M.Sc. DEGREE EXAMINATION
– STATISTICS
THIRD SEMESTER
– APRIL 2006
AC 41
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.
Length (x
1
)
FEMALE
Width (x
2
) Height (x
3
) Length (x
1
)
MALE
Width (x
2
) Height (x
3
)
98
103
103
105
109
123
123
133
133
133
88
92
95
99
81
84
86
86
102
102
44
50
46
51
38
38
42
42
51
51
93
94
96
101
102
103
104
106
107
112
85
81
83
83
74
78
80
84
82
89
38
37
39
39
37
35
35
39
38
40
2.
Given the following trivariate Normal distribution with mean vector and variance covariance matrix.





4 .
45
 9 .
64
.
4
965









2 .
879 10
199
.
.
002
798


3
5
.
1 .
81
.
627
628



 i) Obtain the conditional distribution of X
1
and X
2
given X
3
= 10
(15 marks) ii) Obtain the distribution of CX where
C



5
4
6
3
7
2

 (6 marks) iii) 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 = 2 x
1

3 x
2

5 x
3
Sub. To x
1
 x
2
 x
3

2 x x
1
,
1 x

5 x
2
2 , x
3
 x
3

0
7

10
(18.5 marks)
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b) Solve the following transportation problem:
5 1 8 12


2
3
4
6
0
7


14
4
9 10 11 (15 marks)
4.
a) Solve the following game graphically:
B
1 2 5
A 

8

1
4
5

7
6


(18.5 marks)
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. i) Find the effective arrival rate at the clinic. ii) What is the probability that an arriving patient will not wait? Will find a vacant seat in the room? iii) What is the expected waiting time until a patient is discharged from the clinic? (15 marks)
SECTION – C
Answer any ONE question
726922295
5.
a) Suppose the one step transition probability matrix (tpm) is given as below:
Find Poo
(2)
, foo
(n)
, f
13
(u) and f
33
(u)
.
P


0 .
4




0
0
0
.
.
2
5
.
0
0
0
0
0
.
.
.
.
0
4
0
5
0
0
0
0
.
.
.
.
6
2
5
0
0 .
0

0
0
0 .
.
.
2
0
5




(17 marks)
b) For a three state Markov Chain with states {0,1,2} and tpm.
P =







1
2
0
1
4
0
1
3
1
4
1
2
2
3
1
2







, Find

0
,

1
and

2
.
(17 marks)
6.
a) An infinite Markov Chain on the set of non-negative integers has the transition
function as follows:
P ko
= k
1

2
and P k1 k+1
= k k

1

2
(i) Find whether the chain is positive recurrent, null recurrent or transient.
(ii) Find stationary distribution if it exists.
1
b) For a Branching process with off-spring distribution given by p(0) = p (3) = ,
2
Find the probability of extinction, when (i) X
0
= 1 and (ii) X
0
> 1.
(17+17 marks)
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