Loyola College M.Sc. Statistics April 2006 Applied Regression Analysis Question Paper PDF Download

November 20, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 28

FIRST SEMESTER – APRIL 2006

ST 1811 – APPLIED REGRESSION ANALYSIS

Date & Time : 27-04-2006/1.00-4.00 P.M. Dept. No. Max. : 100 Marks

SECTION – A

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

Define ‘Residuals’ of a linear model.
What is Partial F- test.
What are the two scaling techniques for computing standardized regression coefficients.
Define ‘Externally Studentized Residuals’.
Stae the variance stabilizing tramsformation if V(Y) is proportional to [E(Y)]³.
What is F_OUT in Backward selection process.
How is the multicollinearity trap avoided in regression models with dummy variables.
State any one method of detecting multicollinearit.
Give an example of a polynomial regression model.
Give the motivation for Generalized Linear Models.

SECTION – B

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

Fill up the missing entries in the following ANOVA for a regression model with 5 regressors and an intercept:

Source

d.f

S.S

Mean S.S.

F ratio

Regression

Residual

40.5

13.5

——-

Residual

——–

——-

Also, test for the overall fit of the model.

The following table gives the data matrix corresponding to a model
Y = b₀+b₁X₁+b₂X₂+b₃X₃. Suppose we wish to test H₀: b₂ = b₃. Write down the restrcited model under H₀ and the reduced data matrix that is used to build the restricted model.

1 2 -3 4

1 -1 2 5

1 3 4 -3

1 -2 1 2

X = 1 4 5 -2

1 -3 4 3

1 2 3 1

1 1 2 5

1 4 -2 2

1 -3 4 2

Explain how residual plot are used to check the assumption of normality of the errors in a linear model.

Discuss ‘Generalized Least Squares’ and obtain the form of the GLS estimate.

Explain the variance decomposition method of detecting multicollinearity and derive the expression for ‘Variance Inflation Factor’.
Discuss ‘Ridge Regression’ and obtain the expression for the redge estimate.

Suggest some strategies to decide on the degree of a polynomial regression model.

Describe Cubic-Spline fitting.

SECTION – C

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

Build a linear regression model with the following data and test for overall fit . Also, test for the individual significance of X₁ and of X₂.

Y: 12.8 13.9 15.2 18.3 14.5 12.4

X₁: 2 3 5 5 4 1

X_2:4 2 5 1 2 3

(a)Decide whether “Y =b₀ + b₁X” or “Y² = b₀ + b₁X” is the more appropriate model for the following data:

X: 1 2 3 4

Y: 1.2 1.8 2.3 2.5

(b)The starting salary of PG students selected in campus interviews are given below

along with the percentage of marks they scored in their PG and their academic

stream:

Salary (in ‘000 Rs)

Stream

Gender

% in PG

12.5

7.5

Arts

Science

Commerce

Science

Arts

Commerce

Science

Commerce

Male

Female

Male

Female

Male

Female

It is believed that there could be a possible interaction between Stream and % in

PG and between Gender and % in PG. Incorporate this view and create the data

matrix. (You need not build the model). (10+10)

Based on a sample of size 16, a model is to be built for a response variable with four regressors X₁, …,X₄. Carry out the Forward selection process to decide on the significant regressors, given the following information:

SS_T = 1810.509, SS_Res(X₁) = 843.88, SS_Res(X₂) = 604.224, SS_Res(X₃) = 1292.923, SS_Res(X₄) = 589.24, SS_Res(X₁, X₂) = 38.603, SS_Res(X_1,X₃) = 818.048, SS_Res(X_1,X₄) = 49.84, SS_Res(X₂,X₃) = 276.96, SS_Res(X_2,X₄) = 579.23, SS_Res(X_3,X₄) = 117.14, SS_Res(X_1,X₂,X₃) = 32.074, SS_Res(X_1,X₂, X₄) = 31.98, SS_Res(X_1,X₃,X₄) = 33.89, SS_Res(X_2,X₃,X₄) = 49.22, SS_Res(X_1,X₂,X₃,X₄) = 31.91.

(a) Obtain the likelihood equation for estimating the parameters of a logistic regression model.

(b) If the logit score (linear predictor) is given by –2.4 + 1.5 X₁ + 2 X₂, find the estimated P(Y = 1) for each of the following combination of the IDVs:

X₁: 0 1.5 2 3 -2 -2.5

X₂: 1 0 1.5 -1 2 2.5 (12+8)

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Loyola College M.Sc. Statistics April 2006 Applied Regression Analysis Question Paper PDF Download

November 20, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 52

FOURTH SEMESTER – APRIL 2006

ST 4954 – APPLIED REGRESSION ANALYSIS

Date & Time : 27-04-2006/9.00-12.00 Dept. No. Max. : 100 Marks

SECTION –A

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

State the statistic for testing the overall fit of a linear model with ‘k’ regressors.
Define ‘Extra Sum of Squares’.
Define ‘Studentized’ Residuals.
What is a ‘Variance Stabilizing Transformation’?
State the consequence of using OLS in a situation when GLS is required.
Define “Variance Inflation Factor’.
Give the form of the Ridge Estimate when a constant ‘l’ is added to the diagonal elements of X’X.
What is a hierarchical polynomial regression model?
Mention the components of a ‘Generalized Regression Model’ (GLM).
Define ‘Sensitivity’ of a Binary Logit Model.

SECTION – B

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

The following table gives the data on four independent variables used to build a linear model with an intercept for a dependent variable.

X₁

X₂

X₃

X₄

-2

-3

-1

-2

-3

-1

-2

-3

-1

-2

If one wishes to test the hypothesis H₀: b₁ = b₃, b₂ = 2b₄, write down the reduced

data matrix and the restricted model under H₀. Briefly indicate the test procedure.

Depict the different possibilities that occur when the residuals are plotted against the fitted values. How are they interpreted?

Define ‘Standardized Regression Coefficient’ and discuss any one method of scaling the variables.

Decide whether “Y= b₀ + b₁X” or “Y^1/2 = b₀ + b₁X” is the more appropriate model for the following data:

X	1	2	3	4
Y	3.5	4.7	6.5	9.2

Discuss the issue of ‘multicollinearity’ and its ill-effects.

Eigen Values

of X’X

Singular

Values of X

Condition

Indices

Variance decomposition Proportions

X₁ X₂ X₃ X₄ X₅ X₆

3.4784

2.1832

1.4548

0.9404

0.2204

0.0725

0.0003 0.0005 0.0004 0.0004 ? 0.0350

? 0.0031 0.0001 0.3001 0.0006 0.0018

0.0004 ? 0.0005 0.0012 0.0032 0.2559

0.0011 0.6937 0.5010 0.0002 0.7175 ?

0.0100 0.0000 ? 0.0003 0.0083 0.2845

0.8853 0.3024 0.4964 ? 0.2172 0.0029

Fill up the missing entries in the following table and investigate the presence of collinearity in the data, indicating which variables are involved in collinear relationships, if any.

Explain ‘Cubic Spline’ fitting.

Describe the components of a GLM. Show how the log link arises naturally in modeling a Poisson (Count) response variable.

SECTION – C

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

The observed and predicted values of a response variable (based on a model using 25 data points) and the diagonal elements of the ‘Hat’ matrix are given below:

Y_i	16.68 11.50 12.03 14.88 13.75 18.11 8.00 17.83 79.24 21.50 40.33 21.0 13.5
Y_i^	21.71 10.35 12.08 9.96 14.19 18.40 7.16 16.67 71.82 19.12 38.09 21.59 12.47
h_ii	0.102 0.071 0.089 0.058 0.075 0.043 0.082 0.064 0.498 0.196 0.086 0.114 0.061

Y_i	19.75 24.00 29.00 15.35 19.00 9.50 35.10 17.90 52.32 18.75 19.83 10.75
Y_i^	18.68 23.33 29.66 14.91 15.55 7.71 40.89 20.51 56.01 23.36 24.40 10.96
h_ii	0.078 0.041 0.166 0.059 0.096 0.096 0.102 0.165 0.392 0.041 0.121 0.067

Compute PRESS statistic and R²_prediciton. Comment on the predictive power of the

underlying model.

(a) In a study on the mileage performance of cars, three brands of cars (A, B and C) and two types of fuel (OR and HG) were used. The speed of driving was also observed and the data are reported below:

Mileage(Y)

14.5 12.6 13.7 15.8 16.4 13.9 14.6 16.7 11.8 15.3 16.8 17.0 15.0 16.5

Speed

Car

Fuel

45 60 50 60 55 52 59 50 40 53 62 56 62 55

A B C B A A C A B B C C A B

OR HG OR HG HG OR HG OR OR HG HG OR HG OR

Create the data matrix so as to build a model with an intercept term and interaction terms between Fuel and Driving Speed and also between Car-type and Driving Speed.

(You need not build any model).

(b) Discuss GLS and obtain an expression for the GLS estimate. (14 + 6)

Based on a sample of size 15, a linear model is to be built for a response variable Y with four regressors X₁,…,X₄. Carry out the Forward Selection Process to decide which of the regressors would finally be significant for Y, given the following information:

SS_T = 543.15, SS_Res(X₁) = 253.14, SS_Res(X₂) = 181.26, SS_Res(X₃) = 387.88, SS_Res(X₄) = 176.77, SS_Res(X₁,X₂) = 11.58, SS_Res(X₁,X₃) = 245.41, SS_Res(X₁,X₄) = 14.95, SS_Res(X₂,X₃) = 83.09, SS_Res(X₂,X₄) = 173.77, SS_Res(X₃,X₄) = 35.15, S_Res(X₁,X₂,X₃) = 9.62, SS_Res(X₁,X₂,X₄) = 9.59, SS_Res(X₁,X₃,X₄) = 10.17, SS_Res(X₂,X₃,X₄) = 14.76, SS_Res(X₁,X₂,X₃,X₄) = 9.57

The laborers in a coal-mine were screened for symptoms of pneumoconiosis to study the effect of “number of years of work” (X) on the laborers’ health. The response variable ‘Y’ defined as ‘1’ if symptoms were found and ‘0’ if not. The data on 20 employees are given below:

Y	0 1 1 0 1 1 0 0 0 1 1 1 0 0 1 0 0 1 1 1
X	10 30 28 14 25 35 15 12 20 24 33 27 13 12 18 17 11 28 32 30

The logit model built for the purpose had the linear predictor (logit score) function as – 4.8 + 0.1 X. Construct the Gains Table and compute the KS statistic. Comment on the discriminatory power of the model.

