ST 3814 – STATISTICAL COMPUTING – II

Date : 03-11-10 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

Note : SCIENTIFIC CALCULATOR IS ALLOWED FOR THIS PAPER

Answer any THREE questions

1 a). Let {X_n,n=0,1,2,…} be a Markov chain with state space {0,1,2} and one step transition probabilities (12)

P =

Find (i) P² (ii) (iii) P[X₂ = 0] given X₀ takes the values 0, 1, 2 with probabilities 0.3,

0.4, 0.3 respectively

b). Let {X_n,n=0,1,2,…} be a Markov chain with state space {0,1,2, 3, . . .} and transient function p_xy , where p₀₁= 1 and for x = 1, 2, 3,. . . (22)

Find f₀₀⁽ⁿ⁾ , n = 1, 2, 3 ,. . .
Find mean recurrence time of state 0.
Show that the chain is irreducible. Is it Ergodic?
Find for x = 0 ,1, 2 . . . whenever it exists
Find the stationary distribution, if it exists.

2 a) Consider two independent samples of sizes n₁= 10 , n₂ = 12 from two tri-variate normal

populations with equal variance-covariance matrices. The sample mean vectors and the pooled

variance- covariance matrix are

, and

Test whether the mean vectors of the two populations are equal (16)

b) The distances between pairs of five objects are given below:

1 2 3 4 5

Apply the Single Linkage Algorithm to carry out clustering of the five objects. (18)

Let X ~ B ( 1, θ ); θ = 0.1, 0.2, 0.3. Examine if UMP level 0.05 test exists for H : θ = 0.2 Vs

K : θ = 0.1, 0.3. Otherwise find UMPU 0.05 test. (34)

In a population with N = 4, the Y_i values are 11,12,1 3,1 4,15. Enlist all possible samples of size n = 2, with SRSWOR and verify that E (s²) = S². Also Calculate the standard error of the sample mean.

(34)

5 (a) Marks secured by over one lakh students in a competitive examination were displayed in 39 display boards. In each board marks of approximately 3000 students were given. Kiran, a student who scored 94.86 marks wanted to know how many candidates have scored more than him. In order to estimate the number of student who have scored more than him, he took a SRS of 10 boards and counted the number of students in each board who have scored more than him. The following is the data collected.

13, 28, 5, 12, 0, 34, 14, 41, 25 and 6.

Estimate the number of student who would have scored more than Kiran and also estimate the variance of its estimate. (13)

(b) A sample of 30 students is to be drawn from a population consisting of 230 students belonging to two colleges A and B. The means and standard deviations of their marks are given below:

	Total no. of students(N_i)	Mean	Standard deviation(σ_i)
College X	150	25	7
College Y	80	50	32

How would you draw the sample using proportional allocation technique? Hence obtain the variance of estimate of the population mean and compare its efficiency with simple random sampling without replacement. (21)

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