Entrance Paper - Statistics - University of Hyderabad - Master in Science - 2005, Study notes of Statistics

Download Entrance Paper - Statistics - University of Hyderabad - Master in Science - 2005 and more Study notes Statistics in PDF only on Docsity! D2 Entrance Examination, 2005 M.Sc. (Statistics-OR) | Hall Ticket No. TT a Phos a Answer Part A by circling the ime: ours Max. Marks: 200 correct letter in the array below: 5 : _ a Part B: 75 tlalbleldle Instructions Ajajble|die 3i{albiecl/dje 1. Calculators are not allowed. 4}al/b]c}]dle hfalbteld 2. Part A carries 25 marks. Each cor- “18 ° =I rect answer carries 1 mark and each 6 |alb]eldje wrong answer carries — — mark. 7Tlaflbfeldle So do not gamble. If you want to 8 jalbi]e|dle change any answer, cross out. the old 9lalbi|cl|dje . one and cirele the new one. Over 10}a | byjeldle written answers will be ignored. tilalblelale 3. Part B carries 75 marks. Instructions ial bicld a] for answering Part B are given at the 13 | aib eldje beginning of Part RB. | 14; albiecldfe lalblelale: 4. Do not detach any page from this an- swer book. It contains 18 pages in i6 jajbicid © addition to this top page. Pages 15 Wiajbiclidie to 18 are for rough work. Alalbicidje! ee Wjatbjcjdie : 20) a dje ajalolela| e| 22/albie/dje WBlalbjejdje ajajbleldre 2)/albjef{dle  D2 10. Suppose X is a random variable with probability density function fa) = sagem The mean and variance of X is, respectively, (a) yp ee (b) 1, 1. 1 {c} 1, Se (d) 0, V2. (e} none of the above. 11. For a function io af CeFl, z}21, fx(2) = { 6, otherwise. to be a probability density function, the value of C is e (@) $. (e) none of the above. 12. X and Y are two random variables such that E(X)= 3, E(X|Y = 1) = 2 and P(Y =0) = P(Y =1) = 3. Hence E(X|Y = 0) is {a} 4. (b} 3. (c) 2. (d) 1. (e) nonc of the above. 13. X and Y are independent random variables with variances a7 aud 0? respec- tively. Hence the correlation coefficient hetween X and X — Y is (a) 0. (b) a1 f(of +o)? (©) 92/(of + 03)'”. (a) o3/(o3 + 03)? (e) none of the above. 14. Suppose X and ¥ are independent Binomial random variables with parameters (m4, #1) and (no, p2) respectively. Then Z= X + Y is (a) Binomial(ny + no, 2 + pa). (b) Binomial(m, + 2, (p, 1 pa)/2). (c} Binomial(ny + 2, p14 pa), if py = po =p. (d) Binomiai(n, +2, (pF p2)/2), if pr = py =p. (e) none of the above. 15. Based on 7 iid. observations X,-++,X, from N(u, 77), the maximum likeli- hood estimator of o? is (a) always unbiased and sufficient for o?. (b) unbiased and sufficient when iis known. (c) never unbiased but always sufficient, (d) unbiased but not sufficient when jis known. (e) none of the above. 16. To test for Hp : @ = 1 against Hy : @ = 2, the proposed test procedure is "Reject Ho if X > .5” where X is a random variable which follows Uniform(0,1) distribution. Hence the size of the test and the power of the test. is, respectively, (a) 0.05, 0.95. {b) 0.5, 0.75. (c) 0.05, 0.75. (a) 0.5, 0.5. (e) none of the above. 17. For a frequency data with 7 classes, a normal distribution is filted after es- timating the parameters. If a x?-goodness of fit test is to be used without combining the classes, the degrees of freedom associated with x" test are (a) 4. (b) 5. (c) 6. (d) 7. {e) none of the above. 18. Student’s ¢ test is applied to test the cquality of means of two populations when (a) the two populations are normal. (b) the two populations are normal with equal variances, (c) the (wo populations have cqual variances, hough they may not be nor- mal. D2 (d) the two populations have equal means. (e) none of the above. 1 3 4 19. For the vectors x) = (:} » 2g () and r3 = (:) to be linearly inde- 1 3 4 pendent, (a). the value of a should be 0. (b) the value of a should be 1. (c) the value of a should be 2. (d) no such value of a exists. (e) none of the above. 20. A isa 3x 3 matrix with eigen values 1, 2, and 3. Hence the determinant of A? 2A (a) 12. (b) 24. (c) 32. {d) 64. (e) none of the above. 21. In how many ways can the letters in the word UNIVERSITY can be arranged randomly? (a) ot (b) 10! 10 (9) 10! () a (e) none of the above. biel 22. im S> > > (a) is 0. (b) is }. (c) is 1. ) (d) does not exist. ) {e) is none of the above. 3. For any lwo events A and B, show that P(ANB) 2 1- P(A) — P(B). 4, Let A be the set of « values for which x? + 22 = 8 and B be the set of values for which z? + x = 6, Find AN Band AUB. What would ANB and AUB he if the two equalities were actually inequalities: x?+22 < 8 and 742 < 6? D2 5. If one answers half of questions in an examination correctly, the probability of gaining admission in the course is 0.95. Also, if one does not. answer half of the questions correctly, the probability of not getting admission is 0.80. If for a particular candidate the probability of answering half of the questions correctly is 0.9, evaluate the probability of the candidate getting admission in the course. 6. An urn contains two red balls, two green balls and three blue balls. Two balls are selected at random. Let X = the number of red balls in the balls selected and ¥ = the number of blue balls in the balls selected. Find (a) Prob[X > 1,Y¥ > 1). (b) Prob[X > ¥]. D2 D2 7. In a college, 40% of students know Telugu, 25% know Tamil and 10% know both Telugu and tamil. Lf a student is selected randomly. (a) What is the probability that he doesn’t know either Telugu or Tamil? (b) What is the probability that he knows either Telugu or Tamil, but not both? (c) Tf he knows Telugu, what is the probability that he also knows Tamil? 8. Let X and Y be independent random variables. Suppose that X takes the val- ucs 0,1, and 3 with probabilities i 3, and 4 respectively. Further Y takes the values 0 and 1 with probabilities 4 and 3 respectively. Find the distribution of the random variable Z = X + Y and hence the expectation of 2. 10 D2 13. Suppose X1,-:-.,X, are independent identically distributed random variables (iidrv.s) with common p.d.f. f(e@)= KO +20), -lea<l, -1<0@<1. Find the value of K. Find an unbiased estimator of @ based on al! the random variables. 14. Suppose Xy, Xz, Xq are three iid.c.v.s with common p.d-f. given by 32 a) ut O<e<4, f(x) { 0. otherwise. (a) Find the median of the population. (b) Find the probability that the smallest of the three observations is larger than the population median. 13 J5. Solve the following Lincar Programming Problem graphically Maximize Z = 4a + 5y subject to 3e+y < 27 52 +-5y = 60 62 -+4y > 60 candy > 0. 14 D2