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

November 20, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 25

FIRST SEMESTER – APRIL 2006

ST 1808 – ANALYSIS

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

SECTION- A

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

Define a discrete metric. Show that it satisfies the properties of a metric.

If x_n → x and x_n→ y as n → ∞, show that x =

Define a norm on a vector space and give an example.
For all x, y є R⁽ⁿ⁾, show that x . y = is an inner product.
Check whether or not all the points of any open ball B( a ; r ) are the interior points of B( a ; r ).
Illustrate that an infinite union of closed sets is not closed.
If f is continuous, one-to-one and onto function, then show that f^-1 in general is not continuous.
Show that pointwise convergence does not imply uniform convergence of a sequence of functions.
Let f( x ) = x , 0 ≤ x ≤ 1 . Let D be the partition {0 ,¼ , ½ , ¾ ,1 } of [ 0 , 1 ] . Find the upper sum U( f ; D ) and the lower sum

L( f ; D ) of the function f( x ).

Let R ( g ; a , b ) be the collection of Riemann – Stieltjes integralble

functions with respect to g on [ a , b ] . If f є R (g ; a , b ),

show that kf є R (g ; a , b ) , where k is any constant.

SECTION – B

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

In B[ 1 , 2 ], with ρ( f , g ) = sup | f(x) – g(x) | ,

1≤ x ≤ 2

let f _n be given by f_n(x) = ( 1 + x ⁿ ) ^{1 / n} (1≤ x ≤ 2) .

Show that f_n → f where f(x) = x (1≤ x ≤ 2).

In a metric space (X , ρ ), if x_n → x and y_n→ y as n → ∞,

show that ρ( x_n, y_n) → ρ( x , y ) as n → ∞.

If V is an inner product space, prove that

║ x + y ║² + ║ x – y ║² = 2 [║ x ║² + ║ y ║² ] for all x , y є V.

State three equivalent conditions for a point c є X to be a limit point of E С X .
Show that every convergent sequence in a metric space is a cauchy sequence. Check whether or not the converse is true.
State and prove Banach’s fixed point principle.
Prove that a continuous function with compact domain is uniformly continuous.
State and prove Cauchy’s root test for the absolute convergence or divergence of a series of complex terms.

SECTION – C

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

19(a) With respect to the usual metric , prove that joint convergence of a

sequence is equivalent to the marginal convergence of the

components of that sequence. (10)

19(b) State and prove Cauchy – Schwartz inequality regarding inner

product space. (10)

20(a) Let V , W be the normed vector spaces. Let f : V → W be a linear

transformation. Then prove that the following three statements are

equivalent : (16)

f is continuous on V.
There exists a point x_o in V at which f is continuous.
║f(x)║ ∕ ║x║ is bounded for x є V – { ө }.

What do we conclude from the equivalence of statements (i) & (ii)?

20(b) Show that a compact set in a metric space is complete. (4)

21(a) Prove that the number Λ is the upper limit of the sequence

{x_n , n ≥ 1 } iff for all є > 0

x_n< Λ + є for all sufficiently large n and
x_n> Λ – є for infinitely many n. (10)

21(b) Let {f_n} be a sequence of real functions integrable over the

finite interval [a , b]. If f _n→ f uniformly on [a , b],

prove that f is integrable over [a , b]. (10)

22(a) State and prove the Cauchy’s general principle of uniform

convergence of a sequence of real or complex valued functions. (8)

22(b) State and prove a necessary and sufficient condition for a function

f(x) to be Riemann – Stieltjes integrable on [a , b]. (12)

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Loyola College M.Sc. Statistics April 2006 Advanced Operations Research Question Paper PDF Download

November 20, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 51

FOURTH SEMESTER – APRIL 2006

ST 4951 – ADVANCED OPERATIONS RESEARCH

Date & Time : 27-04-2006/9.00-12.00 Dept. No. Max. : 100 Marks

SECTION A

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

Define linearly independent vectors.
Define a Mixed Integer Programming Problem.
What is the need for Integer Programming Problems?
State Bellman’s principle of optimality.
What is meant by Separable Programming Problem?
Define Goal Programming Problem.
Write down the mathematical formulation of a Geometric Programming Problem.
Explain the stage and state variables in a dynamic Programming Problem.
Name the methods used in solving a Quadratic Programming Problem.
What is the need for Dynamic Programming Problem?

SECTION B

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

Explain the construction of fractional cut in the Gomory’s constraint method.

State all the characteristics of a Dynamic Programming Problem.

In the network given below are different routes for reaching city B from city A passing through a number of other cities, the lengths of the individual routes are shown on the arrows. It is required to determine the shortest route from A to B. Formulate the problem as a Dynamic Programming Problem model, explicitly defining the stages, states and then find the optimal solution.

5 3 2 4

7 4 2 2

Solve the following Non-Linear Programming Problem:

Optimize Z = X ² +Y ² + W ²,

subject to X +Y + W = 1,

X, Y, W ≥ 0.

Derive the Kuhn-Tucker necessary conditions for solving a Generalized Non-Linear Programming Problem with one inequality constraint.

Derive the orthogonality and Normality conditions for solving the unconstrained Geometric Programming Problem.

Convert the following Stochastic Programming Problem into an equivalent deterministic model, max Z = X₁ + 2 X₂ + 5 X₃ ,subject to

P [a₁ X₁ + 3 X₂ + a₃ X₃ ≤ 10 ] ≥ 0.9,

P [ 7 X₁ + 5 X₂ + X₃ ≤ b₂ ] ≥ 0.1,

X₁, X₂, X₃ ≥ 0.

Assume that a₁, a₃ are independent normally distributed random variables with means E (a₁) = 2, E (a₃) = 5, V (a₁) = 9, V (a₃) = 16. Also assume that

b₂ ~ N (15, 25).

The manufacturing plant of an electronics firm produces two types of T.V. sets, both colour and black-and-white. According to past experiences, production of either a colour or a black-and-white set requires an average of one hour in the plant. The plant has a normal production capacity of 40 hours a week. The marketing department reports that, because of limited sales opportunity, the maximum number of colour and black-and-white sets that can be sold are 24 and 30 respectively for the week. The gross margin from the sale of a colour set is Rs. 80, whereas it is Rs. 40 from a black-and-white set.

The chairman of the company has set the following goals as arranged in the order of their importance to the organization.

1. Avoid any underutilization of normal production capacity (on layoffs of production workers).
2. Sell as many T.V. sets as possible. Since the gross margin from the sale of colour T.V. set is twice the amount from a black-and-white set, he has twice as much desire to achieve sales for colour sets as black-and-white sets.
3. The chairman wants to minimize the overtime operation of the plant as much as possible.

Formulate this as a Goal Programming Problem.

SECTION C

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

Solve the following Integer Programming Problem:

Max Z = 3 X₁ + X₂ + 3 X₃

subject to – X₁ + 2 X₂ + X₃ ≤ 4,

4 X₂ – 3 X₃ ≤ 2,

X₁ – 3 X₂ + 2 X₃ ≤ 3,

X₁, X₂, X₃ ≥ 0.

(i) Solve the following Dynamic Programming Problem (DPP):

Min Z = subject to = C , x_j ≥ 0 , j = 1,2, … n. C > 0.

(ii) Solve the following LPP by DPP technique:

Max Z = 3 X₁ + 4 X₂ ,

subject to 2 X₁ + X₂ ≤ 40,

2 X₁ + 5 X₂ ≤ 180,

X₁, X₂ ≥ 0.

Use Kuhn-Tucker necessary conditions to solve the following Generalized Non- Linear Programming Problem:

Max Z = 2 X₁ – X₁² + X₂

subject to 2 X₁ + 3 X₂ ≤ 6,

2 X₁ + X₂ ≤ 4,

X₁, X₂ ≥ 0.

Solve the following Quadratic Programming Problem using Wolfe’s algorithm:

Max Z = 4 X₁ + 6 X₂ – 2 X₁² – 2 X₁ X₂ – 2 X₂² ,

subject to X₁ + 2 X₂ ≤ 2,

X₁, X₂ ≥ 0.

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

November 20, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 38

SECOND SEMESTER – APRIL 2006

ST 2953 – ACTUARIAL STATISTICS

Date & Time : 26-04-2006/9.00-12.00 Dept. No. Max. : 100 Marks

PART-A

ANSWER ALL QUESTIONS 10 ´ 2 = 20

A has invested Rs.1000 in National Defence savings certificate. After 15 years he is entitled to receive Rs.1750. What rate of interest is realized in the transaction.?
Find the nominal rate p.a convertible quarterly corresponding to an effective rate of 6% p.a.
Define a perpetuity due, immediate perpetuity .
Write the formula for (Ia)_n.
Express e_x in terms of l_x .
What is ?
Show that
What is double endowment assurance?
Write the formula for net interest yield of a life insurance company.
Given a complete table of for all values of x and n , how would you find the value of ?

