the Markov chain with state space {0, 1, 2, 3} and one step tpm

 

where q = (1-p), 0<p<1. Comment on the nature of the states.

  1. a) Consider a Markov Chain having the state space {0, 1, 2} and transition matrix

 

 

  • Show that the Markov chain is irreducible.
  • Obtain the period for this Markov chain
  • Obtain lim Pdn.

n→∞

 

  1. b)   An infinite Markov chain on the set of non-negative integers has the transition   matrix as follows:

pk0 = (k+1) / (k+2)       and    Pk, k+1 = 1 / (k+2)

 

  • Find whether the chain is positive recurrent, null recurrent or transient.
  • Find the stationary distribution, in case it exists.

 

 

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