M.Sc Statistics Question Paper

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

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

BA 27

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – November 2008

    ST 3875 – FUZZY THEORY AND APPLICATIONS

 

 

Date : 10-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION – A

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

  1. Define Archimedean t-conorm.
  2. Write the axiomatic skeleton of fuzzy t-norm.
  3. Define Drastic fuzzy union.
  4. Explain increasing generator.
  5. Define arithmetic operations on intervals.
  6. Define membership function and give an example.
  7. Find the core of the fuzzy set whose membership function is given by                                         f(x) = exp [– (x – 3)2]
  8. Give an example of a trapezoidal shaped membership function.
  9. Explain the sigmoid function used for activation.
  10. Present the motivation for fuzzy clustering.

 

SECTION -B

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

  1. Define  &  and prove that , .

 

  1. Prove that ,.

 

  1. Define dual triple and show that given a t-norm i and an involutive fuzzy complement c , the binary operation u on [0,1] defined by  for all  is a t-co-norm such that  is a dual triple.

 

  1. Prove that if  for every ,  then . What will happen when .

 

  1. Prove that A is a fuzzy subset of B if and only if    αA  αB α [0,1].

 

  1. Prove that (i) = α( ) and  (ii)   α ()
  2. Explain the Lagrange interpolation method for constructing membership function from sample data.

 

  1. Describe the architecture of a multi-layer feed-forward network.

 

 

 

 

)

SECTION -C

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

 

  1. (a) Prove that A is a fuzzy number if and only if there exists a closed interval

such that  where

is monotonically increasing continuous from the right such

that  and r is monotonically

decreasing continuous from the left such that r.

(b) Write a short note on Linguistic variables.                                                (15+5)

 

  1. Let  and B

Find the four basic operations for the fuzzy numbers A and B and also find the

corresponding fuzzy numbers.

 

  1. (a) Define equilibrium of a fuzzy complement and show that every fuzzy complement has atmost one equilibrium. Also show that a continuous fuzzy complement has a unique equilibrium.

(b) If a fuzzy complement c has an equilibrium ec, then a ≤ c(a) iff a ≤ ec and

a ≥ c(a) iff a ≥ ec.                                                                                           (14 + 6)

 

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

(b) Let X ={x1, ..,x5} be a universal set and suppose four experts E1, E2, E3, E4

have specified the valuations of the 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 E4
x1

x2

x3

x4

x5

 0

1

1

0

1

 1

0

0

1

0

 1

0

1

0

1

  0

1

1

0

0
Element E1 E2 E3 E4
x1

x2

x3

x4

x5

 1

0

1

0

1

 1

0

1

0

0

 1

0

0

1

0

 0

1

0

1

1

 

 

 

 

 

 

For the set A, the four experts are to be given weights c1 = 1/3, c2 = 1/4, c3 = 1/4,

c4 = 1/6 and for set B, the weights are all equal for the four experts. With these

weights find the degrees of membership of the five elements in A and in B.

Also, evaluate the degrees of membership in A ∩ B using the standard

            intersection and in A U B using the algebraic sum operators.                  (8 + 12)

 

 

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Loyola College M.Sc. Statistics Nov 2008 Applied Regression Analysis Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

BA 22

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – November 2008

    ST 1811 – APPLIED REGRESSION ANALYSIS

 

 

 

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

Time : 1:00 – 4:00

Section A

Answer All the Questions                                                          (10 x 2 = 20 Marks)

 

  1. What are the distributions of the components (yi , ) of a simple regression model?
  2. What do you mean by Hetroscedasticity Property?
  3. Explain the Linear Probability model
  4. What is the Linear Predictor in a Generalized Linear Model?
  5. What is the Identity Link in a Generalized Linear Model?
  6. Explain the Simple Regression Model.
  7. List the four methods of scaling residuals.
  8. Give two examples for nominal variables.
  9. Give an example of a variable that is classified as nominal, ordinal and interval variable.
  1.     Give an example for Poisson Log linear Model.

 

Section B

Answer Any Five Questions                                                          (5 x 8 = 40 Marks)

 

  • Show that the least square estimates ,  of a simple regression

model are unbiased.

  • Explain the procedures for finding the confidence interval forand  of a simple regression model.
  • Discuss multicollinearity with an example.
  • Explain the purpose of Unit Normal Scaling
  • Explain Binomial logit model for the binary data
  • What are the properties of the least square estimates of the fitted regression model?
  • Discuss any two methods of scaling residuals
  • Explain the Logistic regression model with an example

 

Section C

Answer Any Two Questions                                      (2 x 20 = 40 Marks)

 

19 (a) Derive V() of a simple regression model.

(b) Write down the test procedures to test the intercept of a simple

regression model.

 

20 (a) Write down the test Procedures to test H0:  =0 against H1:  ¹ 0

using Analysis of Variance.

(b) Derive the interval estimation of the mean response of a simple

regression model.

 

21 (a) Fit a Logistic regression model

(b) How do you interpret the Poisson Loglinear model for a count data

 

22 (a) Estimate the parameters of a multiple linear regression model by the

methods Maximum Likelihood Estimation.

(b)  Write short notes on Relative Risk, Odds Ratio and cross product ratio.

 

 

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

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

BA 19

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – November 2008

    ST 1808 – ANALYSIS

 

 

Date : 04-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

Section-A                                                        (10X2=20 marks)

Answer ALL the questions.

 

(1)  Show that    ρ(X1, X2) = E| X1-X2| is a metric on the space of all random variables defined on a

probability space.

(2)  If the inner product x.y of a vector y with any vector x is zero, show that y is a null vector.

(3)  If in a metric space, xn→ x as n→ ∞, show that every subsequence of   {Xn, n≥1} converges to x.

(4)  Show that in a metric space with at least two points, all finite sets are closed.

(5)  Prove that in any metric space(X, ρ), both X and the empty set ф are open.

(6)  Give an example of a bijective continuous function, whose inverse is not continuous.

(7)  Examine whether a closed sub – space of a complete metric space is complete.

(8)  Explain the symbols “O” and “o “.

(9)  Let f: X→R(n) , XсR(n)  . If f is differentiable at a, show that f is continuous at a.

(10) Let (X, ρ) be any metric space. Show that a contraction mapping is continuous on X.

 

Section-B                                                        (8X5= 40 marks)

 

Answer any FIVE questions. Each question carries EIGHT marks.

 

(11)  Show that if ρ is a metric on X, then so is σ given by

σ (x,y)= ρ(x,y)

 

1+ ρ(x,y)

and that ρ and σ are   equivalent metrics.

(12) Show that the composition of two continuous functions is continuous.

(13)  Prove that the space R with its usual metric is complete.

(14)  State and prove Banach’s fixed point theorem.

(15)  Show that a metric space is compact if and only if every sequence of points in X has a subsequence

converging to a point in X.

(16)  State and prove Dini’s theorem for a sequence of real valued functions.

(17)  If  f Є R(g ; a, b) on[ a, b] , show that  |f| Є R(g ; a, b) on  [a, b ] and

b                     b

|∫ f dg | ≤   ∫| f |dg

a                     a

(18)  If f is continuous on [a, b] show that f Є R (g; a, b).

 

 

 

 

 

 

Section-C                                            (2 X 20 = 40 marks).

 

Answer any TWO questions. Each question carries 20 marks

 

(19) (a)  Show that a sequence of points in any metric space cannot converge to two distinct limits.

