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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