Part-B

ANSWER ALL QUESTIONS 5 ´ 8 = 40

A has taken a loan of Rs.2000 at a rate of interest 4% p.a payable half-yearly. He paid Rs.400 after 2 years Rs.600 after a further 2 years and cleared all outstanding dues at the end of 7 years from the commencement of the transaction. What is the final payment made by him?

Find the amount of an annuity due of Rs. 300 p.a. payable 12 times a year for 20 years , on the basis of nominal rate oa 6%p.a. convertible 3 times a yaer. Find also the present value of these payment.
Three persons are aged 30,35,40 respectively.Find the probability that One of them dies before age 45 While the others survive to age 55. ii. None of them dies before age 50. iii. Atleast one of them attains age 65. iv. None of them survives upto age 65.
Find the present value of an annuity due of Rs.1000 p.a. for 20 yeare if the rate of interest is 8% p.a. for the first 12 years and 6% p.a. there after. Find also the accumulated value.
Calculate net annual premium under a special endowment assurance for Rs.18000 on (35) for 25 years, the premium being limited to 20 years. In the event of death during the term of assurance, total premium paid are returnable and on survivance to the end of 25 years, the basic sum assured becomes
Calculate office annual premium for a whole life assurance for Rs. 20000 to a person aged 40. provide for first year expenses at 55% of premiums 17 per thousand sum assured; and renewal expenses of 5% of premium and 6 per thousand sum assured.
Calculate the net single premium for an immediate annuity of Rs.1200 per annum payable half yearly in arrear for 15 years certain and thereafter for life to a person aged 60 at entry (basis: a(90) table and 8% interest).
Calculate the net annual premium under a children deferred whole life assurance

for Rs 5000 on the life of a child aged 8,the assurance vesting at age 18.

PART-C

ANSWER ANY TWO QUESTIONS (2 ´ 20 = 40)

a) A loan of Rs 5000 is to be repaid with interest at a rate of 6% p.a. by 18 level annual payment being made at the end of the first year. Immediately after the 10^th payment has been made the borrower requests the lender for extension of the term of the loan by another four years. What is the revised annual payment to be made during the next 12 years on the assumption that the lender to realise an interest of 7% hence forward? b) payments of i. 50/ at the end of each half year for the first 5 years followed by ii. Rs.50/ at the end of each quarter for the next years, one made in to account to which interest is credited at the rate of 9% p.a. convertible half yearly. Find the accumulated value at the end of 10 years.
a) Derive an expression for ii. iii. b) A special policy provides for the following benefits; i. An initial sum of Rs.10,000 with guaranted annual additions of Rs.250 for each year’s premium paid after the first , if death occurs within the term of assurance. ii. Rs.10,000 payable on survivance to the end of the term of assurance. iii. Free paid up assurance of Rs.10,000 at death after expiry of the term of assurance. calculate net annual premium under the policy on the life of (35) for 25 years.
a) A person aged 30 years has approached a life for a special type of policy providing for the following benefits; Rs.1000 on death during the first 5 years ii.Rs.2000 on death during the next 15 years. iii. survival benefit of Rs.500 at the end of the 5^th year. iv. Further payment of Rs.2000 on survivance to 20 years. v. Rs.150 paid for each premium paid. Calculate the yearly premium assuming the paying term is 20 years. b) Derive the formula for decreasing temporary assurance (mortgage redemption assurance).
a) Derive the formulas for net premiums of the various life annuity plans. b) calculate office annual premium for an endowment assurance for Rs.15,000 to a person aged 35 for 25 years. provide for first year expenses at 50% of premium and 15 per thousand sum assured; and renewal expenses of 5%of premiums and 6 per thousand sum assured. A bonus loading of 20 per thousand is also offered. Calculate the office annual premium.

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

November 20, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 M.Sc. DEGREE EXAMINATION – STATISTICS

AB 26

THIRD SEMESTER – NOV 2006

ST 3809 – STOCHASTIC PROCESSES

(Also equivalent to ST 3806)

Date & Time : 27-10-2006/9.00-12.00 Dept. No. Max. : 100 Marks

Section-A (10 × 2=20 marks)

Answer ALL the questions

Define (a) Stationary increments

(b) Independent increments of a stochastic process

Define the period of a state of a Markov chain. Show that an absorbing state is recurrent.
Let j be a state for which f_jj⁽ⁿ⁾ = n/(2⁽ⁿ⁺¹⁾), n>0. Show that j is recurrent.
Write down the postulates for a birth and death process.
Define a Renewal process {N(t),t ≥ 0} and write down its renewal function.
Define a submartingale.

∞

Let {X_n, n≥0} be a Branching process with the off spring mean m<1. Evaluate E[ Σ X_n].

n=0

Define a Brownian motion process.
Show that a Markov Renewal process is a Markov Chain with one step transition probabilities.
Distinguish between wide-sense and strictly stationary processes.

Section-B

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

Show that a Markov chain is fully determined, when its initial distribution and the one step transition probabilities of the Markov chain are known.
Define a transient state and prove that transience is a class property. For any state i and a transient state j, prove that

∞

Σ p_ij⁽ⁿ⁾<∞

n=1

Show that in a two dimensional symmetric random walk, all the states are recurrent.
Assume that a device fails when a cumulative effect of k shocks occur. If the shocks happen according to a Poisson process with the parameter λ, find the density function for the life T of the device.
Obtain the system of differential equations satisfied by the transition probabilities of the Yule process and calculate its transition probabilities when the initial condition is

X(0) = N.

Derive the integral equation satisfied by the renewal function of a Renewal process.
Let {X(t) | t Є[0,∞)} be a standard Brownian motion process. Obtain the conditional distribution of X(t) given X(t₁)= α and X(t₂)=β, where t₁<t<t₂.
If {X_n} is a Branching process and φ_n (s) is the probability generating function of X_n, show that φ_n satisfies the relation φ_n (s)= φ_n-k(φ_k (s) ) for all k such that

k= 1,2,…,n.

Section-C

Answer any TWO questions (2×20 =40)

19.a. Define a recurrent state. (2 marks)

State and prove the Chapman-Kolmogorov equations for a discrete time discrete space Markov Chain.(10 marks)
Consider a random walk on the integers such that p_i,i+1 = p, p_i,i-1=q for all integers i (0<p<1,p+q=1). Determine p₀₀⁽ⁿ⁾.Also find the generating function of p₀₀⁽ⁿ⁾.(8 marks)

20.a. Show that recurrence is a class property.(6 marks)

Show that states belonging to the same class have the same period.(6 marks)
If lm p_jj⁽ⁿ⁾>0, show that j is positive recurrent and aperiodic.(8 marks)

n→∞

21.a Stating the postulates for a birth and death process, derive Kolmogorov backward differential equations.(2+6 marks)

Obtain E[X(t)], where X(t) is a linear birth and death process.(12 marks)

22.a. Define a discrete time Martingale and show that the means of the marginal distributions are equal. (8 marks)

State and prove the prediction theorem for minimum mean square error predictors.

(12 marks)

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

November 20, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 27

FIRST SEMESTER – APRIL 2006

ST 1810 – ADVANCED DISTRIBUTION THEORY

(Also equivalent to ST 1806/ST 1803)

Date & Time : 20-04-2006/AFTERNOON Dept. No. Max. : 100 Marks

Section – A (2×10 = 20 marks)

Answer ALL the questions

If X and Y are independent Binomial variates with same parameters (n, p), show that the conditional distribution of X given by X+Y is a Hyper geometric distribution.
Let X_n be discrete uniform on {1/n, 2/n, 3/n …1}, n Є N. Find the moment generating function (MGF) of X_n.
Define truncated Poisson distribution, truncated at zero and hence find its mean.
State and prove the additive property of bivariate Binomial distribution.
Show that for a random sample of size 2 from N(0, σ²) population, E[X₍₂₎] = σ/√п
If (X₁, X₂) is bivariate normal, show that (X₁-X₂) is normal.
Define bivariate exponential distribution.
Show that in the case of bivariate exponential distribution, marginal distributions are exponential.
Write down the density function of non-central t-distribution. What is its non-centrality parameter?
Find the mean of non-central χ²– distribution.

Section – B (5×8 = 40 marks)

Answer any FIVE questions

Find the MGF of power series distribution. Show that Binomial and Poisson distributions are particular cases of power series distribution.
Establish the recurrence relation satisfied by raw moments of log-series distribution. Hence or otherwise, obtain the mean and variance of log-series distribution.
In a trinomial distribution with the parameters (n, p₁, p₂), show that the marginal distributions are Binomial. Also, find the correlation coefficient between X₁ and X₂.
If (X₁, X₂) is bivariate Poisson, obtain the conditional distributions and the regression equations.
For lognormal distribution, show that mean > median > mode.

Let X₁ and X₂ be independent and identically distributed random variables with positive variance. If (X₁ +X₂) and (X₁-X₂) are independent, show that X₁ is normal.
Show that the ratio of two independent standard normal variates is a Cauchy variate. Is the converse true?
State and prove the additive property of Inverse Gaussian (IG) distribution.

