B4.1-R3: COMPUTER BASED STATISTICAL AND NUMERICAL TECHNIQUES
: JANUARY 2006

1.a) Compute

using Simpson’s rule of integration with h=0.2.

 

b) Let f(x) = ln x. Given the table of values

Estimate the value of ln (0.60).

 

c) An infinite sequence of independent trails is to be performed. Each trial results in a success with probability p and a failure with probability 1 – p. What is the probability that

i) at least 1 success occurs in the first n trails;

ii) exactly k successes occur in the first n trails;

iii) all trails result in successes?

d) Suppose that the joint density of X and Y is given by

Find E[X] and P(Y>1).

e) The average working-set size X of a program is normally distributed with unknown mean and a known variance . The program was executed 36 times and the average working-set size for each run recorded. The sample mean was computed to be 100 page frames. Assuming successive runs of the program are independent, find the 95 percent confidence interval for the mean average working-set size.

f) Write down probability density functions of

i) Uniform Variate

ii) Exponential Variate

iii) Normal Variate

iv) Gamma Variate

g) State the axioms of probability. Using these axioms, show that P(A) £ P(B) if A Í B.

(7x4)

 

2.a) Consider the set of equations:

6 x + 2 y- z = 4

x + 5 y + z = 3

2x + y+ 4 z = 27

Determine approximate solution of the system of equations using Gauss-Seidel method taking the initiate solution as x=y=z=1.

b) For the data given below, write the Newton interpolating polynomial of degree 3 for f(x) = x 2 e -x/2.

 

Hence, find the error of the interpolate for x=1.75. (Given the tabulate value f(1.75) = 1.28611)

(9+9)

 

3.a) In answering a question on a multiple choice test, an examinee either knows the answer or he guesses or he copies. Suppose each question has four choices. Let the probability that examinee copies the answer is 1/6 and the probability that he guesses is 1/3. The probability that his answer is correct given that he copied the answer is 1/8. Suppose an examinee answers a question correctly, what is the probability that he really knows the answer?

b) If X and Y are independent Poisson random variables with respective parameters l 1 and l 2 , calculate

P(X=k | X+Y=n).

(8+10)

 

4.a) The chances of three persons Mr. X, Y and Z becoming managers of a company are 4:2:3. The probabilities that bonus scheme shall be introduced if X, Y and Z become managers are 0.3, 0.5, 0.8 respectively. Find the probability that the bonus scheme will be introduced. What is the probability that X is appointed as the manager?

b) State central limit theorem. Using this theorem argue that if X is binomially distributed with parameters n and p, the distribution of approaches as standard normal Variate as n ® ¥ .

c) If and are the standard deviations of independent random samples of size n 1=61 and n 2=31 from normal populations with and , find

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5.a) Let X i, i=1,…,10, be independent random variables, each uniformly distributed over (0, 1). Calculate an approximation to P( å X i > 6).

b) If is a random sample from a Normal population with mean m and variance unity, then show that is an unbiased estimator of m 2 + 1.

c) X is Poisson Variate with parameters l . Obtain its Mgf. Hence find E[2X+3].

(6+6+6)

 

6.a) Let X denote the main memory requirement of a job as a fraction of the total user-allocatable main memory of a computing center. The density function of X has the from:

f(x) =

 

A large value of k implies a preponderance of large jobs. If k =0, the distribution-of-memory requirement is uniform. Suppose you have a sample of size 8 given by:

 

0.25, 0.45, 0.55, 0.75, 0.85, 0.95, 0.90

 

Estimate the value of k, using the method of moments.

b) It is suspected that the number of errors discovered in a system program is Poisson distributed. The number of errors discovered in each one-week period is given as follows:

 

 

Assume that the total number of errors observed in the 50 weeks was 95. Estimate the rate parameter l for the Poisson probabilities given above. Test the hypothesis that errors occur according to Poisson distribution using significance level of 0.05.

(8+10)

 

7.a) Find the coefficient of linear correlation between the variables X and Y given in the table.

 

 

Also obtain the regression equation of Y on X for the data.

b) State principle of least squares.

c) Show that cov(aX,Y) = a cov(X,Y).

(10+4+4)

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