(6 marks)

(b)  Give an example of a normed vector space, which is not an inner product space.                 (8 marks)

(c)   State and prove Cauchy –Schwartz inequality.                                                                      (6 marks)

 

 

(20)   (a)  Let (X, ρ)  and (Y, σ) be the metric spaces and let f:X → Y. Prove that f is continuous on X if and

only if f-1 (G) is open in X whenever G is open in Y.                                                 (10 marks)

(b)  Let G be an open subset of the metric space X. Prove that G ‘=X-G is closed. Conversely, if F is a

closed subset of X,  prove that  F’ = X-F   is open.                                                            (10marks)

 

(21)  (a)  Define uniform convergence. Let (X, ρ) and (Y, σ) be two metric spaces. Let f n: X → Y be a

sequence of functions  converging  uniformly  to a function  f:X →Y. If each f n is continuous at c,

show that f is also continuous at c .                                                                                      (10 marks)

(b)  State and prove Weirstrass M- test for absolute convergence and uniform convergence.

(10 marks)

(22)  (a)  What is meant by Riemann – Stieltjes integral? Establish the necessary and sufficient condition

for a bounded real valued function f Є R(g ; a, b).                                                              (12 marks)

(b)  If f is a continuous function on [a, b], show that there exists a number c lying between a and b

such that

b

∫ f dg  =    f(c) [g(b)-g(a)].                                                     (8 marks)

a

 

 

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

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BA 21

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – November 2008

    ST 1810 – ADVANCED DISTRIBUTION THEORY

 

 

 

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

Time : 1:00 – 4:00

SECTION – A                                                                       

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

 

  1. Define a truncated distribution and give an example.
  2. Find the MGF of a power series distribution.
  3. Define lack of memory property for discrete random variable.
  4. If X is distributed as Lognormal, show that its reciprocal is also distributed as Lognormal.
  5. Let (X1, X2) have a bivariate Bernoulli distribution. Find the distribution of X1 +  X2.
  6. Find the marginal distributions associated with bivariate Poisson distribution.
  7. Show that Marshall – Olkin bivariate exponential distribution satisfies bivariate lack of memory property.
  8. Define non-central chisquare – distribution and find its mean.
  9. Let X1, X2, X3, X4 be independent standard normal variables. Examine whether

X12 +3 X22 + X32 +4 X42 – 2 X1X2 + 6 X1X3 + 6 X2X4– 4 X3X4  is distributed as chi-square.

  1. Let X be N(q, 1), q = 0.1, 0.5. If q is discrete uniform, find the mean of the compound

distribution.

 

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

 

  1. State and establish a characterization of Poisson distribution.
  2. Derive the pdf of a bivariate binomial distribution. Hence, show that the regressions are

linear.

  1. Let (X1, X2) follow a Bivariate normal distribution with V(X1) = V(X2). Examine

whether  X1 + X2  and (X1 – X2)2  are independent.

  1. Show that the mean of iid Inverse Gaussian random variables is also Inverse Gaussian.
  2. Let (X1, X2) follow a Bivariate exponential distribution . Derive the distributions of Min{X1, X2} and Max{X1, X2}.
  3. Find the mean and the variance of a non-central F – distribution.
  4. Let X1, X2, X3,…, Xn be iid N(0, σ2), σ > 0 random variables.Find the MGF of X /AX/ σ2.

Hence find the distribution of X1X2.

  1. Illustrate the importance of the theory of quadratic forms in normal variables in ANOVA.

 

 

 

SECTION – C

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

 

  1. a) Let X1, X2, X3,…, Xn be iid non-negative integer-valued random variables. Show that X1

is geometric if and only if Min{X1, X2, X3,…, Xn} is geometric.

 

  1.  b) State and establish the additive property of  bivariate Poisson distribution.

 

  1. a) Let (X1, X2) have a bivariate exponential distribution of Marshall-Olkin. Find the

cov(X1, X2).

 

  1.  b) Let (X1, X2) follow a bivariate normal distribution. State and establish any two of its

properties.

 

  1. a) Define non-central t – variable and derive its pdf.

 

  1.  b) Let X  be a random variable with the distribution function F given by

0 ,                 x < 0

F(x) =   (2x + 1)/4,    0  ≤ x < 1

1,                   x  ³ 1.

Find the mean, median and variance of X.

 

 

  1. a) State and establish a necessary and sufficient condition for a quadratic form in normal variables to

have a chi-square distribution.

 

  1. b) Let (X1, X2) follow a trinomial  distribution with index n and cell probabilities θ1 ,θ2. If the prior

distribution is uniform, find the compound distribution. Hence find the means of  X1 and X2.

 

 

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Loyola College M.Sc. Statistics April 2009 Testing Statistical Hypothesis Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 37

SECOND SEMESTER – April 2009

ST 2812 / 2809 – TESTING STATISTIACAL HYPOTHESIS

 

 

 

Date & Time: 22/04/2009 / 1:00 – 4:00       Dept. No.                                                       Max. : 100 Marks

 

 

 

SECTION A

Answer all questions.                                                                                    (10 x 2 = 20)

 

  1. Define level and power of a test.
  2. Let X be a random variable with pdf .

Obtain the Most Powerful Test of size for testing H0: θ = 1 Vs H1: θ = 2.

  1. Give the general form of (k+1) parameter exponential family of distributions.
  2. Define Uniformly Most Powerful Test.
  3. Let. Consider the test function

for testing H0: θ = 0.2 Vs H1: θ > 0.2.Obtain the value of power function at

θ = 0.4.

  1. What are the circumstances under which Locally Most Powerful test is used?
  2. What is meant by shortest length confidence interval?
  3. Define maximal invariant function.
  4. What is meant by nuisance parameter? Give an example.
  5. Define Likelihood Ratio Test.

 

 

SECTION B

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

 

  1. Let  denote a random sample fromDerive a Most Powerful test of  level 0.05 for testing Vs. Also obtain the       cut-off point.
  2. Show that the family of densities possesses MLR property.
  3. Let denote a random sample of size n from. Consider the problem of testing Vs. Show that UMP test of  does not exist.
  4. For (k+1) parameter exponential family of densities, derive an unconditional UMPUT of level for testing  Vs  clearly stating the assumptions.
  5. State and prove the sufficient part of Generalized Neyman-Pearson lemma.
  6. Show that any test having Neyman structure is similar. Also show that the converse is true under certain assumptions (to be stated).

 

 

  1. Derive the Locally Most Powerful test for testing Vs based on a random sample of size n from, where  and  are known pdf’s.
  2. Find maximal invariant function under the group of i.) Location transformations and ii.) Scale transformations.

 

SECTION C

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

 

  1. a.) Derive a UMP test of level  for testing  Vs  for the family of densities that possess MLR in T(x). Show that the power function of the above testing problem increases in

b.) Show that any UMP test is always UMPUT.                                          (16+4)

  1. Consider a one parameter exponential family with density. Assume  is strictly increasing in. Derive a UMP test of level  for testing  Vs.
  2. Let X and Y be independent Binomial variables with parameters and  respectively, where m and n are assumed to be known. Derive a conditional UMPUT of size  for testing  Vs.
  3. Let anddenote independent random samples from  and respectively. Derive the Likelihood Ratio Test for testing Vs.

 

 

 

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Loyola College M.Sc. Statistics April 2009 Stochastic Processes Question Paper PDF Download

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    LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 42

THIRD SEMESTER – April 2009

ST 3809 – STOCHASTIC PROCESSES

 

 

 

Date & Time: 23/04/2009 / 1:00 – 4:00 Dept. No.                                                      Max. : 100 Marks

 

 

PART-A

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


  1.  Define a stochastic process with an example.
  2. Define a process with independent increments.

 

  1. Show that communication between two states i and j satisfies transitive relation.

 

  1. Define (i) transcient state (ii) recurrent state.

 

  1. Define a Markov process.

 

  1. Obtain the PGF of a Poisson process.

 

  1. Define a renewal function. What is the relation between a renewal function and the

distribution functions of inter occurrence times?