Section- C

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

19.a. Show that in the case of multinomial distribution, multiple regressions are linear. Hence find the partial correlation coefficient. (10 marks)

State and establish the additive property of trinomial distribution. (10 marks)

20.a. Obtain the MGF of bivariate Poisson distribution with the parameters (λ₁, λ₂, λ₃). Also find the covariance of bivariate Poisson distribution. (10 marks)

Let (X₁, X₂₎ be bivariate Poisson. Find the necessary and sufficient condition for X₁ and X₂ to be independent. (10 marks)

21.a. Let X₁, X₂ X₃, X₄be independentN(0,1) random variables. Find the distribution of (X₁X₄– X₂ X₃) (10 marks)

Let (X₁, X₂) have bivariate normal distribution with the parameters (0,0,1,1,ρ) . Find the correlation coefficient between X₁² and X₂². (10 marks)

22.a. Derive the density function of non-central F-distribution. (10 marks)

Find the mean and variance of non-central F-distribution.(10 marks)

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Loyola College M.Sc. Statistics Nov 2006 Statistical Computing – II Question Paper PDF Download

November 20, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 M.Sc. DEGREE EXAMINATION – STATISTICSTHIRD SEMESTER – NOV 2006ST 3810 – STATISTICAL COMPUTING – II

Date & Time : 30-10-2006/9.00-12.00 Dept. No. Max. : 100 Marks
(i) Choose either 1 or 2(ii) 3 is compulsory(iii) Choose either 4 or 5(iv) 1. Compare the performances of SRS-HTE strategy and SRS-HARTELY ROSS UNBIASED RATIO TYPE ESIMATOR strategy in estimating total population during the year 2006 using the following population data assuming the sample size is 2 (Treat 2004 data as auxiliary information)
Area : 1 2 3 4
Population in 2004 : 37 36 48 51 (in ‘000)
Population in 2006 : 41 49 51 57 (in ‘000)
2. Certain characteristics associated with a few recent US presidents are listed below:
President Birth region Elected first time Party Prior congressional experience Served as vice presidentReagen Midwest Yes Republican No NoCarter South Yes Democrat No NoFord Midwest No Republican Yes YesNixon West Yes Republican Yes YesJohnson South No Democrat yes Yes Define suitable binary variables to convert the above data into categorical data. Form clusters using single and complete linkage methods with suitable similarity measure. Draw dendograms and compare your results.
3. (a) It is decided to estimate the proportion of students in a college having the habit of indulging in malpractice during examinations. Two random experiments were deviced. Device 1 when conducted will result in either the question “Do you indulge in copying during examinations ? “ or “Do you know the first prime minister of India ?” with probabilities 0.4 and 0.6 respectively. Device 2 also results in one of those two questions with probabilities 0.45 and 0.55. The following is the data collected from 2 independent SRSWRs of sizes 10 and 15. Responses from the first and second samples which used device 1and device 2 are
yes,no,yes,yes,no,no,yes,yes,no,no
and no, no, yes,yes,no,yes,yes,no,yes,no,no,no,yes,yes,no
Estimate the proportion of students in the college who got the habit of using unfair means during exams and also estimate the variance of your estimate.
(b) Given the normal distribution Np , where
= and =
(i) Find the distribution of CX, where C = (1, -1 , 1 )(ii) Find the conditional distribution of   [X1, X2] X3 = 190; [ X1, X3] X2 = 160 ; X1 [ X2 = 150 , X3 = 180]
4. (a) A certain genetic model suggests that the probabilities of a particular trinomial    distribution are respectively P1 = p 2, P2 = 2p(1-p) and P 3 = (1-p2) , 0 < p < 1.    If x1, x2 and x3 represent the respective frequencies in n independent trials, how     we would check on the adequacy of the genetic model given x1 = 25 , x2 = 35   and x3 = 40. (b) The following table gives the probabilities and the observed frequencies in 4   phenotypic classes AB, Ab, aB, ab in a genetical experiment. Estimate the    parameter  by the method of maximum likelihood and find the standard error.
Class : AB Ab aB ab Prabability : Frequency : 102 25 28 5 (16+17)
5. (a) A markov chain with state space has tpm given by Find (i) equivalence classes. (ii) recurrent and transient states (iii) mean recurrence time for recurrent states (iv) periodicity of the states.
(b) A Markov chain with state space  Obtain the steady state distribution of the Markov chain.

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

November 20, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 M.Sc. DEGREE EXAMINATION – STATISTICS

AB 21

FIRST SEMESTER – NOV 2006

ST 1812 – STATISTICAL COMPUTING – I

Date & Time : 04-11-2006/1.00-4.00 Dept. No. Max. : 100 Marks

Answer any THREE questions.

a.) The following data relates to the family size(X) and average food expenditure per week (Y) of 8 persons randomly selected from a small urban population.

Y: 40 50 50 70 80 100 110 105

X: 1 1 2 3 4 2 5 6

Assuming there is a linear relationship between Y and X, perform a regression of Y on X and estimate the regression coefficients. Also find the standard error of the estimate.

b.) Consider the following ANOVA table based on OLS regression.

Source of Variation df Sum of Squares

Regression ? 800

Residual 45 ?

Total 49 1200

How many observations are there in the sample?
How many independent variables are used in the model ?
Find an unbiased estimate of the variance of the disturbance term?
Calculate the value of the coefficient of determination and interpret it.
Test the overall significance of the model at 5% level.

(20+14)

a.) Consider the following information from a 4 variable regression equation:

Residual sum of squares = 94;

Y = 10, 12, 14, 9, 7, 8, 2, 22, 4, 12.

i.) Find TSS and ESS.

ii.) Test the hypothesis that R² = 0 Vs R² # 0 at 5% level.

b.) Test whether there is structural change in the model Y = β₀ + β₁X + u

between the two groups where the observations under group I and group

II are as given below:

Group I Y: 10 15 17 14 12

X: 3 5 4 6 7

Group II Y: 12 14 13 15 18

X: 5 3 7 6 4

Use 5% level.

c.) Consider the following OLS regression results:

Y = 16.5 + 2.1X₁ + 50X₂

(10) (0.5) (20) n = 28

where the numbers in the parenthesis are the standard error of the

regression coefficients.

i.) Construct a 95% confidence interval for β₁.

ii.) Test whether in intercept is significantly different from zero at 5%

level.

(7+20+7)

a.) Consider the following data on annual income (in 000’s $) categorized by

gender and age.

Income: 12 10 14 15 6 11 17

Gender: 0 1 1 0 0 1 1

Age: 1 1 0 1 0 0 1

where Gender = 1 if male; 0 if female

Age = 1 if less than or equal to 35; 0 if greater than 35.

Perform a linear regression of Income on Gender and age. Interpret the results.

What is the benchmark category for the above model ?

b.) Fit a Poisson distribution for the following data relating to the number of

printing mistakes per page in a book containing 200 pages:

Number of mistakes: 0 1 2 3 4 5

Frequency: 60 50 40 30 15 5

(17+17)

Fit a normal distribution for the following heights (in cms) 0f 200 men

randomly selected from a village.

Height: 144 – 150 150 – 156 156 – 160 160 – 164

frequency: 3 10 25 50

Height: 164 – 168 168 – 172 172 – 176

Frequency: 63 30 19

Also test the goodness of fit at 5% level. (33)

a.) Fit a truncated binomial distribution to the following data and test the

goodness of fir at 5% level.

X: 1 2 3 4 5 6 7

f: 6 15 18 12 9 8 2

b.) Fit a negative binomial distribution to the following data and test the

goodness of fit at 5% level.

X: 0 1 2 3 4 5

f: 180 120 105 90 40 12

(20+14)

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

November 20, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AB 18

FIRST SEMESTER – NOV 2006

ST 1809 – MEASURE AND PROBABILITY THEORY

Date & Time : 28-10-2006/1.00-4.00 Dept. No. Max. : 100 Marks

Part A

Answer all the questions. 10 X 2 = 20

Define minimal s – field.
Explain Lebesgue measure with an example.
What is a set function?
What is positive part and negative part of a borel measurable function?
State Randon – Nikodym theorem
Show that a random variable need not necessarily be a discrete or continuous type.
Define almost everywhere convergence.
State Holder’s inequality.
Describe a simple function with an example.
If X_n X and g is continuous then show that g(X_n) g (X).

Part B

Answer any five questions. 5 X 8 = 40

Show that countable additivity of a set function with m(f) = 0 implies finite additivity of a set function.
Show that a counting measure is a complete measure on a s – field.
Let F be the distribution function on R given by

0 if x < -1

1 + x if -1 £ x < 0

F(x) = 2 + x²if 0£ x < 2

9 if x ³ 2.

If m is the Lebesgue – Stieltjes measure corresponding to F, compute the measure

of the set { x: ÷ x÷ + 2x² > 1}.

Let f be B-measurable and if f = 0 a.e. [m]. Then show that f dm = 0.
State and establish additivity theorem of integral.
State and establish Minkowski’s inequality.

If X_nX then show that (X_n² + X_n) (X² + X).
Describe Central Limit theorem and its purpose.

Part C

Answer any two questions. 2 X 20 = 40

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

A_i = (A _i – A _{i – 1}).

b). Show that a monotone class which a field is s – field. (10 +10)

a). State and establish basic integration theorem.

b). If hdm exists then show that ½hdm ½£ ïh ïdm (12 + 8)

a). State and establish monotone class theorem.

b). If X_n X then show that E½X_n½^r E½X½^r as n ® ¥. (12+ 8)

a). Show that Liapunov’s Central Limit theorem is a particular case of

Lindeberg’s Central Limit theorem.

b). State and establish Levy’s theorem. (8 + 12)

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Loyola College M.Sc. Statistics Nov 2006 Investment Management Question Paper PDF Download

November 20, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 M.Sc. DEGREE EXAMINATION – STATISTICS

AN 28

THIRD SEMESTER – NOV 2006

EC 3900 – INVESTMENT MANAGEMENT

Date & Time : 03-11-2006/9.00-12.00 Dept. No. Max. : 100 Marks

Part – A

Answer any FIVE questions in about 75 words each. (5 x 4 = 20 marks)

Distinguish between security and non-security forms of investment.
What is a turn around stock?
Why is standard deviation of returns suggested as a measure of risk?
What is the role of correlation in reducing portfolio risk?
Briefly explain the benefits of diversification.
What is opportunity-threat analysis?
What is meant by week-end effect?

