QP-IES-ISS-24-STATISTICS-PAPER-I-240624

LE.S/LS.S. Examination, 2024 DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE TOLD TO DO SO T.B.C. : SGSE-E-STS Test Booklet Series ow Zz S TEST BOOKLET ag STATISTICS o a 4 Paper—I Time Allowed : Two Hours Maximum Marks : 200 9. 10. INSTRUCTIONS IMMEDIATELY AFTER THE COMMENCEMENT OF THE EXAMINATION, YOU SHOULD CHECK THAT THIS TEST BOOKLET DOES NOT HAVE ANY UNPRINTED OR TORN OR MISSING: PAGES OR ITEMS, ETC. IF SO, GET IT REPLACED BY A COMPLETE TEST BOOKLET. Please note that it is the candidate’s responsibility to encode and fill in the Roll Number and Test Booklet Series A, B, C or D carefully and without any omission or discrepancy at the appropriate places in the OMR Answer Sheet. Any omission/discrepancy will render the Answer Sheet liable for rejection. You have to enter your Roll Number on the Test Booklet in the Box provided alongside. DO NOT write anything else on the Test Booklet. This Test Booklet contains 80 items (questions). Each item comprises four responses (answers). You will select the response which you want to mark on the Answer Sheet. In case you feel that there is more than one correct response, mark the response which you consider the best. In any case, choose ONLY ONE response for each item. You have to mark all your responses ONLY on the separate Answer Sheet provided. See directions in the Answer Sheet. All items carry equal marks. Before you proceed to mark in the Answer Sheet the response to various items in the Test Booklet, you have to fill in some particulars in the Answer Sheet as per instructions sent to you with your Admission Certificate. After you have completed filling in all your responses on the Answer Sheet and the examination has concluded, you should hand over to the Invigilator only the Answer Sheet. You are permitted to take away with you the Test Booklet. Sheets for rough work are appended in the Test Booklet at the end. Penalty for wrong answers : THERE WILL BE PENALTY FOR WRONG ANSWERS MARKED BY A CANDIDATE IN THE OBJECTIVE TYPE QUESTION PAPERS. (i) There are four alternatives for the answer to every question. For each question for which a wrong answer has been given by the candidate, one-third of the marks assigned to that question will be deducted as penalty. (ii) If a candidate gives more than one answer, it will be treated as a wrong answer even if one of the given answers happens to be correct and there will be same penalty as above to that question. (iii) If a question is left blank, i.e., no answer is given by the candidate, there will be no penalty for that question. DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE TOLD TO DO SO 1. If 2), 22) ‎ . ‎ are order statistics from a population with pdf f(x) = 2 2%, x > 0, then what is the density function of 2)? () 102-3020 -22(1) (b) 1-e (© 1- 100 10 270‏ Consider the following for the next two (2) items that follow : Let 2. Xo, X3,.... X, be a random sample from 77 (1, 2) distribution and n = ‎ ( -X)?, X=- DX 121 Mea ஒட்டம்‌ #⁄5- 1 0 2. What is the moment-generating function of 527 == hy 11 (a) ~ 1_ ( = 1£ (0) |1- =) 202 - © < ¢ (9 ‎ Fe ‎ (a) ) SGSE-E-STS/62B then what is the value of [] 5 (1- 17 (a) . = (hiss nr Consider the following for the next three (3) items that follow : Let 20, 22, X3,..., Xn be iid 0 1) variates and T = = In (212223 ... Xp). 4. What is E(T) equal to? (a) n (0) 27 (c) 4n (0 7/2 5. What is Var (7) equal to? 1 8. What is E न தனரா equal to? ( 47 21 +22 + 53 + 24 () 37 (a) 0 (4 27 (b) 1 (0 7 (9 1/2 6. What is the moment-generating function 0‎ of T? (a) (1-2t)-2" 9. What is 2 a = ae BB equal to? X2 + 72 + 2 4X? -2n (b) (1-1) (a) 0 © (b) 1 (c) 2 (0) (1-2) "/2 1 (0) 3 Consider the following for the next four (4) items that follow : 10. If Let 2. Xo, 23, ..., X7 be iid random variables, each having the 7(0 1) distribution. + 264 + 265 + 26 + 27 2527 7. What is the distribution of the following? ~ Cauchy (0, 1) distribution V2(X5 - 26 + 27) [ 22 4558) 322 then what is the value of 512? 