 

  1. When do you say that  is a martingale with respect to ?

 

  1. What is a branching process?

 

  1. What is the relationship between Poisson process and exponential distribution?

 

PART-B

            Answer any 5 questions:                                                                                    (5 X 8 = 40)

 

  1. State and prove Chapman – Kolmogorov equation for a discrete time Markov chain.

 

  1. Obtain the equation for in a Yule process with X(0) = 1.

 

  1. Let and  be i.i.d random variables with mean 0 and variance.

Show that   is a martingale with respect to .

 

  1. Show that the matrix of transition probabilities together with the initial distribution

completely specifies a Markov chain.

 

  1. Show that the renewal function satisfies

 

 

  1. Establish the relationship between Poisson process and Binomial distribution.

 

  1. Obtain the stationary distribution for the Markov chain having transition probability

matrix

 

 

 

  1. If a process has stationary independent increments and finite mean show that

 

where     and  .

 

PART-C

                 Answer any 2 questions:                                                                          (2 X 20 = 40)

 

  1. a) State and prove the necessary and sufficient condition required by a state to be                               recurrent.

 

b.)  Verify whether state 0 is recurrent in a symmetric random walk in three dimensions.                                                                                                                                         (10+10)

 

  1. a) State the postulates of a Poisson process. Obtain the expression for.

 

b.)  Obtain the distribution for waiting time of k arrivals for a Poisson process.                                                                                                                                                                    (15+5)

 

  1. a) Obtain the generating function for a branching process. Hence obtain the mean and                        variance.

 

  1.       b)  Let   be the probability that an individual in a generation generates k

off springs. If  obtain the probability of extinction.

(15+5)

  1. a.) Obtain the renewal function corresponding to the lifetime density.

 

 

b.)  Show that the likelihood ratio forms a martingale.

 

c.)  Let be a martingale with respect to.  If  is a convex function with

 

show that   is a sub martingale with respect to .

                                                                                                                             (10+5+5)

                                        

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Loyola College M.Sc. Statistics April 2009 Statistical Process Control Question Paper PDF Download

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      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 48

FOURTH SEMESTER – April 2009

ST 4806 – STATISTICAL PROCESS CONTROL

 

 

 

Date & Time: 21/04/2009 / 9:00 – 12:00 Dept. No.                                                     Max. : 100 Marks

 

 

SECTION – A

 Answer ALL the questions                                                                                         10×2 =20

  1. Define quality improvement
  2. Explain six-sigma quality
  3. Discuss the statistical basis underlying the general use of 3 – sigma limits on control charts.
  4. How is lack of control of a process is determined by using control chart technique? .
  5. Write down the expression for process capability ratio (PCR) when only the lower specification is known.
  6. What information is provided by the OC curve of a control chart?
  7. Give an expression for AOQ for a single sampling plan.
  8. Write a short note on Multivariate Quality Control.
  9. Define a) Specification limits b) Natural tolerance limits.
  10. Explain double sampling plan.

SECTION- B

Answer any FIVE questions                                                                                      5 x 8= 40

  1. What are the major statistical methods for quality improvement? .
  2. A normally distributed quality characteristic is monitored by a control chart with K sigma

control limits. Develop an expression for the probability that a point will plot outside the

control limits when the process is really in control .

  1. Sampes of n=6 items are taken from a manufacturing process at regular interval. A normally

distributed quality characteristic is measured and x-bar and S values are calculated for each

sample. After 50 subgroups have been analyzed, we have

 

  1. a) Calculate the control limits for the x-bar and S control charts.
  2. b) Assume that all the points on both charts plot within the control limits .What are the natural

tolerance limits of the process? .

  1. Write a detailed note on the moving average control chart.
  2. 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 0.50 of detecting a shift in the process to 0.26?.

  1. 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.

 

 

  1. What are acceptance and rejection lines of a sequential sampling plan for attributes?. How

are the OC and ASN values obtained for this plan? .

  1. What are chain samplings and skip-lot sampling plans?

 

SECTION- C

 

Answer any two questions                                                                                            2 X 20 = 40

 

  1. a) Describe the procedure of obtaining the OC curve for a p-chart with an example .
  2. b) Explain process capability analysis with an illustration.                               ( 12+8 )

20.a) What are modified control charts?. Explain the method of obtaining control limits for these

charts.

  1. b) A control chart for non-conformities per unit uses 0. 95 and 0.05 probability limits .The

center line is at  u=14. Determine the control limits if the size of the sample is 10.     (14+6)

21.a) Discuss the purpose of cumulative sum chart .

  1. b) Outline the procedure of constructing V-mask.                                 (8+12)
  2. a) Explain with an illustration the method of obtaining the probability of acceptance

for a triple sampling plan.

  1. b) What are continuous sampling plans?. Mention a few situations where these plans are

applied.                                                                                                                      (10 + 10)

 

 

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Loyola College M.Sc. Statistics April 2009 Statistical Computing – III Question Paper PDF Download

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      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

YB 50

M.Sc. DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – April 2009

ST 4808 – STATISTICAL COMPUTING – III

 

 

 

Date & Time: 25/04/2009 / 9:00 – 12:00  Dept. No.                                                    Max. : 100 Marks

 

 

Answer any three questions            

                                                                                               

  1. a) The data shown here are  and R values for 24 samples of size n=5 taken from a process producing bearings.  The measurements are made on the inside diameter of the bearing, with only the last three digits recorded.

 

Sample number 1 2 3 4 5 6 7 8 9 10 11
34.5 34.2 31.6 31.5 35.0 34.1 32.6 33.8 34.8 33.6 31.9
R 3 4 4 4 5 6 4 3 7 8 3
Sample number 12 13 14 15 16 17 18 19 20 21 22
38.6 35.4 34 37.1 34.9 33.5 31.7 34 35.1 33.7 32.8
R 9 8 6 5 7 4 3 8 4 2 1
Sample number 23 24
33.5 34.2
R 3 2

 

(i). Sep up  and R charts on this process.  Does the process seem to be in statistical           control?  If necessary, revise the trial control limits.

 

(ii). If specifications on this diameter are 0.50300.0010, find the percentage of nonconforming bearings produced by this process.  Assume that diameter is normally distributed.

 

b). In the semiconductor industry, the production of microcircuits involves many steps.  The wafer fabrication process typically builds these microcircuits on silicon wafers and there are many microcircuits per wafer.  Each production lot consists of between 16 and 48 wafers.  Some processing steps treat each wafer separately, so that the batch size for that step is one wafer.  It is usually necessary to estimate several components of variation: within-wafer, between-wafer, between-lot and the total variation. A critical dimension (measured in mm) is of interest to the process engineer. Suppose that five fixed position are used on each wafer (position 1 is the center) and that two consecutive wafers are selected of each batch. The data that results several batches are shown below.

 

(i) What can you say about over all process capability?

 

(ii)  Can you construct control charts that allow within- wafer variability to be evaluated?

 

(iii) What control charts would you establish to evaluate variability between wafers? Set

up these charts and use them to draw conclusions about the process.

 

 

Lot No. Wafer No. Position
1 2 3 4 5
1 1 2.15 2.13 2.08 2.12 2.10
2 2.13 2.10 2.04 2.08 2.05
2 1 2.02 2.01 2.06 2.05 2.08
2 2.03 2.09 2.07 2.06 2.04
3 1 2.13 2.12 2.10 2.11 2.08
2 2.03 2.08 2.03 2.09 2.07
4 1 2.04 2.01 2.10 2.11 2.09
2 2.07 2.14 2.12 2.08 2.09
5 1 2.16 2.17 2.13 2.18 2.10
2 2.17 2.13 2.10 2.09 2.13
6 1 2.04 2.06 2.00 2.10 2.08
2 2.03 2.10 2.05 2.07 2.04
7 1 2.04 2.02 2.01 2.00 2.05
2 2.06 2.04 2.03 2.08 2.10
8 1 2.13 2.10 2.10 2.15 2.13
2 2.10 2.09 2.13 2.14 2.11
9 1 2.00 2.03 2.08 2.07 2.08
2 2.01 2.03 2.06 2.05 2.04
10 1 2.04 2.08 2.09 2.10 2.01
2 2.06 2.04 2.07 2.04 2.01

(17 +17)

 

  1. a). Bath concentrations are measured hourly in a chemical process. Data (in PPM) for the

last 32 hours are shown below (read down from left).