Part – B

Answer any FOUR questions in about 300 words each. (4 x 10 = 40 marks)

Explain the advantages of investing in equity shares.
Compute the risk and return from the following data of price and dividend of a scrip

Year : 1987 1988 1989 1990 1991

Price: 11.50 11.50 19 29.50 31.50

Dividend: – 1.20 1.50 1.50 1.50

State and explain Samuelson’s continuous equilibrium model.
Explain Markowitz diversification and classification of risk.
Distinguish Simulation test, Serial correlation test and filter rules.
Briefly explain the methods of forecasting.
Explain equilibrium of a risk-averse and risk-loving investor diagrammatically.

Part – C

Answer any TWO questions in about 900 words each. (2 x 20 = 40 marks)

Explain the significance of Macro Economic environment in security analysis.
State and explain Random Walk theories.
Explain the role of variance, covariance analysis in security analysis using appropriate illustrations.
Discuss the approaches to measurement of portfolio risk.

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Loyola College M.Sc. Statistics Nov 2006 Fuzzy Theory And Applications Question Paper PDF Download

November 20, 2017 by 01 prakash

LOYOLA COLLEGE AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AB 28

THIRD SEMESTER – NOV 2006

ST 3875 – FUZZY THEORY AND APPLICATIONS

Date & Time : 06-11-2006/9.00-12.00 Dept. No. Max. : 100 Marks

SECTION – A

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

Define a fuzzy set. Give an example.
Define α–cut and strong α –cut of a fuzzy set.
Define height of a fuzzy set. What is a normal fuzzy set?
What is the axiomatic skeleton for fuzzy complements?
Give two examples of fuzzy t-conorm that are frequently used as fuzzy unions.
If X = {0, 1, 2, 3, 4} and A is a fuzzy set defined by the membership function

A(x) = x / 4, find the scalar cardinality of A

Give an example of fuzzy set operations that constitute a dual triple.
Distinguish between direct and indirect methods of constructing membership

functions.

Define an ‘Artificial Neural Network’.
State the formal definition of ‘Knowledge’.

SECTION – B

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

Prove that a fuzzy set A on R is convex if and only if A(λx₁ + (1 – λ) x₂) ≥

min[A(x₁), A(x₂)] for all x₁, x₂ R and all where min denotes the

minimum operator.

Let A_i F(X) for all iI, where I is an index set. Then prove that

and .

Explain the extension principle for fuzzy sets.
Prove that the standard fuzzy intersection is the only idempotent t-norm.
Let X = R and let A be a fuzzy set defined by the membership function

x – 1, 1 ≤ x ≤ 2

A (x) = 3 – x , 2 ≤x ≤ 3

0, otherwise

Plot the membership function and the ½ -cut and ¼ -cut of A. Also find the support and core and state whether it is a normal fuzzy set.

Define an increasing generator and decreasing generator and their Pseudo-inverses.

Give an example for both and find their Pseudo-inverses.

Discuss the indirect method of constructing membership functions with one expert.
Describe a multilayer feed forward network with a neat diagram.

SECTION -C

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

(a) Let A, B F(X). Then prove that the following properties hold good for all

(b) Give an example to show third decomposition theorem. (15 + 5)

(a) State and prove First decomposition theorem.

(b) Prove that every fuzzy complement has at most one equilibrium. (12 + 8)

Let X ={x₁, ..,x₄} be a universal set and suppose three experts E1, E2, E3 have

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

in the following table:

Membership in A Membership in B

Element

x₁

x₂

x₃

x₄

Element

x₁

x₂

x₃

x₄

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

weights as c₁ = ½, c₂ = ¼, c₃ = ¼ and for set B as equal weights, find the degree of

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

membership in A∩B using the Standard intersection and Bounded difference function and that in AUB using the Standard union and Drastic Union..

(a)Describe the basic model of a neuron with a neat diagram, labeling its elements

and explaining the notations.

(b)Discuss the three basic types of activation functions.

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Loyola College M.Sc. Statistics Nov 2006 Applied Regression Analysis Question Paper PDF Download

November 20, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI 600 034

M.Sc. Degree Examination – Statistics

I Semester – November 2006

ST 1811 – APPLIED REGRESSION ANALYSIS

02 / 11/ 2006 Time: 1.00. – 4.00 Max. Marks: 100

SECTION – A

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

Define ‘residuals’ and ‘residual sum of squares’ in a linear model.
State the test for the overall fit of a linear regression model.
Define Adjusted R² of a linear model.
Give an example of a relationship that can be linearized.
What is the variance stabilizing transformation used when σ² is proportional to E(Y)[1 – E(Y)]?
State any one criterion for assessing and comparing performances of linear models.
State any one ill-effect of multicollinearity.
Illustrate with an example why both X and X² can be considered for inclusion as regressors in a model.
Define the logit link used for modeling a binary dependent variable.
Define any one measure of performance of a logistic model.

SECTION – B

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

Discuss “No-Intercept Model” and give an illustrative example where such a model is appropriate. State how you will favour such a model against a model with intercept. Indicate the ANOVA for such a model.

A model (with an intercept) relating a response variable to four regressors is to be built based on the following sample of size 10:

X₁

X₂

X₃

X₄

13.8

22.9

23.7

16.8

21.6

25.5

16.6

17.4

19.9

24.6

Write down the full data matrix. Also, if we wish to test the linear hypothesis H₀: β₄ = 2β₁ + β₂, write down the reduced model under the H₀ and also the reduced data matrix.

Give the motivation for standardized regression coefficients and explain anyone method for scaling the variables.

The following residuals were obtained after a linear regression model was built:

0.17, – 1.04, 1.24, 0.48, – 1.83, 1.57, 0.50, – 0.32, – 0.77

Plot the ‘normal probability plot’ on a graph sheet and draw appropriate conclusions.

Describe the Box-Cox method of analytical selection of transformation of the dependent variable.

Discuss the role of dummy variables in linear models, explaining clearly how they are used to indicate different intercepts and different slopes among categories of respondents /subjects. Illustrate with examples.

The following is part of the output obtained while investigating the presence of multicollinearity in the data used for building a linear model. Fill up the missing entries and point out which regressors are involved in collinear relationship(s), if

any:

Eigen Value (of X’X)	Singular value (of X)	Condition Indices	Variance Decomposition Proportions X₁ X₂ X₃ X₄ X₅ X₆
2.429	?	?	0.0003 0.0005 0.0004 0.0000 0.0531 ?
1.546	?	?	0.0004 0.0000 ? 0.0012 0.0032 0.0559
0.922	?	?	? 0.0033 0.9964 0.0001 0.0006 0.0018
0.794	?	?	0.0000 0.0000 0.0002 0.0003 ? 0.4845
0.308	?	?	0.0011 ? 0.0025 0.0000 0.7175 0.4199
0.001	?	?	0.9953 0.0024 0.0001 ? 0.0172 0.0029

Discuss ‘Spline’ fitting.

SECTION – C

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

(a)Depict the different possibilities that can arise when residuals are plotted against the fitted (predicted) values and explain how they can be used for detecting model inadequacies.

(b) Explain ‘partial regression plots’ and state how they are useful in model building. (13 + 7)

The following data were used to regress Y on X₁, X₂, X₃ and X₄ with an intercept term and the coefficients were estimated to be β₀^{^} = 45.1225, β₁^{^} = 1.5894, β₂^{^} = 0.7525, β₃^{^} = 0.0629, β₄^{^} = 0.054. Carry out the ANOVA and test for the overall significance of the model. Also test the significance of the intercept and each of the individual slope coefficients.

Y(Heat in calories)	X1(Tricalium Aluminate)	X2(Tricalcium Silicate)	X3(Tetracalcium alumino ferrite)	X4(Dicalium silicate)
78.5	7	26	6	60
74.3	1	29	15	52
104.3	11	56	8	20
87.6	11	31	8	47
95.9	7	52	6	3
109.2	11	55	9	22
102.7	3	71	17	6
72.5	1	31	22	44
93.1	2	54	18	22
115.9	21	47	4	26

The following is also given for your benefit:

15.90911472	-0.068104115	-0.216989375	-0.042460127	-0.165914393
-0.068104115	0.008693142	-0.001317006	0.007363424	-0.000687829
-0.216989375	-0.001317006	0.003723258	-0.001844902	0.002629903
-0.042460127	0.007363424	-0.001844902	0.009317298	-0.001147731
-0.165914393	-0.000687829	0.002629903	-0.001147731	0.002157976

(X’X)^{– 1} =

Build a linear model for a DV with a maximum of four regressors using Stepwise Procedure, based on a sample of size 25, given the following information:

SS_T = 5600, SS_Res(X₁) = 3000, SS_Res(X₂) = 3300, SS_Res(X₃) = 3600,

SS_Res(X₄) = 2400, SS_Res(X₁,X₂) = 2300, SS_Res(X₁,X₃) = 2760,

SS_Res(X₁,X₄) = 2100, SS_Res(X₂,X₃) = 2600, SS_Res(X₂,X₄) = 2040,

SS_Res(X₃,X₄) = 1980, SS_Res(X₁, X₂, X₃) = 2000, SS_Res(X₁, X₂, X₄) = 1800,

SS_Res(X₁,X₃, X₄) = 1700, SS_Res(X₂,X₃,X₄) = 1500, SS_Res(X₁,X₂,X₃, X₄) = 1440.

(a) Briefly indicate the Wilk’s Likelihood Ratio Test and the Wald’s Test for testing the significance of a subset of the parameters in a Logistic Regression model.