1‏ )1 70 ) £2) (b) 2‏ . © 4‏ 0( (0) 5‏ 0( SGSE-E-STS/62B 3 [ P.T.O. 11. Consider the following pairs : 13. If X is a normally distributed random 7 7 > variable with |} = 1 and 2 =2, then 1. ಇದ್ದ +भ + 2८95 शु + 2592 7202 the asymptotic distribution of sample median for n= 20 is 11. 5 ¬ 2122 -x2; 2u? + ‎ ¬ 2 (a) 20, 1/10) 111. > + 2122 - 2x2; 232 + 9152 +2//2 /2) 70, 7/20) Iv. xe + ‎ - 2; + ‎ - 52 © 2/(1, 7/10 How many of the above are non- (0 770, 7/2) orthogonal pairs? Daly pair 14. Suppose X and Y are independent (b) Only two pairs random variables having chi-square distributions with m and n degrees of (c) Only three pairs freedom. Which of the following statements is/are correct? () All four pairs follows "‏ Suppose X and Y are zero-mean jointly mY‏ .12 normally distributed random variables 17 h that 2 =4, 2 =—", 7007 = 2. ‎ 7. ” Fy m- 17% 0 If Z=2X-3Y, then what is the pdf of Z? (a) Select the answer using the code given 127 below. ட்டது 187 (a) 1 only =z (0) ‌ (b) 11 only (c) Both I and II ‏ ® V8n () Neither I nor 11 SGSE-E-STS/62B 4 15. It is known that 8X-10Y +66=0 and 18. If 402 = 902 =25, where x and y are 40X-18Y = 214 are two regression independently distributed, then what is equations. Consider the following 2x +3y statements : the value of Var ( | 1. The regression coefficient of X on Y is 0.8. (a) 25 Il. The correlation coefficient is 0-6. (b) 50 . The most probable value of Y, when = 30, is 30-6. © 625 Which of the statements given above are correct? ( 75 (a) 1 and II only (b) 11 and III only 19. If Cov (2x, y=4, Var(x/4)=1 and Var (/9) = 7, then what is the value of 111 01 ୬୩ ୦୩୭୪ (18r(x, 9)2 equal to? () 1, 11 and 111 7 16. The moment-generating function of X, (a) when X follows 1310, is ) 1/7 (0 (1-6) 3 () 4 ¢} 0130) 10 (0 (1-2) 3 6:4 0 20. If , 202) 2(3)› 214) 25) are order statistics from U[0, 2], then what is the 17. The pdf 9700 of X,) in a random value of the pdf 72) 0? sample of size n from the exponential distribution with pdf f(x) = € ×, x > 0, is (a) 1/16 (a) 02-1 16 (b) 01 5714 (b) (0 1/32 ~<) (0 5/32 SGSE-E-STS/62B 5 ৮১৮০, 21. A point (x, is to be selected from the square 5 containing all points (x, y such that 0 < x < land 0 < y< 1. What is the probability that the selected point belongs to subset of points such that 0-5<(x+y<1-5? (a) 0-15 (b) 0:25 (© 0.45 (0 0-75 22. Let (20. 22, X3,..., 2) be a random sample from (0 1) distribution. In X follows (a) gamma distribution (b) negative exponential distribution () half-normal distribution (0) None of the above 23. If X, Y and 2 are iid 77 (0, 1) variates, then X+YZ. the characteristic function of ———— is 1-2 (a) in (b) ew 1 14:02 (6) 1 : +2 SGSE-E-STS/62B 24. Two random variables X and Y are such that P(X=+1)=0-5 and Y~ 770 3. What is the characteristic function of 2XY? (a) ae (b) ett, 1 4+t? (6) 1 (6) ‌ 4-2 25. 1120 and 202 are iid 700, 1) variates, then the characteristic function of X;X is (a) et 1 14:02 (b) ] (0) 67211 (0 26. Let U(t) and V(t) be the moment- generating functions of two random variables X and Y respectively, and Vit) = U (log 000). If E(X) = 2, then what is the value of E(Y)? (a) 1 (b) 2 4 2 10 Consider the following for the next two (2) items that follow : Gambles are independent and each one results in the player being equally likely to win or lose 1 unit. Let W denote the net winnings of a gambler whose strategy is to stop gambling immediately after his first win. 27. If the first win is on trial N, then the winnings is (a) 2/N (c) 2N (b) 2-N (a) (2-N/N 28. What is P(W< 0) equal to? (a) 1/2 (b) 1/4 (c) 3/4 (d) 0 Consider the following for the next two (2) items that follow : Suppose that a balanced, six-faced dice is rolled repeatedly until the same number appears on two successive rolls and let X denote the number of rolls that are required. 29. What is the value of P(X = 10)? 