160 186 190 206
158 195 189 210
150 179 185 216
151 184 182 212
153 175 181 211
154 192 180 202
158 186 183 205
162 197 186 197

The process target is =175 PPM.

(i). Estimate the process standard deviation.

 

(ii). Construct a tabular cusum for this process using standardized values of h = 5 and

k =  .

 

b). A product is shipped in lots of size N = 2000.  Find a Dodge-Romig single-sampling plan for which the LTPD = 1%, assuming that the process average is 0.25% defective.  Draw the OC curve and ATI curve for this plan.  What is the AOQL for this sampling plan?                                                                                                                                 (20+14)

 

 

 

 

 

 

3)    (a)  Analyze the following 32 factorial design                                                                 (24)

Replicate I                                            Replicate II

 

a0b0

20

 a1b0

32

 a0b2

40
a1b1

60

 a0b1

48

 a2b0

55
a2b1

60

 a1b2

31

 a2b2

51
a1b1

42

 a1b2

60

 a0b1

40
a2b0

25

 a0b0

62

 a1b0

45
a2b2

61

 a2b1

31

 a0b2

42

 

(b) Construct BIBD using the following :

V = 7, b =7, r = 3, k = 3, λ=1                                                                                      (10)

 

4)  (a) Analyze the following 23 factorial experiment in blocks of 4 plots, involving three fertilizers N,

P and K each at two levels.                                                                                     (17)

Replicate I                                                  Replicate II

Block 1 np

88

 npk

90

 (1)

115

 k

75
Block 2 p

101

 n

111

 pk

75

 nk

55
Block 3 (1)

115

 npk

95

 nk

90

 p

80
 Block 4 np

125

 k

95

 pk

80

 n

100

 

Replicate III

Block 5 pk

53

 nk

76

 (1)

65

 np

82
Block 6   n

75

 npk

100

 P

55

 k

92

 

(b) Use the Kuhn-Tucker conditions to solve the following Non-Linear Programming Problem:

Maximize z =  2x1 + x2 -x12

Subject to the constraints:

2x1+ 3x2 ≤ 6,

5x1+ 2x2 ≤ 10

x1, x2 ≥ 0                                                                                (17)

 

5)  (a)  Use Penalty method to solve the following L.P.P:

Minimize = 9x1 + 10x2

Subject to the constraints:

2x1 + 4x2  ≥ 50,

4x1 + 3x2  ≥ 24,

3x1 + 2x2   ≥ 60

x1, x2 ≥ 0                                                                                                      (17)

(b)   Use Beale’s method to solve the following Q.P.P:

Minimize z = 6- 6x1 + 2x12 – 2x1x2 + 2x22

Subject to x1 + x2 ≤ 2

x1, x2 ≥ 0                                                                                 (17)

 

 

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Loyola College M.Sc. Statistics April 2009 Statistical Computing – I Question Paper PDF Download

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     LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 35

FIRST SEMESTER – April 2009

ST 1812 – STATISTICAL COMPUTING – I

 

 

 

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

 

 

Answer ALL the questions.  Each carries THIRTY FOUR marks

                                                                                        

  1. a). In a population containing 490 live birds of same weight and age, the birds were divided into 70 equal groups.  They were then given a stimulus to increase the growth rate.  The following data gives the frequency distribution of birds with significant weights at the end of 6 weeks.  Fit a truncated binomial distribution and test the goodness of fit.
            No of birds 1 2 3 4 5 6 7
Frequency 6 15 21 17 8 2 1

 

b). The following are the marks of 135 students of B.Com in a city college.

Marks No. of Students
0-10 5
10-20 15
20-30 30
30-40 40
40-50 25
50-60 10
60-70 6
70-80 4

Fit a normal distribution by ordinate method and test the goodness of fit at                                    5% level of significance.

(OR)

c). Fit a truncated Poisson distribution to the following data and test the                            goodness of fit at 1% level of significance.

x 1 2 3 4 5 6
f 86 52 26 8 6 1

d). Fit a distribution of the form, where

, x=0,1,2,…,  > 0 and

; x=1,2,…, 0<p<1,

to the following data:

 

x 0 1 2 3 4 5 6 7 8
f 72 113 118 58 28 12 4 2 2

 

 

  1. a). Find the inverse of the following symmetric matrix using partition method.

 

b). Obtain the characteristic roots and vectors for the matrix

 

 

 

 

(OR)

c). Find  for the following matrices:

i).

ii).

d). Draw a random sample of size 10 from the exponential distribution having the    density function

.

Also find the mean and variance of the sample observations.

 

  1. a) The following data were collected on a simple random sample of 10 patients with hypertension.

 

Serial No. mean arterial blood pressure (mm/Hg) weight (kg) heart beats / min
1 105 85 63
2 115 94 70
3 116 95 72
4 117 94 73
5 112 89 72
6 121 99 71
7 121 99 69
8 110 90 66
9 110 89 69
10 114 92 64

i). Fit a regression model and estimate effect of all variables / unit of measurement,        taking blood pressure as the dependent variable.

ii). Find R2 and comments on it

(OR)

  1. b) The following table explains a company monthly income based on their advertisement

on V-Slicer product.

 

Serial No. Monthly Income on sales ($’000) Advertisement on TV ($ ‘000) Advertisement on News Paper ($ ‘000)
1 5.5 0.2 0.1
2 6.7 0.5 0.2
3 8.0 1.2 0.8
4 10.1 2.0 0.9
5 15 3.0 1.4
6 18.0 4.0 2.0
7 23 5.0 2.5
8 28 6.2 3.8
9 32 8.0 4.1
10 35 10.0 5.2

 

  1. Draw a scatter diagram for the above data.
  2. Fit a regression mode taking TV advertisement and News paper advertisement as independent variables and estimate monthly income when TV advertisement is 15 and News paper advertisement is 7 in 1000 dollars.

 

 

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Loyola College M.Sc. Statistics April 2009 Multivariate Analysis Question Paper PDF Download

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     LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 41

THIRD SEMESTER – April 2009

ST 3808 – MULTIVARIATE ANALYSIS

 

 

 

Date & Time: 16/04/2009 / 1:00 – 4:00         Dept. No.                                                       Max. : 100 Marks

 

 

PART – A

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

 

  • Give an example in the bivariate situation that the marginal distributions are normal but the bivariate distribution is not.
  1. Let X, Y and Z have trivariate normal distribution with null mean vector and covariance matrix

2     3      4

3     2     -1

4    -1      1   ,

 

find the distribution of  Y+X.

  1. Mention any two properties of multivariate normal distribution.
  2. Explain the use of partial and multiple correlation coefficients.
  3. Define Hotelling’s T2 – statistics. How is it related to Mahlanobis’ D2?
  4. Outline the use of discriminant analysis.
  5. What are canonical correlation coefficients and canonical variables?
  6. Write down any four similarity measures used in cluster analysis.
  7. Write the c.f. of X where

X~N2 { ,   }.

10.Write  a short  note on data mining.

 

PART B

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

 

  1. Obtain the maximum likelihood estimator S of p-variate normal distribution with

mean vector known.

  1. Let X1, X2,…, X n be independent N( 0 , 1 ) random variables. Show that X’ A X

is chi-square if A is idempotent, where  X= ( X1,X2,…,X n )’.    

  1. 13. How will you test the equality of covariance matrices of two multivariate normal

distributions on the basis of independent samples drawn from two populations?.

