M.Sc.(Mathematics with Applications in Computer Science) Probability and Statistics

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1 MMT-008 Assignment Booklet M.Sc.(Mathematics with Applications in Computer Science) Probability and Statistics (Valid from 1 st July, 013 to 31 st May, 014) It is compulsory to submit the assignment before filling in the exam form. School of Sciences Indira Gandhi National Open University Maidan Garhi New Delhi

2 Dear Student, Please read the section on assignments in the Programme Guide for Elective courses that we sent you after your enrolment. A weightage of 30 per cent, as you are aware, has been earmarked for continuous evaluation, which would consist of one tutor-marked assignment for this course. The assignment is in this booklet. Instructions for Formatting Your Assignments Before attempting the assignment please read the following instructions carefully. 1) On top of the first page of your answer sheet, please write the details exactly in the following format: ROLL NO: NAME: ADDRESS: COURSE CODE:. COURSE TITLE:. ASSIGNMENT NO.. STUDY CENTRE:.... DATE:.... PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE EVALUATION AND TO AVOID DELAY. ) Use only foolscap size writing paper (but not of very thin variety) for writing your answers. 3) Leave 4 cm margin on the left, top and bottom of your answer sheet. 4) Your answers should be precise. 5) While solving problems, clearly indicate which part of which question is being solved. 6) This assignment is valid only upto May, 014. If you have failed in this assignment or fail to submit it by May, 014, then you need to get the assignment for the July, 014 and submit it as per the instructions given in that assignment. 7) It is compulsory to submit the assignment before filling in the exam form. We strongly suggest that you retain a copy of your answer sheets. We wish you good luck.

3 TUTOR MARKED ASSIGNMENT Course Code: MMT-008 Assignment Code: MMT-008/TMA/ Maximum Marks: 100 Q.1 Suppose that {Xn : n 1} is a random walk with X 0 = 0 and probability p of a step to the right, find: (i) P{X 4 = } (ii) P{X 3 = 1, X 6 =,} (iii) P{X 5 = 1, X 10 = 4, X 16 = } (iv) Var (-3+ X 4 ) (v) Var ( + 3X X 5 ) (vi) Var(3X 4 X 5 + 4X 10 ). (10) Q. a) Let P be a three state Markov Chain with transition matrix P = Suppose that chain starts at time 0 in state. i) Find the probability that at time 3 the chain is in state 1. ii) Find the probability that the first time the chain is in state 1 is time 3. iii) Find the probability that the during times 1,,3 the chain is ever in state. (3) b) Suppose that 3 white balls and 5 black balls are distributed in two urns in such a way that each contains 4 balls. We say that the system is in state i is the first urn contains i white balls, i = 0,1,,3. At each stage 1 ball is drawn from each urn at random add the ball drawn from the first urn is placed in the second, and conversely the ball of the second urn is placed in the first urn. Let Xn denote the state of the system after the n th stage. Prove x is a Markov chain. Find the matrix of transition = probabilities. (4) that { n } n 1 c) Let E = {1,}, transition matrix p 1 p p = 1 p p Prove by induction that: 1 1 n 1 1 n + ( p 1) ( p 1) p = 1 1 n 1 1 n ( p 1) + ( p 1) (3) Q.3 a) Let E = {1,} and let α 1 α p = β 1 β Show that if 0 < α, β < 1, the Markov chain is irreducible and ergodic. Find the limiting probabilities. (5) 3

4 b) Consider the Markov chain with state space E = {1,,3,4} and transition Matrix p = Find the communicating classes. Classify the communicating classes as transient or recurrent. Also, find the period d of each communicating class. (5) Q.4 a) A repair man fixes broken televisions. The repair time is exponentially distributed with a mean of 30 minutes. Broken televisions arrive at his repair shop according to a Poisson stream, on average 10 broken televisions per day (8 hours). i) What is the fraction of time that the repair man has no work to do? ii) iii) How many televisions are, on average, at his repair shop? What is the mean throughput time (waiting time plus repair time) of a television? (5) b) In a branching process, the offspring distribution is given by its characteristic Function P(s) = as + bs + c where a; b; c > 0. i) Find the extiction probability for this branching process. ii) Give a condition for sure extinction. (5) Q.5 A single repairperson looks after the two machines 1 and. Each time it is repaired, machine i stays up for an exponential time with rate λ i, where i = 1,. When machine i fails, it requires an exponentially distributed amount of work with rate i to complete its repair. The repairperson will always service machine 1 when it is down. For instance, if machine 1 fails while is being repaired, then the repairperson will immediately stop work on machine and start on 1. i) Write down all the states. ii) What is the probability that the machine is down. (10) Q.6 a) An urn contains 4 red chips, 3 white chips, and 1 blue chip. Consider the following -step experiment: Step 1: Thoroughly mix the contents of the urn. Choose a chip and record its color. Return the chip plus two more chips of the same color to the urn. Step : Thoroughly mix the contents of the urn. Choose a chip and record its color. Let X be the number of red chips, and Y be the number of white chips, chosen. Construct a table showing the joint (X; Y ) distribution, and the marginal X and Y distributions. (5) b) Let X = [ X,X ] be a random vector with mean vector [, ] 1 X = 1 and variance-covariance matrix σ11 σ1 Σ x = σ1 σ Find the means and covariance matrix for the linear combinations 4

5 Z = X X 1 1 Z = X1 + X or Z1 1 1 X1 Z = CX Z = = 1 1 X in terms of x and Σ x. (5) beq7. a) Calculate the least square estimates, the residuals, and the residual sum of squares for a straight line model y = b0 + b1x + e fit to the data given below: x y (4) b) 9 Let A =. 6 Check whether A is a variance-covariance matrix (3) c) Give a precise statement of the Cochran s theorem. (3) Q.8 Suppose the random variables X 1,X, and X 3 have the covariance matrix 1 0 Σ = Find first, second and third principal components. (10) Q.9 Let the data matrix for a random sample of size n = 3 from a bivariate normal population be X Evaluate the observed T for 0 = [ 9,5]. What is the sampling distribution of T in this case? (10) Q.10 Which of the following statements are true or false? Justify your answer with a short proof or counter example. (10) i) A branching process starts from 10 individuals, and each reproduces according to according to the probability distribution (p 0 ; p 1 ; p ; : : :), where p 0 = 1/4, p 1 = 1/4, p = 1/, p n = 0, for n >. The extinction probability for the whole population is equal to 1/104. ii) The relation of accessibility in states is transitive. iii) The matrix given by is not a transition probability matrix of a Markov Chain. iv) Every variance-covariance matrix is a non-negative definite matrix. v) In linear regression model the extent of fit is measured by partial correlation coefficient. 5