58 5? (a) 6° (b) 610 510 sil (c) 610 (a) 612 30. What is the value of E(X)? (a) 17 (b) 7 (c) 18/35 (@) 17/25 SGSE-E-STS/62B 31. Two events A and B are such that P(A)=1/3, P(B)=1/5 and P(A| B)+P(B|A)=2/3. What is the value of 12P(A UB)? (a) 23/2 (b) 11 (c) 7 (a) 5 32. Let the joint distribution of X and Y be given by ele ty), x>0, y>0 = |x! x, = ise 0 ; otherwise With reference to the above, consider the following statements : I. The value of c for f(x, y to be a proper pdf is 4. Il. The probability P(X? > 2Y?)is 1/3. Ill. X and Y are independently distributed. Which of the statements given above are correct? (a) I and II only (b) Il and III only (c) 1 and III only (d) 1, Wand [ P.T.O. 33. Let X and Y have joint probability mass function 672 ‌ ± = 0 1 2 ..., yand y=1 23... With reference to the above, consider the following statements : 2( = = † = 0) = 1. 17 follows Poisson distribution with parameter 1/2. . E(X)=1 Which of the statements given above is/are correct? (a) I only (b) 1 only (c) Both I and II (d) Neither I nor II 34. The probability mass function of a random variable X is given by ‌ = 23; n=] 2 Define a random variable ” if X is odd +1, if Xiseven With reference to the above, consider the following statements : 1. 01 = 1 = 2 / 3 Il. 80)=-1/3 . 1) = 8/9 Which of the statements given above are correct? (a) 1 and 11 only (0) II and 111 only (c) 1 and 111 only (0 1, 11 and 111 SGSE-E-STS/62B 35. Let X and Y be jointly distributed with joint probability mass function x P(x, भन्दा) ; y=012... 233, 1) (2752 With reference to the above, consider the following statements : 1. The marginal distribution of Y is geometric distribution. 1. Var (£) = 1 Which of the statements given above is/are correct? (a) 1 only (b) 11 only (c) Both I and II (d) Neither I nor II 36. Let X and Y be jointly distributed as 1870, 0, 36 16 1/2). Consider the following statements : 1. EW|X=7=2 11. Cov (-, 7-20 = -20 . V(X|¥ = 1 = 27 Which of the statements given above are correct? (a) 1 and II only (0) 11 and 111 only (c) I and III only (0) 1, and 37. Let X and Y be two independent random 41. Consider the following statements in variables having moment-generating respect of debugger : functions I. Debugger is a program used to 0-8) end My (t) = 0-0 detect errors and bugs in a 4 4 program. What is P[(X 41) < 2] equal to? 11. In order to debug the program, a 27 2 07 ‎ debugger helps us perform step- (a) 4 (0) nee by-step execution of a program. 81. ہے‎ | 27 > ग्रा. In order to debug the program, (0 32° () 32° a debugger helps perform stopping the execution of the 38. Let ), 2, ..., 200 be a sample of size 10 program until the errors are from an exponential distribution with corrected. mean 6. The lower bound to the Which of the statements given above are probability 2(0 < < 26), where correct? 10 xe 7 22 (a) 1 and 1! only 21 (b) 11 and 111 only (© 1 and 111 only 8-1 09 ‎ 1, Wand 111 (0 01 (6) 0.9 42. Consider the following statements in 39. Suppose Y = X 2 and 14 = 1, 0 < × < 1. respect of Operating System : What is the pdf of Y? 1. It is the principal component of Jy system software. , 0 < {¥ < 1 ---, 0 < }/ < 1 ( Jy | ) 2 11. It is responsible for overall management of the computer ot yen 2Jy resources. 111. It provides an interface between 40. If the moment-generating function of a the computer and user and helps distribution is in implementing the application My (=~ for 7. ‎ -1 Which of the statements given above are then what are mean and variance of the correct? distribution respectively? (a) 1 and 11 only (a) ped 22 (b) 1. ¬. (b) ]] and III only A-t 0-0 9: 72 (c) I and III only Oa, (0) 1, 22 ‎ 1, 11 00 SGSE-E-STS/62B 9 [ P.T.O. 43. Consider the following tasks : I. Search and replace Il. Encryption Ill. Virus scanner IV. Backup V. Data recovery How many of the above are tasks of Utility Programs? (a) Only two (b) Only three (c) Only four (ad) All five 44. Which of the following security services prevents either sender or receiver from denying a transmitted message? (a) Authentication (b) Confidentiality (c) Non-repudiation (d) Integrity SGSE-E-STS/62B 10 45. At which layer of TCP/IP do ICMP and ARP function? (a) (b) (c) (a) Network layer Application layer Transport layer Physical layer 46. What is the representation of — 59 in the 2’s complement system? (a) (b) (c) (a) 11000101 11000100 10000101 11000001 47. Consider the following statements : I. Il. Both compiler and interpreter can translate the program written in high-level language into machine- level instructions. The interpreter translates the whole program into machine language program before executing any of the instructions. Which of the statements given above is/are correct? (a) (b) (c) (a) I only II only Both I and II Neither I nor II 49. Which 48. Consider the following statements in respect of Webcam : I. It is an input device. Il. The quality of the video taken through the Webcam largely depends on frame rate and resolution. Ill. For a video to be smooth and clear, the frame rate is maximum and resolution is minimum. Which of the statements given above is/are correct? (a) II only (b) 1 and II only (c) I and III only (d) 1, II and Ill of the following are main operations of CPU? I. Fetching instructions from the memory Il. Decoding the instructions Ill. Executing the instructions IV. Storing the results back in the memory Select the correct answer using the code given below. (a) I, Ill and IV only (b) 1, II and IV only (c) Il and III only (d) 1, Il, Il and IV 50. In an e-mail service, the protocols used are SMTP, POP3 and IMAP. What are the port numbers in sequence? (a) 110, 143, 25 (b) 25, 143, 110 (c) 143, 110, 25 (da) 25, 110, 143 SGSE-E-STS/62B 11 . What are the functions of BIOS in computer system? I. It performs power-on self-test. Il. It boots the computer system. Ill. It provides hardware-independent access to the physical devices. Select the correct answer using the code given below. (a) I and II only (b) II and III only (c) 1 and III only (da) I, 1 and 52. Sending messages using IPSEC protocol on the internet between two IPSEC compliant router servers is also termed as (a) tunneling (b) leased line (c) packet filtering (d) message switching 53. Which one of the following methods is used for interprocess communication in distributed systems? (a) Shared cache memory (b) Message passing (c) Shared region (a) Cache memory [ P.T.O. 54. Consider the following statements : 56. Which of the following are network connecting devices? I. The hexadecimal equivalent of (1101010101), is 355. I. Repeater Il. Bridge Il. The complement of (47);9 is (53);9. eT Which of the statements given above IV. Bluetooth is/are correct? Select the correct answer using the code given below. (a) I only (a) I, Il and Ill (b) Il only (b) 1, Il and IV fc) 1, Ill and IV (c) Both I and II ) (a) Il, Ill and IV (d) Neither I nor II 57. Consider the following statements regarding Database Management 55. Which of the following are network layer System (DBMS) : a sana I. It helps the user in maintaining 1: HIT data integrity while performing the various operations on the database Il. FTP values. Il. It helps the user in maintaining Ill. NMP consistency in the information stored in the database. Ve ot III. It allows the user to maintain the security of the confidential Select the correct answer using the code information by password-protecting given below. ; the database. Which of the statements given above are (a) 1, Il and Ill correct? (b) 1, Il and IV (a) 1 and II only () 1, Mand Iv (b) Il and III only ec}. A, an (c) 1 and III only (d) Ul, Wl and Iv (d) 1, Wand Wl SGSE-E-STS/62B 12 58. Consider the following statements : I. Light pen is an_ electro-optical pointing device which is generally connected to the Visual Display Unit (VDU) of the computer. Il. Touchscreen is a pointing device that enables us to enter data such as text, pictures and images by directly touching the screen. Ill. Joystick is a pointing device which controls the movement of the cursor on the screen by pointing in