  1. Let (Xi, Yi)’ , i = 1, 2, 3 be independently distributed each according to bivariate

normal with mean vector and covariance matrix as given below. Find the joint

distribution of six variables. Also find the joint distribution of  and .

Mean vector: (m, t)’, covariance matrix:

  1. Outline single linkage and complete linkage clustering procedures with an

example.

  1. Giving suitable examples explain how factor scores are used in data analysis.
  2. Consider a multivariate normal distribution of X with

m =      ,       S =                    .

 

Find i )  the conditional distribution of ( X1, X3 ) / ( X2, X4 )

  1. ii) s42  
  2. a) Define i ) Common factor  ii) Communality  iii) Total variation

b)Explain classification problem into two classes and testing problem.

PART C

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

 

  1. a) Derive the distribution function of the generalized T2 – statistic.
  2. 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 20.                                                                                 (15 + 5)

  1. a) What are principal components?. Outline the procedure to extract principal

components   from a given covariance matrix.

  1. b) Define partial correlation between Xi and Xj .Also prove that

______   ______

r12.3=  ( r12-r13r23)/ {Ö(1-r223) Ö(1-r213)}.                           (  12+8)

21.a) Consider the two data sets

X1=     and   X2 =

for which         .

1) Calculate the linear discriminant function.

2) Classify the observation x0‘= ( 2  7 ) as population π1 or  population π2 using

the decision rule with equal priors and equal costs.

  1. b) Explain how the collinearity problem can be solved in the multiple regression.

( 14+6)

22.a)  Explain the method of extracting canonical correlations and their variables

from a dispersion matrix.

  1. b) Prove that under some assumptions (to be stated), variance and covariance can

be written as S = LL’ + y in the factor analysis model. Also discuss the effect

of an orthogonal transformation.                                                              (8 + 12)

 

 

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Loyola College M.Sc. Statistics April 2009 Fuzzy Theory And Applications Question Paper PDF Download

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      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

YB 44

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – April 2009

ST 3875 – FUZZY THEORY AND APPLICATIONS

 

 

 

Date & Time: 27/04/2009 / 9:00 – 12:00     Dept. No.                                                       Max. : 100 Marks

 

 

SECTION – A

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

  1. Write the axiomatic skeleton of fuzzy t-conorm.
  2. Define Archimedean t-norm.
  3. Define Drastic fuzzy intersection.
  4. Write a short note on fuzzy number.
  5. If , find
  6. Define a fuzzy variable and give an example.
  7. Define scalar cardinality of a fuzzy set.
  8. State the role of a ‘knowledge engineer’ in constructing fuzzy sets.
  9. Distinguish between direct methods and indirect methods of constructing membership functions.
  10. Define an artificial neural network.

 

SECTION – B

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

  1. Define &  and prove that ,.

 

  1. Prove that , where  and  denote drastic and yager class of t-norm.

 

  1. Let the triple be a dual generated by an increasing generator. Prove that fuzzy operations  satisfy the law of excluded middle and the law of contradiction. Also prove that  does not satisfy distributive law.

 

  1. Let and B. Find the 4 basic operations for the fuzzy numbers A and B.

 

  1. Prove under usual notations: (i) α(Ac) = ( (1 – α) +A)c (ii)  = α+ ()
  2. State the axiomatic skeleton and desirable requirements for fuzzy complements. Prove that if the monotonic and involutive axioms are satisfied, then the boundary and continuity conditions are satisfied.

 

  1. Let X ={x1, x2 ,x3} be a universal set and suppose two experts E1 and E2 have specified the valuations of these three as elements of two fuzzy sets A and B as

given in the following table:

Membership in A                   Membership in B

Element E1 E2
 x1 0.6 0.5
x2 0.2 0.3
x3 0.8 0.6
Element E1 E2
x1 0.2 0.4
x2 0.9 0.7
x3 0.6 0.3

 

 

 

 

 

Assuming that for set A, the experts have to be given weights as c1 = 0.7 and c2 = 0.3 and that for set B, the weights are c1 = 0.2, c2 = 0.8, find the degree of membership of the three elements in A and in B. Also, find the degree of membership in AUB by bounded sum operator.

 

  1. State the three different classes of network architectures and briefly describe any one of them with a diagram.

 

SECTION -C

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

  1. (a) Let i be a t-norm and strictly increasing and continuous function in (0,1) such that g(0)=0, g(1)=1. Prove that the function ,  where denotes pseudo inverse of g is a t-norm.

(b) Prove that the triples  and  are dual with

respect to any fuzzy complement.                                                                    (15+5)

 

  1. Let MIN and MAX be binary operations on the set of all fuzzy numbers. Prove that for any fuzzy numbers A, B, C the following properties hold:

(a) MIN(A,MIN(B,C))=MIN(MIN(A,B),C)

(b) MAX(A,MAX(B,C))=MAX(MAX(A,B),C)

(c) MIN(A,MAX(A,B))=A

(d) MAX(A,MIN(A,B))=A

(e) MIN(A,MAX(B,C))=MAX(MIN(A,B), MIN(A,C))

(f) MAX(A,MIN(B,C))=MIN(MAX(A,B), MAX(A,C))

 

  1. (a)Explain the indirect method of constructing a membership function with one expert.

(b) State the role of ‘activation function’ in neural networks. Describe the three basic types of activation functions.                                                               (10 + 10)

 

  1. (a) Briefly explain the three practical issues in ‘Pattern Recognition’.

(b) State the problem of ‘Fuzzy Clustering’ and present the Fuzzy c-means

algorithm.                                                                                                 (6 + 14)

 

 

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Loyola College M.Sc. Statistics April 2009 Reliability Theory Question Paper PDF Download

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      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 40

SECOND SEMESTER – April 2009

ST 2957 / ST 2955 – RELIABILITY THEORY

 

 

 

Date & Time: 27/04/2009 / 1:00 – 4:00   Dept. No.                                                   Max. : 100 Marks

 

 

SECTION –A (10 x 2 = 20)         

Answer any TEN questions. Each question carries TWO marks

 

  1. Define the following: (i) Mean time before failure(MTBF)

(ii) Steady state availabilty

  1. If the hazard function r(t)=3t2, t>0, obtain the corresponding probability distribution of

time to failure

  1. Obtain the reliability of a parallel system consisting of n components, when the

reliability of each component is known. Assume that the components are non-repairable.

  1. Explain in detail an n unit standby system.
  2. What is meant by reliability allocation?
  3. Define a coherent structure and give two examples.
  4. If x is a path vector and y ≥ x, show that y is also a path vector.
  5. Write down the structure function for a two out of three system.
  6. Let h ( p ) be the system reliability of a coherent structure. Show that h(p)is strictly

increasing in each pi, whenever 0<pi<1, i=1,2,3,…,n .

  1. Give an example of a distribution, which is IFR as well as DFR.

 

SECTION-B (5×8 =40 marks)

Answer any FIVE questions. Each question carries EIGHT marks

 

  1. Obtain the reliability function, hazard rate and the system MTBF for the following

failure time density function

f(t) = 12 exp(-4 t3)t2,  t>0.

  1. 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 the parameter λi, i= 1,2,…,n. Find the system reliability.

  1. Find the system MTBF for a (k,n) system, when the lifetime distribution is

exponential with the parameter λ. Assume that the components are non-repairable.

  1. Assuming that the components are non-repairable and the components have identical

constant failure rate λ, obtain the MTBF of the series-parallel system.

  1. Let Φ be a coherent structure. Show that

Φ(x .y ) ≤ Φ(x ) Φ(y )

Show that the equality holds for all x and y if and only if the structure is series.

  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.

Also, show that the equality holds if and only if the system is parallel.

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

the set of associated random variables.