(b) The following data were used to build a logistic model:

DV	1	1	0	1	0	0	1	0	1	1	1	0	0	1	0	1	1	0	0	0
X1	2	4	1	0	-1	3	5	-2	3	-2	3	0	-4	2	-3	1	-1	3	4	-2
X2	-2	-4	2	0	4	-2	1	3	-4	2	1	3	0	-2	-4	-3	1	-1	2	0

The estimates were found to be β₀ = 2.57, β₁ = 3.78, β₂= – 3.2. Construct the Gains Table and compute KS Statistic. (8+12)

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Loyola College M.Sc. Statistics Nov 2006 Analysis Question Paper PDF Download

November 20, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 M.Sc. DEGREE EXAMINATION – STATISTICS

AB 17

FIRST SEMESTER – NOV 2006

ST 1808 – ANALYSIS

Date & Time : 26-10-2006/1.00-4.00 Dept. No. Max. : 100 Marks

.SECTION – A

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

Define a metric and give an example.

Let ρ be a metric on X. Define σ = 2ρ. Show that ρ and σ are equivalent.

Define Norm on a Vector Space. Give two examples.

Write two equivalent definitions of a limit point of a set.

Explain Linear function with an example.

Define a contraction mapping and verify whether a contraction mapping is continuous .

Suppose { x_n } and { v_n } are sequences in R¹. State the conditions under which we can write

( i ) x_n = O ( v_n) ( ii ) x_n= o ( v_n).

State D’Alembert’s ratio test regarding convergence of a series.

State the general principle of uniform convergence of a sequence of real / complex valued functions.

Let D₁ be any partition of [ a , b ]. If D is the partition containing all the points of division of D₁ , then show that the lower sums satisfy the inequality s (D , f , g ) ≥ s ( D₁,f , g ).

SECTION – B

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

———————————-

State and prove Cauchy – Schwartz inequality regarding inner product.
Prove that a necessary and sufficient condition for the set F to be closed is that lim x_nÎ F whenever { x _n} is a convergent sequence of points in F.

ⁿ

Let X = R² , E = R² – { (0,0) } and Y = R¹ .

Define g : E → R¹ as

g ( x , y ) = x ³ / ( x ² + y ²) , (x , y ) Î E

Show that g ( x , y ) → 0 as ( x , y ) → ( 0 , 0 ).

Prove that pointwise convergence does not imply uniform

convergence of a sequence { f_n} of functions.

Prove that a linear function f : R^m→ Rⁿ is everywhere continuous.

Show that R¹with usual metric is complete.

Establish the following relations :

( i ) O ( v_n ) + O ( w_n ) = O ( v_n + w_n )

( ii ) O ( v_n ) + O (v_n ) = O ( v_n )

( i ) O ( v_n ) O ( w_n ) = O ( v_n w_n )

Let f : X → Rⁿ ( X C R^m) be differentiable at ξ Î X. Then show that all the partial derivatives D_i f_j (ξ ) , i = 1,2, . . . , m ; j = 1,2, . . . , n exist and obtain the linear derivative Df (ξ ).

SECTION – C

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

———————————–

( a ) Let X = R^2. Take x_n = ( 3n / (2n + 1) , 2n² / (n² – 2 ) ) ,

n = 1, 2, 3, . . . .

Show that ( i ) x _n –|→ ( 1/2 , 2 ) as n → ∞

( ii ) x _n→ ( 3/2 , 2 ) as n → ∞

( 8 marks)

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

show that ( i ) σ is a metric

( ii ) ρ and σ are equivalent. ( 12 marks)

Let ( X , ρ ) be a metric space and let f _i , i = 1,2, … , n be

functions form X to R¹ .

Define f = ( f ₁ , … , f_n ) : X → Rⁿ as

f ( x ) = ( f ₁( x ), . . . , f _n( x ) ). Then show that f is continuous

at x₀ Î X iff f is continuous at x_{0 ,} for all i = 1, 2, 3, … , n.

( a ) State and prove Banach’s fixed point theorem ( 16 marks)

( b ) State any two properties of compact sets. ( 4 marks)

( a ) State and prove Cauchy’s root test regarding convergence of series of compex terms. ( 10 marks )

( b ) State and prove Darboux theorem regarding Riemann – Stieltje’s integral.

( 10 marks )

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Loyola College M.Sc. Statistics Nov 2006 Advanced Distribution Theory Question Paper PDF Download

November 20, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AB 19

FIRST SEMESTER – NOV 2006

ST 1810 – ADVANCED DISTRIBUTION THEORY

(Also equivalent to ST 1806/1803)

Date & Time : 31-10-2006/1.00-4.00 Dept. No. Max. : 100 Marks

SECTION – A

Answer all the questions (10 x 2 = 20)

Define truncated distribution and give an example.
Show that geometric distribution satisfies lack of memory property.
Define bivariate binomial distribution.
If (X₁,X ₂) is bivariate Poisson, find the marginal distributions.
If (X₁,X ₂) is bivariate normal, find the distribution of X₁ – X ₂ .
Define bivariate exponential distribution of Marshall – Olkin.
Find the mean of non-central chi-square distribution.
Explain compound distribution.
Let X_{1 ,}X_{2 ,}X₃ be independent N(0,1) random variables. Examine whether

X₁²+ X₂²+ 2X₃² – X₁X₂+ 2X₂X₃ has a chi-square distribution.

Let X_{1 ,}X_{2 ,}X₃,X₄ be independent N(0,1) random variables. Find the MGF of X₁X₂+ X₃X₄.

SECTION – B

Answer any five questions (5 x 8 = 40)

For a power series distribution, state and establish a recurrence relation satisfied by the

cumulants.

For a lognormal distribution, show that mean > median > mode.
State and establish the additive property for bivariate binomial distribution.
Derive the conditional distributions associated with bivariate Poisson distribution.
If X = (X₁,X ₂)^/ is bivariate normal with mean vector m and dispersion matrix S , then show that

a^/ X and b^/ X are independent if and only if a^/S b = 0.

If X = (X₁,X ₂)^/ is bivariate exponential, find the distribution of Min{ X₁,X ₂}.
State and establish the additive property for noncentral chi-square distribution.
If X has N_p(m , S) distribution, then show that ( X – m )^/ S ^-1(X – m ) is distributed as chi-square.

SECTION – C

Answer any two questions (2 x 20 = 40)

19 a) State and establish a characterization of exponential distribution.

b) Let X₁, X₂, …,X_n denote a random sample from IG(m, l). Show that

= S X_i /n follows IG distribution
ii) lV = l (S 1/X_i – 1/ ) follows chi-square distribution

and iii) and V are independent.

20 a) State and establish a relation between bivariate binomial and bivariate Poisson distributios.

b) Define bivariate beta distribution.Derive its probability density function.

21 a) State and establish a characterization of bivariate exponential distribution.

b) Define non-central F distribution and derive its mean and variance.

22 a) State and prove Cochran theorem.

b) Given a random sample from normal distribution, using the theory of quadratic forms, show

that the sample mean and the sample variance are independent.

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Loyola College M.Sc. Statistics April 2007 Testing Statistical Hypothesis Question Paper PDF Download

November 20, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 33

SECOND SEMESTER – APRIL 2007

ST 2809/ST 2807/ST 2802 – TESTING STATISTICAL HYPOTHESIS

Date & Time: 19/04/2007 / 1:00 – 4:00 Dept. No. Max. : 100 Marks

SECTION-A (10 x 2 = 20)

Answer ALL the questions. Each carries 2 marks.

Distinguish between randomized and non-randomized tests.
What are the two types of errors in testing of hypothesis?
Give an example of a family of distributions, which has MLR property.
State the necessary condition of Neyman Pearson Fundamental Lemma.
Use Graphical illustration to differentiate between MPT and UMPT.
Define the (k+1) parameter exponential family and give an example.
What do you mean by Unbiasedness?
When do you say that a test function is similar?
When do you say that a function is maximal invariant?
Explain briefly the principles of LRT.

SECTION-B (5 x 8 = 40)

Answer any FIVE questions. Each carries 8 marks.

If X ≥ 1 is the critical region for testing H₀: θ = 1 against H₁: θ = 2 on the basis of a single observation

from the population with pdf

f(x ,θ) = θ exp{ – θ x }, 0 < x <∞; 0 otherwise.

Obtain the size and power of the test.

State and prove MLR theorem of Karlin-Rubin.

Suppose there exists UMPT of size a for testing a composite H₀against composite H₁ then show that it is

unbiased.

Let X_1,X₂, …, X_n be i.i.d random variables each with density

f(x, θ) = exp { – (x_i-θ)}, θ< x < ∞, -∞<θ<∞

0, elsewhere.

Find the UMPT of size α for testing H₀: θ≤ θ₀ against H₁: θ > θ₀.

Also, obtain the cut-off point when α = 0.05, n=15 and θ₀ = 5.

Let X₁,X₂,…,X_n be iid C(θ, 1). Derive LMPT of size a for testing H₀:θ ≤ 0 against H₁: θ > 0 and show

that it is biased.

Show that a function T is invariant under G if and only if T is a function of the maximal invariant.

Let the p.d.f. of X be f(x) = (2/θ²) (θ-x) ; 0< x < θ ,

0, otherwise

Construct 100(1-α)% confidence interval for θ.

Let X be a binomial variate with parameters n and p. Derive the likelihood ratio test of level α for testing

H₀: p ≤ p₀ against H₁: p > p₀.

