LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2003

ST-3803/S918 – COMPUTATIONAL STATISTICS – III

10.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

Answer any THREE  questions.                                                                     

 

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

MAXIMIZE         Z = 12 X₁ + 15X₂ +14 X₃

subject to                     X₁ + X₂ + X₃  100

-0.01 X₂ + 0.02 X₃  0

-0.01 X₁ + 0.01 X₂  0

X₁, X₂,      X₃   0

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

 

Player -B

B₁    B₂

Player-A

(18+15.5)

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

ACTIVITY:                       A         B         C         D         E          F          G

t­_{o      :}                              3          2          2          2         1         4          1

t_m    :                      6          5          4          3         3         6          5

t_{P      :}                             9          8          6          10        11        8         15

PREDECESSOR:               –           –           A         B         B        C,D       E

 

 

Calculate

• The expected time and SD for each activity
• The CRITICAL PATH
• The probability that the project will be completed by 18 weeks.

1. b) Consider the inventory problem with 3 items (n = 3), the parameters of the problem are

shown below

Item                 K_I                    b_I                            h_I                     a

1                      $10                  2 units             $ 0.3                1 sq.ft

2                      $5                    4 units             $ 0.1                1 sq.ft

3                      $15                  4 units             $ 0.2                1 sq.ft

 

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

optimal order quantity for the three items.

 

(20+13.5)

 

 

 

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

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

sample of   N = 47 males

 

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

(20)

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

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

 

1     2    3    4

4. a) Consider a Markov chain with TPM

• Examine whether the Markov chain is irreducible.
• Also check whether the state 4 is ergodic
• Find the stationary distribution (3.5+4+6)

1. b) Let A = B =   =    and

to be distributed independently according to trivariate normal population with

respective parameters.

 

 

 

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

 

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

 

(4×5=20marks)

 

 

 

 

 

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

step TPM given by

 

P  =

Find the equivalence classes and the periodicity of the states.

 

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

 

Find if the states are transient or recurrent.

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2004

ST 3803 – COMPUTATIONAL STATISTICS – III

02.11.2004                                                                                                           Max:100 marks

1.00 – 4.00 p.m.

 

SECTION – A

 

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

 

1. a) Use two phase method to solve

Max. z = 5x  – 2y + 3z

Subject .to

2x + 2y – z   ≥  2

3x – 4y         £  3

y + 3z <  5

x, y, z ≥ 0                                                        (17 marks)

 

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

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

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

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

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

 

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

 

(17 marks)

2. a) Solve the following unbalanced transportation problem:

 

To

1      2      3   Supply

From

Demand          75    20   50                                                                  (17 marks)

 

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

shown in the table.

 

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

 

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

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

 

SECTION – B

 

 

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

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

 

 

.

(17 marks)

 

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

 

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

 

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

as follows:

p_k0 = (k+1) /(k+2)        and  p_k,k+1 1/(k+2)

 

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

 

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

that m₂ = l₀.  Using the forward equation, find p_0y (t), y = 0,1,2.                   (17 marks)

 

SECTION – C

 

 

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

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

correlation coefficient at 5% level of significance.

 

X1 No. of cycles to failure	X2 Charge rate in (amps)	X3 Depth of discharge
101	0.375	60.0
141	1.000	76.8
96	1.000	60.0
125	1.000	43.2
43	1.625	60.0
16	1.625	76.8
188	1.00	100.0
10	0.375	76.8
386	1.00	43.2
160	1.625	76.8
216	1.00	70.0
170	0.375	60.0

(20 marks)

 

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

coefficient between X₁ and X₂.                                                                          (14 marks)

 

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

 

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

 

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

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

(34 marks)

 

 

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 41

THIRD SEMESTER – APRIL 2006

                                            ST 3803 – COMPUTATIONAL STATISTICS – III

 

 

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

 

 

Answer THREE questions choosing one from each section.

SECTION – A

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

 

FEMALE			MALE
Length (x1)	Width (x2)	Height (x3)	Length (x1)	Width (x2)	Height (x3)
98	81	38	93	74	37
103	84	38	94	78	35
103	86	42	96	80	35
105	86	42	101	84	39
109	88	44	102	85	38
123	92	50	103	81	37
123	95	46	104	83	39
133	99	51	106	83	39
133	102	51	107	82	38
133	102	51	112	89	40

2. Given the following trivariate Normal distribution with mean vector and variance covariance matrix.

 

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

(15 marks)

1. Obtain the distribution of CX where

(6 marks)

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

 

SECTION – B

Answer any ONE question

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

Max Z =

Sub. To

 

(18.5 marks)

 

1. b) Solve the following transportation problem:

 

9   10  11                                                                              (15 marks)

4. a) Solve the following game graphically:

B

(18.5 marks)

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

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

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

1. Find the effective arrival rate at the clinic.
2. What is the probability that an arriving patient will not wait? Will find a vacant seat in the room?

• What is the expected waiting time until a patient is discharged from the clinic?                                                              (15 marks)

SECTION – C

 

Answer any ONE question

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

 

 

(17 marks)

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

P = ,   Find m₀, m₁ and m₂.

(17 marks)

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

function as follows:

P_ko =    and P_{k1 k+1} ⁼

 

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

1. b) For a Branching process with off-spring distribution given by p(0) = p (3) =

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

(17+17 marks)

 

 

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