  1. Show that Wiebull distribution is a DFR distribution.

 

 

SECTION-C (2X20=40 marks)

Answer any two questions. Each question carries TWENTY marks

 

  1. a) Obtain the reliability function, hazard rate and the system MTBF for exponential

failure time distribution with the parameter λ.                                                 (8 marks)

  1. b) Obtain the system failure time density function for a (m, n) system. Assume that

the components are non-repairable.                                                      (12 marks)

20.a) Define the terms: (i) Hazard rate  and (ii) Interval reliability                (4 marks)

  1. b) For a simple 1 out of 2 system with constant failure rate λ and constant repair rate

μ, obtain the system of  differential-difference equations. Also, obtain

the system reliability and system MTBF.                                              (16 marks)

21.a) Define: (i) Dual of a structure (ii) Minimal path vector and (iii) Minimal cut vector

(6 marks)

  1. b) Let h be the reliability function of a coherent system. Show that

h( p . p‘) ≤ h( p ) . h( p‘) for all 0  ≤ p , p‘ ≤ 1.                        (10 marks)

Also show that the equality holds if and only if the system is series.

  1. c) If X1, X2, …, Xn are associated binary random variables, show that

(1-X1), (1-X2),…,(1-Xn) are  also  associated binary random variables.(4 marks)

22.a) If the probability density function of F exists, show that F is an IFR

distribution iff  r(t)↑t.                                                                            (10 marks)

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

 

 

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Loyola College M.Sc. Statistics April 2009 Probability Theory Question Paper PDF Download

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     LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

YB 52

M.Sc. DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – April 2009

S  815 – PROBABILITY THEORY

 

 

 

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

 

 

SECTION-A (10 × 2=20)

Answer ALL the questions.

 

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

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

∞

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

n=1

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

F(x) =   0         if x<0

x/2      if o≤ x<1

  • if x ≥ 1.
  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. State Glivenko-Cantelli theorem.
  2. Φ is the characteristic function (CF) of a random variable X, find the CF of (2X+3).
  3. State Kolmogorov’s strong law of large numbers(SLLN).
  4. State Lindeberg-Feller central limit theorem.

 

SECTION-B (5 × 8 = 40)

Answer any FIVE questions.

 

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

defining properties.

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

variables.

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

convergence in distribution.

  1. a) Define “Convergence in quadratic mean” for a sequence of random variables.
  2. b) 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.      (2+6)

 

  1. Establish the following:

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

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

P [lim sup │ ‌Xn – X│> є ]= 0

  1. 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 (2 x 20 = 40 marks)

Answer any TWO questions

 

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

distribution function.

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

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

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

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

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

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

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

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

distribution, prove that (Xn2 + Yn2) → (X2+Y2) in distribution.            (10 marks)

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

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

probability one.                                                                          (10 marks)

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

functions.(12 marks).

b)Let {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).

 

 

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Loyola College M.Sc. Statistics April 2009 Sampling Theory Question Paper PDF Download

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      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 38

SECOND SEMESTER – April 2009

ST 2813 / 2810 – SAMPLING THEORY

 

 

 

Date & Time: 24/04/2009 / 1:00 – 4:00  Dept. No.                                                     Max. : 100 Marks

 

 

SECTION – A

 

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

 

  1. Define a parameter and a statistic.  Give an example for both.
  2. Give an example for an estimator which is unbiased under a sampling design.
  3. Show  that

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

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

  1. Prove that an unbiased estimator for the population total can be found if and only if the first order inclusion probabilities are positive for all N units in the population.
  2. Prove that E p (  s y ­2  )  =  S y 2    under  SRSWOR  Design.
  3. Define Midzuno Sampling Design.  Verify whether or not this design is a probability sampling design.
  4. Describe Random Group Method for selecting a sample and write the estimator for population total under this method.
  5. List all possible Modified Systematic Samples of size 8 when the population size is 40.
  6. Check whether LR is more efficient than   R .
  7. Prove that the Desraj ordered estimator is unbiased for the population total.

 

SECTION – B

 

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

     

  1. Write the unit drawing mechanism for implementing SRSWOR Design and show that this mechanism implements the design.

 

  1. 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. Check whether or not LSS is more efficient  than SRS for population with linear trend.

 

  1. Show that the usual expansion estimator is unbiased for the population total in CSS when there is a linear trend in the population.
  2. Check whether the estimated variance v( HT  ) is  non-negative under MSD for all “ s ” receiving positive probabilities.

 

  1. Explain Simmon’s unrelated randomized response model and obtain the estimate of ΠA when ΠY is unknown.

 

  1. Derive the estimated variance of DR.
  2. Derive the formula for n h under Cost Optimum Allocation.

 

SECTION – C 

 

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

 

19 ( a ) Illustrate that an estimator can be unbiased under one design but biased under

another design.                                                                                         ( 10 )

( b )  Derive  HT   and  V (HT ) using the formula for Π i  and  Π ij  under SRSWOR

Design.                                                                                                     ( 10 )

20 ( a ) Describe Warner’s randomized response technique and explain the procedure

For estimating the proportion Π A .                                                         ( 10 )

( b ) Deduce the expressions for   St ,   V (St )   and  v (St ) when samples are

drawn   independently from different strata using    ( i )  SRSWOR,  and

( ii )  PPSWR Designs.                                                                              ( 10 )

  1. Find 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 Designs.                                                                 ( 20 )

22 ( a ) Verify whether or not the  Hansen-Hurwitz estimator dhh  under double

sampling is unbiased  for Y and derive its variance.                                 ( 10 )

( b ) Find the mean and variance of TS ,  the estimator for population total, under

Two – Stage Sampling with SRS in both stages.                                    ( 10 )

 

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Loyola College M.Sc. Statistics April 2009 Measure And Probability Theory Question Paper PDF Download

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      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 32

FIRST SEMESTER – April 2009

ST 1809 – MEASURE AND PROBABILITY THEORY

 

 

 

Date & Time: 25/04/2009 / 1:00 – 4:00  Dept. No.                                                     Max. : 100 Marks

 

 

SECTION A

      Answer all questions.                                                                              (10  x 2 = 20)

 

  1. Define limit inferior of a sequence of sets.
  2. Mention the difference between a field and a σ – field.
  3. Give an example for counting measure.
  4. Define Minimal σ – field.
  5. Show that a Borel set need not be an interval.
  6. Define Signed measure.
  7. State Radon – Nikodym theorem.
  8. Show that the Lebesgue measure of any interval is its length.
  9. State Borel-Cantelli lemma.
  10. Mention the various types of convergence.

 

SECTION B

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

 

  1. Let be an increasing sequence of real numbers and let. What is the connection between a.) and b.) and ?

 

  1. Show that every finite measure is a σ – finite measure but the converse need not be true.

 

  1. State and prove the order preservation property of integrals and hence show that if exists then.

 

  1. Show that ifis finite, then  is finite for.

 

  1. State and prove Monotone convergence theorem for conditional expectation given a random object.

 

  1. Show that the random variable X having the distribution function is neither discrete nor continuous.
  2. State and prove Chebyshev’s inequality.

 

  1. If e , show that

a.)  a.e   and

b.)  a.e .

 

                                                     

SECTION C

 

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

 

  1. ) Let andbe two increasing sequences of sets defined
    on. If  then show that.

b.) If  exists, show that where ‘c’ is a constant.
(6+14)

 

  1. State and prove basic integration theorem.

 

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

 

b.) State and prove Minkowski’s inequality.                                      (10+10)

 

  1. ) Derive the defining equations of the conditional expectation given a random
    object and given a -field.

 

b.) Let Y1,Y2,…,Yn be iid random variables from U(0,θ), θ > 0. Show that
where Xn = max{Y1,Y2,…,Yn}.                              (10+10).

 

 

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Loyola College M.Sc. Statistics April 2009 Estimation Theory Question Paper PDF Download

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    LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 36

SECOND SEMESTER – April 2009

ST 2811 / 2808 – ESTIMATION THEORY

 

 

 

Date & Time: 20/04/2009 / 1:00 – 4:00       Dept. No.                                                       Max. : 100 Marks

 

 

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

 

01.Give an example of a parametric function for which unbiased estimator is unique.