SECTION-C (2 x 20 =40)

Answer any TWO questions. Each carries 20 marks.

a) State and prove the sufficiency part of Neyman- Pearson Generalized theorem. (12)

b) Show that UMPT of size α does not exist for testing H₀: μ= µ₀ against H₁: μ ≠ µ₀

when the sample of size ‘n’ is drawn from N(μ, 1). (8)

Let X and Y be independent Poisson variates with parameters λ and μ respectively. Derive the

unconditional UMPUT of size a for testing H₀: λ ≤ aμ against H₁: λ> aμ, where a > 0. (20)

a) Consider the ( k+1) parameter exponential family. Suppose there exists a function

V =h(u,t) such that V is independent of T when q = q₀ and V is increasing in U for every fixed T then

derive the UMPT of size a for testing H₀ : q £ q₀against H₁ : q > q_0. (10)

b) Why do we require Locally optimal tests? How do you derive it using the

Generalized Neyman-Pearson theorem? (10)

22 a) Let X_1,X_2,…..X_nbe iid N(m,s²). Consider the problem of testing H₀: s £ s₀ against H₁: s > s_0.

Derive UMPIT for the above testing problem under the appropriate group of transformations. (12)

b) Let X₁, X₂, …, X_nbe iid U(0, θ) random variables. Construct (1-α) – level UMA

confidence interval for θ. (8)

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Loyola College M.Sc. Statistics April 2007 Stochastic Processes Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AC 44

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – APRIL 2007

ST 3809/3806/3800 – STOCHASTIC PROCESSES

Date & Time: 26/04/2007 / 9:00 – 12:00 Dept. No. Max. : 100 Marks

SECTION-A (10 × 2 = 20 marks)

Answer ALL the questions. Each question carries TWO marks.

Define the term “Stochastic Process” with an example.

Let { X_n, n=0,1,2,…} be a Markov chain with state space S = {1,2,3} and transition probability matrix

1/2 1/4 1/4

P = 2/3 0 1/3

3/5 2/5 0

Compute P[X₃=3 │X₁=1]

Explain the terns:

Recurrence time
Mean recurrence time.

For any state i and a transient state j, find the value of

lim p_ij⁽ⁿ⁾

n→∞

Under the condition X(0)=1 , obtain the mean of Yule process.
Define renewal function and find the same when the inter occurrence times are independent and identically distributed exponential.
Find the probability of ultimate extinction of a Branching Process with offspring distribution having the probability generating function 0.5s²+0.5.
Define a Brownian motion process.
Show that a Markov Renewal process is a Markov Chain with one step transition probabilities.
Give an example of a stationary process, which is not covariance stationary.

SECTION- B (5 × 8=40marks)

Answer any FIVE questions. Each question carries EIGHT marks

When do you say that two states of a Markov Chain communicate with each other? Show that communication is an equivalence relation.

Show that in a two dimensional symmetric random walk, all the states are recurrent.
State and establish Kolmogorov forward differential equations satisfied by a birth-death process.
Show that the sum of two independent Poisson processes is a Poisson process. Is the difference of two independent Poisson processes a Poisson process?
Derive the integral equation satisfied by the renewal function of a Renewal process.
Define: (i) Sub martingale and (ii) Super martingale. Give an example of a martingale which is not a Markov Chan.

Derive the recurrence relation satisfied by the probability generating function, where { X_n, n=0,1,2,… } is a Branching Process with X₀=1.
Show that an AR process can be represented by a MA process of infinite order.

SECTION – C (2 × 20=40)

Answer any TWO questions. Each question carries TWENTY marks

a) State and prove Chapman- Kolmogorov equations for a discrete time Markov

chain. (8 marks)

Define a recurrent state j. Show that a state j is recurrent or transient according

∞

∑ p_jj⁽ⁿ⁾ = + ∞ or < ∞ ( in usual notation). (12 marks)

n=1

a) State and prove the Basic limit theorem of Markov chains. (12 marks)

If lim p_jj⁽ⁿ⁾ > 0, show that j is positive recurrent and aperiodic. (8 marks)

n→∞

a) Obtain E[X(t)], where X(t) is a linear birth and death process. (10 marks)

Define M│M│1 queue. Obtain E(W_Q) in this case, when the steady state solution exists. (10 marks)

a) If {X_n, n=0,1,2,… } is the Galton-Watson Branching process, obtain E(X_n) and

Var(X_n). (12 marks)

State and prove the prediction theorem for minimum mean square error

predictors. (8 marks)

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Loyola College M.Sc. Statistics April 2007 Statistics For Competitive Examinations Question Paper PDF Download

November 20, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AC 51

M.Sc. DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – APRIL 2007

ST 4804 – STATISTICS FOR COMPETITIVE EXAMINATIONS

Date & Time: 23/04/2007 / 9:00 – 12:00 Dept. No. Max. : 100 Marks

SECTION A

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

The events A = {1, 2}, B = {2, 3} and C= {2, 4} are exhaustive and A and B are independent .If P (A) = ½ and P (B) = ⅓, what must be P(C)?

(A) 1/6 (B) ⅔ (C) ½ (D) 5/6

If P AUB) =5/6 and P (A) = ½, then P(B/A^C) is

(A) 2/3 (B) 3/5 (C) 1/3 (D) ½

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

F(x) = 0 if x < -1

1-λe ^–x/2 if x ≥ -1

is continuous?

(A) 1 (B) 1 /√e (C) ½ (D) √e

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

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

If X₁ and X₂are independent and identically distributed Geometric random variables with parameter 1/3, then the distribution of Y= min (X₁, X₂) is Geometric with the parameter

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

A box contains 7 marbles of which 3 are red and the rest are green. If 4 marbles are drawn from the box at random without replacement, what is the probability that 3 marbles are green?

(A) 1/35 (B) 18/35 (C) 4/35 (D) 12/35

A random variable X distributed uniformly is such that P(X < 9) =1/8 and

P(X > 22) = 1/3. What is P (11 < X < 19)?

(A) 1/3 (B) 1/8 (C) ½ (D) 4/15

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

(A) -1 (B) 1 (C) -ρ (D) ρ

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

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

If X is an exponential random variable with E (e^X) = 2, then E(X) is

(A) 1/2 (B) 1 (C) 2 (D) 6

A random variable X has the probability density function

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

= 0 otherwise

The lower bound for Pr (0 < X < 2(m+1)) is

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

If R_1.23=1, then the value of R_2.13is

(A) 0 (B)-1 (C) ½ (D) 1

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

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

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

(A) Variables can be unrestricted

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

(D) All variables must be nonnegative

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

the objective function’s value will be

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

The number of basic vells in any IBS solution for a TP is

(A) m+n+1 (B) m+n-1 (C) m-n+1 (D) n-m+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 0 2 1 0 1

x4 0 -1 1 0 1 2

The above table indicates

(A) several optima (B) degeneracy

An LPP has 6 variables and 3 constraints. How many sets of basic variables are possible ?

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

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

(A) Strictly increasing in (B) Strictly decreasing

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

(A) (B)

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

of all

(A) Unbiased tests (B) Similar tests

Two phase sampling is resorted when

(A) variance of an estimator can not be estimated

(B) we can not use systematic sampling schemes

(D) sensitive information is to be gathered

Among the following which can not use Yates-Grundy estimated variance

(A) Simple random sampling (B) PPSWR

When N=20 and n=4 which of the following represents ideal group size under

random group method ?

(A) 4 each (B) 5 each (C) 4,6,5,5 (D) 6,6,2,2

Choose the correct statement

(A) Ratio estimator is always unbiased

(B) Regression estimator is always unbiased

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

Choose the correct statement

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

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

(D) Proportional allocation is better then optimum allocation

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

(A) (B)

The function f(x) = | x + 1| is NOT differentiable at

(A) 0 (B) 1 (C) –1 (D) f is differentiable everywhere

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

a = Probability of getting an odd number of heads

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

Then

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

The system of equations

x – y +3z = 3

4x – 3y +2z = 7

(3m – 1)x – y – 4 = 2 m² – 1

has infinitely many solutions if m equals

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

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

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

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

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

In a bivariate dataset {(X_i, Y_i), i =1, 2, …,n}, X assumed only two values namely 0 and – 1 and the correlation coefficient was found to be –0.6. Then , the correlation coefficient for the transformed data {(U_i, V_i), i =1, 2, …,n}, where U_i = 4 – 2 X_i³ and V_i = 3Y_i + 5, is

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

Which one of the following cannot be the 1^st and 2^nd raw moments for a Poisson distribution?

(A) 2, 6 (B) 4, 12 (C) 5, 30 (D) 6, 42

Let X and Y be random variables with identical means and variances. Then

(A) X + Y and X – Y are uncorrelated

(B) X + Y and X – Y are independent

(D) X + Y and X – Y are identically distributed if X and Y are uncorrelated

If X₁, X₂, …, X_n is a random sample form N (q ,1), – µ < q < µ, which of the following is a sufficient statistic?

(A) S X_i² (B)S (X_i – )²(C) (SX_i , SX_i²) (d) None of these

Let be an unbiased estimator of a parameter. The Rao-Blackwell Theorem is used to

(A) get an improved estimator of by conditioning upon any sufficient statistic

(B) get an equally good estimator of by conditioning upon any sufficient

statistic

(D) get the UMVUE of by conditioning upon a complete sufficient statistic

The upper control limit of a c- chart is 40. The lower control limit is

(A) 0 (B) 10 (C) 20 (D) Cannot be determined

The linear model appropriate for two-way classification is

(A) Y_ij = a_i + b_j + e_ij (B) Y_ij = m + a_i + e_ij

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

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

SECTION B

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

Let the joint probability mass function of (X,Y) be

e ^{– (a+b)} a^xb^y-x/ [x! (y – x)!] , x = 0,1, 2,…,y ; y = 0,1 ,2,….

f(x, y) =

0 otherwise, where (a, b) > 0.

Find the conditional probability mass functions of X and Y.

