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

10.11.2003
1.00 - 4.00
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
M.Sc., DEGREE EXAMINATION - STATISTICS
THIRD SEMESTER – NOVEMBER 2003
ST-3803/S918 - COMPUTATIONAL STATISTICS - III
Max:100 marks
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
b) Solve the following game using graphical method:
Player-A
Player -B
B1 B2
A1  1 2
A2  5 4
A3  7 9


A4  2 1
(18+15.5)
2. a) Construct a network based on the following data:
ACTIVITY:
A
B
C
D
to :
3
2
2
2
tm :
6
5
4
3
tP :
9
8
6
10
PREDECESSOR:
A
B
E
1
3
11
B
F
4
6
8
C,D
G
1
5
15
E
Calculate
(i)
The expected time and SD for each activity
(ii)
The CRITICAL PATH
(iii)
The probability that the project will be completed by 18 weeks.
b) Consider the inventory problem with 3 items (n = 3), the parameters of the problem are
shown below
Item
KI
I
hI
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)
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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
1 0.6  0.36  0.53

1  0.46  0.37


1
0.37 


1 

(i)
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.
(ii)
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)
3
1 3 1


b) Let X ~ N  ,  with the mean vector   10 and   3 16 2 . What is the
 


 8 
1 2 4
conditional distribution of X2 (X1 = 8, X3 = 5).
(13.5)
1 2 3 4
1 0 0 1 0


2 0 0 0 1
4. a) Consider a Markov chain with TPM  0 1 0 0 
3
1 1 1 1
4

4 8 8 2
(i)
Examine whether the Markov chain is irreducible.
(ii)
Also check whether the state 4 is ergodic
(iii)
Find the stationary distribution
(3.5+4+6)
1 1 1 
2  1  1
b) Let A = 
B = 1 1 0  x = ( x1 x 2 x3 )1 and y  ( y1 y2 y3 )1



0 1 1 
0 1 1
to be distributed independently according to trivariate normal population with
respective parameters.
 2
  2
2
3 2 1 
1  2 4 1
1 1 2
3
 2  4
2
4 2 0
 2  2 4 2
0 2 4
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')
(4x5=20marks)
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5. a) Consider a Markov chain {xn, n  0 }with the state space S = {0 , 1, 2, 3, 4, 5} and one
step TPM given by
0 1 / 2 2 / 3 0
0
0
0 
1 2 / 3 1 / 3 0
0
0
0 
2 0
0 1/ 4 3 / 4 0
0 
P =


3 0
0 1/ 5 4 / 5 0
0 
4 1 / 4
0 1 / 4 0 1 / 4 1 / 4


5 1 / 6 1 / 6 1 / 6 1 / 6 1 / 6 1 / 6
Find the equivalence classes and the periodicity of the states.
b) For a Morkov chain with one step TPM
0
1
2
0 1 / 2 0 1 / 2 
1  0 1 / 3 2 / 3
2 1 / 4 1 / 4 1 / 2 
Find if the states are transient or recurrent.

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