02.State any two loss functions for simultaneous estimation problem.

03.Show that UMVUE of a parametric function is unique.

04.Define Fisher information for the multiparameter situation.

05.Define bounded completeness and give an example.

06.Given a random sample of size 2 from E(0, σ), σ>0, suggest two ancillary statistics.

07.Define a scale equivariant estimator and give an example.

08.Let X follow N( θ,1), θ = 0, 0.1. Find the MLE of  θ .

09.If  δ is consistent for θ, show that there exists infinitely many consistent estimators of θ.

10.Describe Conjugate family and give an example.

 

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

 

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

Hence find the class of unbiased estimators of N and N2.

12.State Cramer-Rao inequality for the multiparameter case. Hence find the Cramer- Rao

lower bound for estimating  σ/μ  based on a random sample from N(μ,σ2), μ ε R, σ > 0.

  1. Discuss the importance of Fisher information in finding a sufficient statistic.
  2. Let X1,X2,…,Xn be a random sample from U(0, θ), θ >0. Find a minimal sufficient

statistic and examine whether it is complete.

15.State and establish Basu’s theorem.

16.Given a random sample from N(0, τ2), τ > 0, find MREE of τ 2  with respect

to  standardized squared error loss. Is it unbiased ?

17.Find  MREE of the location parameter with respect to absolute error loss based on a

random sample from E(ξ, 1), ξ ε R.

  1. Let X1,X2,…,Xn be a random sample from P(θ), θ > 0. If the prior distribution is E(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 any two properties of Fisher information.

  1. b) Let X have the pdf

P( X = x) = (1- θ)2 θx , x = 0,1,…  ; 0< θ < 1

=  θ,  x = -1.

Using Calculus approach examine whether UMVUE of the following parametric functions

exist:  i) θ     ii) (1 – θ)2.

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

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

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

21 a) Show that the bias and the risk associated with a location equivariant estimator do not depend

on the parameter.

  1. b) Show that a location equivariant estimator δ is an MREE if and only if E0(δu) = 0 for each

invariant function u.
22 a) Given a random sample from N(μ,σ2), μ ε R, σ > 0, find the maximum likelihood

estimator of (μ,σ2). Examine whether it is consistent.

  1. b) Stating the regularity conditions, show that the likelihood equation admits a solution which

is consistent.

 

 

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Loyola College M.Sc. Statistics April 2009 Applied Regression Analysis Question Paper PDF Download

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    LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 34

FIRST SEMESTER – April 2009

ST 1811 – APPLIED REGRESSION ANALYSIS

 

 

 

Date & Time: 30/04/2009 / 1:00 – 4:00 Dept. No.                                                   Max. : 100 Marks

 

 

SECTION – A

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

  1. What is a multiple linear regression model ?

2.Why do regressions have negative signs. Give reasons.

  1. Explain BLUE.
  2. Explain the co-efficient of determination
  3. State any two ways in which ‘specification Error’ occurs.
  4. What is multi collinearity?

7.What is the formula for finding the adjusted r-square?

  1. What is Residuals ?
  2. Why do we use Dummy variables in a model?
  3. What are response and explanatory variables?

SECTION – B

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

  1. What are the three components specified in a generalized linear model? Explain in detail.
  2. Explain in detail categorical data analysis with examples. What are the two primary types of scales of categorical variables? Give example.
  3. What is the form of logistic regression model? Also give link function for a logistic regression model?
  4. Explain the four methods of scaling the Residuals ?
  5. Write short notes on Residual Plot.
  6. Estimate bo , b1, and s of  simple linear regression model by MLE
  7. Give an application scenario to illustrate the simple regression model .
  8. Write a short note on detecting multi collinearity.

SECTION –C

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

  1. Give an illustration and explain the following in detail :

a)Binomial logit models for binary data

b)Poisson log linear model for count data                   (10+10 Marks)

  1. a) Explain the procedure of standardizing the regression model using the

(i) Unit normal scale and (ii) Unit length scale                 (5+5 Marks)

  1. b) Explain probit and complementary log-log model (10 Marks)
  2. Explain the following methods for scaling the residuals.

(i)  Standardized residuals

(ii) Studentized residuals

(iii) Press residuals

(iv) R Student residuals                                  (5 Marks each)

  1. a) Derive the procedure for testing the hypothesis that all of the regression slopes are zero.

(10 Marks)

  1. b) Derive the least square estimates of the parameters of a simple regression model.(10 Marks)

 

 

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Loyola College M.Sc. Statistics April 2009 Applied Experimental Design Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AF 01

FOURTH SEMESTER – April 2009

ST 4805 – APPLIED EXPRIMENTAL DESIGN

 

 

 

Date & Time: 18/04/2009 / 9:00 – 12:00       Dept. No.                                                       Max. : 100 Marks

 

 

 

SECTION – A

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

 

  1. What is meant by non statistical principle of experimental design?
  2. Briefly explain the term Random effect model with an example .
  3. When do we go in for factorial design ?
  4. State the minimal function for 52 factorial design.
  5. Define the “term irreducible polynomial”.
  6. State the formula for a missing value in a LSD ?
  7. Define the term Aliases with an example .
  8. Distinguish between RLSD and LSD.
  9. Give any two advantages of BIBD.

10.Write the homogeneous equation for the highest order interaction in the case of 24

       factorial design.

 

SECTION-B

 

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

 

  1. Explain the term “Effiency of LSD relative to RBD” with suitable illustration
  2. Define linear contrast show that in 25 designs the main effects and interaction

effects are mutually orthogonal.

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

hypothesis, ANOVA and conclusions.

14.Develop the analysis of variance for a 24 fractional design in which the highest

order interaction is confounded in all the replications

  1. If G (F) = pn when p = 11 and n = 1 list the elements of the finite field

and explain all the operations with suitable example.

  1. State clearly the model used in the case of Youden Square and construct a real life

example.

  1. Derive the minimal function for 23 experiment and hence list the power cycle.
  2. State and prove all the parametric conditions of a BIBD.

 

 

 

 

 

 

SECTION-C

 

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

 

19.a) Discuss in detail the applications of the finite field with suitable illustrations.

  1. b) Define the term key BLOCK in the case of 25 factorial design of size 23 in

which 2 independent interactions and 1 generalized interaction are Confounded

Discuss in detail using the required linear equations for the confounded effects.

(8 +12-Marks)

20.a) When do we go for Split plot design? Explain with an example.

 

  1. b) Develop the Analysis of variance for the Split-plot design stating all the

hypothesis ANOVA and Inferences.                                            (8 +12-Marks)

 

21 a) Distinguish between Lattice Square and Lattin square designs.

  1. b) Explain the m-ple Lattice Square design and hence construct lattice square design

when the block size k=5.                   (8+12-Marks)

 

22 Write shorts on the following

  1. Critical difference
  2. PBIBD
  3. Response surface design
  4. Primitive root (5+5+5+5-Marks)

 

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Loyola College M.Sc. Statistics April 2009 Analysis Question Paper PDF Download

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   LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 31

FIRST SEMESTER – April 2009

ST 1808 – ANALYSIS

 

 

 

Date & Time: 17/04/2009 / 1:00 – 4:00 Dept. No.                                                     Max. : 100 Marks

 

SECTION-A (10X2=20 marks)

                                                                    Answer ALL the questions.

 

(1)Define a metric space and distinguish between bounded and unbounded metric spaces.

1

(2) Show that ρ (f, g) =    ∫| f(x)- g(x)| . dx is a metric on the class of all bounded , continuous real

0

functions on  [0,1].

(3)  Examine if the set of all vectors (x1, x2, x3) with x1+x 2= 1 is a vector space, where x1, x2, x3  Î R.