Using central limit theorem , prove that

∞ e^-t t ^n-1

Lim ∫ ———— dt = ½.

n→∞ 0 (n-1)!

Consider a Poisson process with the rate λ (>0). Let T₁ be the time of occurrence of the first event and let N (T₁) denote the number of events in the next T₁ units of time.

Show that E [N (T₁).T₁] = 2/λ and find the variance of N(T₁).T₁.

Explain how will you solve the following game theory problem using linear
programming technique (Complete solution needed)

B1 B2 B3

A1 3 -1 -3

A2 -2 4 -1

A3 -5 -6 2

Show that family of binomial densities has MLR in
Develop Hartley-Ross ratio type unbiased estimator under simple random

sampling.

47(a) Let X₁, X₂, X₃, X₄ have the multinomial distribution with parameters q₁,q₂,q₃,

q₄ and q₅ where q₅ = 1 – (q₁ + q₂+ q₃+ q₄) and n = 30. If the observed values of

the random variables are X₁= 7, X₂ = 4, X₃ =6, X₄ = 9, find the MLEs of the

parameters.

(b) Obtain the MLE of q based on a random sample of size 7 from the double

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

(7 + 3)

48 (a) If X₁,….,X_n is a random sample from U(0, 1), show that the nth order statistic converges in

probability to 1.

(b) Let T₁ and T₂ be stochastically independent unbiased estimators of q and let V(T₁) be four

times V(T₂). Find constants c₁ and c₂ so that c₁T₁ + c₂T₂ is an unbiased estimator of q with

the smallest possible variance for such a linear combination. (5 + 5)

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

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

P{Atmost ‘x’ samples are to be taken for ‘r’ points to go out of the control

limits}

(b) For samples of size n =2, give the theoretical justification for the value of A₂

used to determine the control limits for the Chart for means. (4 + 6)

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Loyola College M.Sc. Statistics April 2007 Statistical Process Control Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 50

FOURTH SEMESTER – APRIL 2007

ST 4806/ST 4803 – STATISTICAL PROCESS CONTROL

Date & Time: 18/04/2007 / 9:00 – 12:00 Dept. No. Max. : 100 Marks

Part A

Answer all the questions. 10 X 2 = 20

Discuss the statistical basis underlying the general use of 3-sigma limits on control charts.
What is a control chart?
Define rational subgroup concept.
How is lack of control of a process determined by using control chart techniques?
What is process capability ratio (PCR)?
Why is the np chart not appropriate with variable sample size?
Describe an attribute single sampling plan.
Give an expression for AOQ for a single sampling plan.
Write a short note on multivariate quality control.
Define a) Specification limit. b) Natural tolerance limit.

Part B

Answer any five questions. 5 X 8 = 40

What are the major statistical methods for quality improvement?
A control chart with 3 sigma control limits monitors a normally distributed quality characteristic. Develop an expression for the probability that a point will plot outside the control limits when the process is really in control.
Samples of n = 6 items are taken from a manufacturing process at regular interval. A normally distributed quality characteristic is measured and x and S values are calculated for each sample. After 50 subgroups have been analyzed, we have

_i = 1000 and _i = 75

a) Calculate the control limits for the x and S control charts.
b) Assume that all the points on both charts plot within the control limits. What

are the natural tolerance limits of the process?

Write a detailed note on the moving average control chart.

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 be 0.26?

Estimate process capability using and R charts for the power supply voltage data . If specifications are at 350 ± 5 V, calculate PCR, PCR_k and PCR_km. 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

Find a single sampling plan for which p₁ = 0.05, a = 0.05 p₂ = 0.15 and b = 0.10.
What are chain sampling and skip-lot sampling plans?

Part C

Answer any two questions. 2 X 20 = 40

a). Explain the procedure of obtaining the OC curve for a p-chart with an illustration.

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 modified control charts.

b). A control chart for non-conformities per unit uses 0.95 and 0.05 probability limits.The center line is at u = 1.4.Determine the control limits if the sample size n =10

(14+6)

a) Outline the procedure of constructing V-mask.
b) What is Exponentially Weighted Moving Average control chart?. (15+5)

a) Write a detailed note on six-sigma quality.

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 2007 Statistical Computing – III Question Paper PDF Download

November 20, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 55

FOURTH SEMESTER – APRIL 2007

ST 4808 – STATISTICAL COMPUTING – III

Date & Time: 23/04/2007 / 9:00 – 12:00 Dept. No. Max. : 100 Marks

Answer Any three Questions:

Analyse the following 2³ Confounded factorial design

Replication -1

Block-1	N 120	P 121	K 141	Npk 151
Block-2	(1) 121	Nk 145	Np 167	Kp 211

Replication -2

Block-1	Knp 131	K 101	Np 141	(1) 51
Block-2	Nk 62	N 83	P 43	Pk 32

Replication -3

Block-1	Knp 142	Pk 123	N 195	(1) 143
Block-2	Np 143	Nk 105	P 165	K 212

2) Analyse the following Repeated Latin Square design, stating all the Hypotheses, Anova and Inferences. The data represent the Production in Millions of five different soft drinks in five seasons at five different Companies( for the first two weeks).

WEEK-1

Company/Season	1	2	3	4	5
S1	A125	B338	C345	D563	E233
S2	B635	C453	D634	E784	A345
S3	C455	D901	E344	A124	B466
S4	D781	E443	A235	B 948	C452
S4	E245	A378	B565	C712	D344

WEEK-2

Company/Season	1	2	3	4	5
S1	A255	B385	C455	D156	E273
S2	B165	C454	D645	E748	A734
S3	C475	D903	E354	A124	B456
S4	D078	E432	A253	B498	C455
S5	E485	A782	B556	C142	D534

3 a). The data given below are temperature readings from a chemical process in a

degrees centigrade, taken every two minutes.

853 985 949 937 959

945 973 941 946 939

972 955 966 954 948

945 950 966 935 958

975 948 934 941 963

The target value for the mean is m₀ = 950

i). Estimate the process standard deviation.

ii). Set up and apply a tabular CUSUM for this process, using standardized values

h = 5 and k = 0.5. Interpret this chart.

Reconsider the above data. Set up and apply an EWMA control chart to these

data using l = 0.1 and L =2.7.

b). Find a single sampling plan for which p₁ = 0.05, a = 0.05, p₂ =0.15 and

b = 0.10.

a). A paper mill uses a control chart to monitor the imperfections in finished rolls of paper. Production output is inspected for 20 days, and the resulting data are shown below. Use these data to set up a control chart for nonconformities per roll of paper. Does the process appear to be in statistical control? What center line and control limits would you recommend for controlling current production?

Day: Number of rolls produced Total number of imperfection

1 18 12

2 18 14

3 24 20

4 22 18

5 22 15

6 22 12

7 20 11

8 20 15

9 20 12

10 20 10

11 18 18

12 18 14

13 18 9

14 20 10

15 20 14

16 20 13

17 24 16

18 24 18

19 22 20

20 21 17

(b) Solve the following IPP :

Maximize

Subject to

are nonnegative integers

Consider the design of an electronic device consisting of three main components. The three components are arranged in series so that the failure of one component will result in the failure of the entire device. The reliability of the device can be enhanced by installing standby units in each component. The design calls for using one or more standby units, which means that each main component may include upto three units in parallel. The total capital available for the design of the device is $10,000. The data for the reliability, cost for various components for given number of parallel units are summarized below. Determine the number of parallel units for each component that will maximize the reliability of the device without exceeding the allocated capital. You should use dynamic programming technique to solve the given problem.



1	0.6	1	0.7	3	0.5	3
2	0.8	2	0.8	5	0.7	4
3	0.9	3	0.9	6	0.9	5

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ST 1811 – APPLIED REGRESSION ANALYSIS

SECTION – B

SECTION – C

ST 4954 – APPLIED REGRESSION ANALYSIS

ST 1808 – ANALYSIS

ST 4951 – ADVANCED OPERATIONS RESEARCH

ST 2953 – ACTUARIAL STATISTICS

ANSWER ALL QUESTIONS 10 ´ 2 = 20

Part-B

ANSWER ALL QUESTIONS 5 ´ 8 = 40

ST 3809 – STOCHASTIC PROCESSES

ST 1810 – ADVANCED DISTRIBUTION THEORY

ST 1812 – STATISTICAL COMPUTING – I

ST 1809 – MEASURE AND PROBABILITY THEORY

Part A

Answer all the questions. 10 X 2 = 20

Part B

Answer any five questions. 5 X 8 = 40

Part C

Answer any two questions. 2 X 20 = 40

EC 3900 – INVESTMENT MANAGEMENT

ST 3875 – FUZZY THEORY AND APPLICATIONS

ST 1808 – ANALYSIS

SECTION – B

SECTION – C

ST 1810 – ADVANCED DISTRIBUTION THEORY

ST 2809/ST 2807/ST 2802 – TESTING STATISTICAL HYPOTHESIS

SECTION-B (5 x 8 = 40)

SECTION-C (2 x 20 =40)

ST 3809/3806/3800 – STOCHASTIC PROCESSES

ST 4804 – STATISTICS FOR COMPETITIVE EXAMINATIONS

SECTION B

ST 4806/ST 4803 – STATISTICAL PROCESS CONTROL

Part A

Answer all the questions. 10 X 2 = 20

Part B

Answer any five questions. 5 X 8 = 40

Part C

Answer any two questions. 2 X 20 = 40

ST 4808 – STATISTICAL COMPUTING – III

3 a). The data given below are temperature readings from a chemical process in a

degrees centigrade, taken every two minutes.

853 985 949 937 959

945 973 941 946 939

972 955 966 954 948

945 950 966 935 958

975 948 934 941 963

The target value for the mean is m0 = 950

The target value for the mean is m₀ = 950