(4)  Examine whether the set {1, ½, 1/3, 1/4…} is closed.

(5)  Examine if the classes of closed and open sets are mutually exclusive and exhaustive.

(6) Show that the intersection of two open sets is open.

(7)  Show that every convergent sequence in a metric space is a Cauchy sequence.

(8)  If  Ω:X →  X is defined  as Ω (x)=x2 , where X=[0,1/3] , show that Ω  is a contraction mapping  on

[0,1/3].

(9)  Prove that any continuous image of a compact space is compact.

(10)For any sequence (xn) in R, show that

lim inf (-xn) =  – lim sup xn.

 

SECTION-B (8 X 5=40 marks)

Answer any FIVE questions. Each question carries EIGHT marks.

(11)  Let X and Y be two metric spaces with ρ1 and ρ2 as the respective metrics. Show that

ρ { (x 1, x2),(y1, y2 )} = max { ρi, (xi, yi) | i=1, 2}

is a metric on the Cartesian  product XxY.  Further, show that if X and Y are complete, then X×Y is

also complete.

(12)  Show that (a) the union of any collection of open sets is open.

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

(13)  Let X and Y be two metric spaces and f a mapping of X into Y. Prove that f is continuous if and only

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

(14)  Let (X, ρ) be any metric space. Let a be a fixed point of X and let the function g: X → R be defined

by the equation g(x) =ρ (a, x) for all xЄ X. Show that  g is continuous on X.

(15)  State and prove Banach’s fixed point theorem.

(16)  Define uniform convergence. Let (X, ρ) and (Y, σ ) be two metric spaces.  Let fn: X→ Y be a

sequence of functions converging uniformly to a function f: X →Y. If each fn is continuous at c,

show  that f is continuous at c.

(17) State and prove Cauchy’s necessary and sufficient condition for the uniform convergence of a

sequence of functions.

(18)  State and prove Dini’s theorem for a sequence of real valued functions.

 

 

SECTION-C (2×20=40 marks).

Answer any TWO questions. Each question carries TWENTY marks.

(19)(a)   Prove that, if V is an inner product space, then for all x,y Є V,

IIx+y II 2+ II x-yII2 = 2[IIxII 2+ II yII2].                                             (4 marks)

(b)  The sequences {xn}, {yn} in the normed vector space V converge to x, y respectively and the

numerical sequences { αn}, { βn } converge to α, β respectively. Show that

αn  xn + βn  yn → αx + βy.

Prove also that, if V possesses an inner product, then   xn. yn → x. y                  (8marks)

(c)   Prove that the metrics ρ, σ on X are equivalent if there are constant λ, μ>0 such that

λ .ρ(x, y) ≤ σ(x, y) ≤ μ. ρ(x, y) for all x, y Є X. Give an example to show that the converse is not

true.                                                                                                                      (8 marks)

(20)(a)   State and establish the necessary and sufficient condition for a set F to be closed. (10 marks)

(b)  Prove that the set of real numbers is complete.                                                   (10 marks)

(21) (a)  Let (X, ρ) be a metric space and let E с X. Show that

(i)  if E is compact, then E is bounded and closed.                                                         (6 marks)

(ii) if X is compact and  E is closed , then E is compact.                                                 (6 marks)

(b)  Show that a continuous function with compact domain is uniformly continuous.(8 marks)

(22)(a)  State and prove the necessary and sufficient condition for a bounded real valued function

f Є R (g; a, b).                                                                                                         (10 marks)

(b)   If f1, f2 Є R (g; a, b), prove that (f1 + f2) ЄR (g; a, b) and

b                               b                b

∫ (f1+f2) dg =    ∫ f1dg + ∫ f2dg                                                            (10marks)

a                               a                a

 

 

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Loyola College M.Sc. Statistics April 2009 Advanced Operations Research Question Paper PDF Download

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     LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 49

FOURTH SEMESTER – April 2009

ST 4807 – ADVANCED OPERATIONS RESEARCH

 

 

 

Date & Time: 23/04/2009 / 9:00 – 12:00  Dept. No.                                                    Max. : 100 Marks

 

 

SECTION -A                                                                                                                                 

Answer all the questions                                                                  10 x 2 = 20 marks

 

  1. When a solution to an LPP is called infeasible?
  2. How dual simplex method differs from other simplex methods ?
  3. Define holding and penalty costs.
  4. Write basic components of a queuing model.
  5. Write the significance of integer programming problem.
  6. Define Dynamic Programming Problems.
  7. Differentiate goal programming from other programming problems.
  8. Write a note on complementary slackness condition.
  9. Provide any two  applications for parallel  and sequence service systems.
  10. For a single item static model if D = 100 , h = $0.02 , K = $100 and lead

time is 10 days,find the economic order quantity and re order point.

 

                                                                SECTION -B                                                                                                                                 

Answer any five questions                                                                 5 x 8 = 40 marks

 

  1. Use the graphical method to solve the following LPP:

Maximize Z = 2x1 + 3x2

Subject to the constraints:                                                                                                           x1 + x2 ≤ 30 ,  x1 – x2 ≥ 0 , x2 ≥ 3 , 0≤ x1 ≤ 20 and 0 ≤ x2 ≤ 12.

  1. Write big M method algorithm.
  2. Use duality to solve the following LPP:

Maximize Z = 2x1 + x2

Subject to the constraints:

x1 + 2x2 ≤ 10  ,   x1 + x2 ≤ 6 ,  x1 – x2 ≤ 2 , x1 – 2x­2 ≤ 1 ; x1,x2 ≥ 0 .

  1. Write briefly about inventory management.
  2. Derive the steady state measures of (M/M/1) : (GD/∞/∞) queuing model.
  3. Write Beale’s algorithm to solve Quadratic Programming Problem.
  4. Obtain the set of necessary and sufficient conditions  for the following NLPP.

Minimize Z = 2x12 – 24x1 + 2x22 – 8x2 + 2x32 – 12x3 + 200

Subject to the constraints:

x1 + x2 + x3 = 11 ,  x1,x2, x3 ≥ 0 .

  1. Solve the following NLPP using Kuhn- Tucker conditions :

Maximize Z =  –x12 – x22 – x32 + 4x1 + 6x2

Subject to the constraints:

x1 + x2  ≤ 2  ,   2x1 + 3x2 ≤ 12 ; x1, x2 ≥ 0

 

 SECTION -C                                                                                                                              

Answer any two questions                                                                    2x 20 = 40 marks

 

19.(a) Use two-phase simplex method to

Maximize Z =  5x1 + 8x2

Subject to the constraints:

3x1 + 2x2 ≥ 3 , x1 + 4x2 ≥ 4 , x1 + x2 ≤ 5 ; x1, x2 ≥ 0

 

  • Use dynamic programming to solve:

Minimize Z = x12 + 2x22 + 4x3

Subject to the constraints:

x1 + 2x2 + x3 ≥ 8 ;  x1 ,x2 , x3 ≥ 0.

(12 + 8 )                                                                 20(a)  Derive probabilistic EOQ model.

 

(b)  Electro uses resin in its manufacturing process at the rate of 1000 gallons

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 optimum ordering policy for Electro.

(10 + 10)

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

Maximize Z = 6x1 + 3x2 – 4x1x2 – 2x12 – 3x22

Subject to the constraints:

x1 + x2 ≤ 1  ,  2x1 + 3x2 ≤ 4 ; x1, x2 ≥ 0 .

 

  1. Use cutting plane algorithm to solve the following LPP:

Maximize Z = 200x1 + 400x2 + 300x3

Subject to the constraints:

30x1 +  40x2 + 20x3 ≤  600

20x1 + 10x2 + 20x3  ≤  400

10x1 + 30x2 + 20x3  ≤  800

x1, x2, x3 ≥ 0  and are integers.

 

 

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