a particular direction only. Which of the statements given above are correct? (a) 1 and II only (b) II and III only (c) 1 and III only (d) 1, Il and Ill 59. Which of the following are looping statements in general of a computer programming language? I. While Il. Break Il. For IV. Continue V. Do while Select the correct answer using the code given below. (a) I and II (b) Il and IV only (c) Il, Il and IV (a) I, Ill and V 60. In which of the following data models are the records arranged in the form of a tree? (a) Entity model (b) Relational model (c) Hierarchical model (ad) Network model SGSE-E-STS/62B 13 61. If y satisfies the initial value problem dy =1l+xy; y(2)=0 i xy; y(2) then what is the approximation of y at x = 3? (Use second approximation of Picard’s method) (a) 3-0 (b) 2:33 (cc) 4:33 (a) 0 62. Consider the following statements : I. Forward Euler method = and backward Euler method for initial value problem have same order. Il. Classical fourth order Runge-Kutta method for initial value problem has higher order than forward Euler method. Ill. Classical fourth order Runge-Kutta method for initial value problem has higher order than backward Euler method. Which of the statements given above are correct? (a) 1 and II only (b) Tl and fl only (c) 1 and III only (@) I, Il and Ill 63. Use Simpson’s one-third rule to evaluate f x‘*dx by dividing [1, 3] into two parts. What is the value of the error? (a) 0-12 (b) 0-19 (oc) 0:27 (a) 0:33 [ P.T.O. 64. Runge-Kutta method of order four is 67. What is the value of V°y,? used to find the solution of the differential equation () V°un_s 80, _5 = fix); 050 (4 2590 ( 252,-5 6 68. To find the approximate value of y(h=h [aro + 2) +of 2 : ydx, where b=a+8h, which of the ம்க்‌ the wales ofia 8, ८ following rules can be applied? respectively? (a) Trapezoidal rule and Simpson’s () 1/6, 2/3, 2/3 three-eighths rule (b) 2/3, 2/3, 1/6 2/3, 1/6, 1/6 : £ wer / / ( Simpson’s one-third rule and (d) 1/6, 2/3, 1/6 Weddle rule (97 Simpson’s three-eighths rule and Weddle rule (b) Trapezoidal rule and Simpson’s one-third rule 65. { = 23 + 2× + ‎ and Yo, ‎ 2 are the values of y for x=a, a+h, a+2h respectively. What is the value of a+2h 69. The value of the integral fr (x)dx is 5 8 ydx? (Use trapezoidal rule) by using trapezoidal rule and 3 by using h Simpson’s one-third rule dividing [1, 3] (a) 2 *28 + 2) into minimum number intervals. What is the value of f(2)? 0) “(20 + ‎ + 292) 80‎ < (a) 1/2 h © ` 2 (/0 +2u + 292) 07 (4 3/2 h (a) 2 + + ) (0) 2 66. What is the value of the following? 70. The third divided difference of a cubic polynomial is 480 + 6550 40 90 + &3 yp a (a) always constant but not necessarily (0 2-271-18 zero /2) ‎ + 1 + + ‎ (b) always zero (© ‎ + - 92 - ‎ (c) equal to sum of arguments (0 + ¬ | - ‎ (d) equal to product of arguments SGSE-E-STS/62B 14 71. If A(ab*)=ab*, the interval 01 74. Which one of the following is the best differencing being 1, then what is the possible choice for the step size h to value of a? tabulate f(x)=sinx on [0 7/4] that results in truncation error for the linear (a) alb- 12 interpolation at most 1075? (a) 2/520 (b) a (b) 0 (0 ( - 12 ( 10 (0) -1 (0 7/92 a .18 Ga 50218, the interval of 75. The roots of the least degree polynomial differencing being 1, then what are the satisfied by (0, 1), (1/2, 3/2) and (1, 3) values of and 3 respectively? are (a) 6,6 (a) real and distinct 0) 6,0 (0) complex with real part zero (c) real and repeated (c) 0,6 (0) complex with non-zero real part ( 33 76. The bound on the truncation error 73. If 8=aVE", the interval of differencing pf Lagrange polynomial f(x) on [a, b] is What is being 1, then what are the values of the bound on the truncation error % and n respectively? of Newton forward interpolating polynomial of f(x) on [a, ]? @ 1, 1/2 (a) /2 (b) 1, 1/2 (b) ( 1/2, -1/2 ‎ (0 -1/2,1/2 (0) 4 SGSE-E-STS/62B 15 [1.9 77. If the formula 79. Consider the following data with regard to depth (y at a distance (x) of a river [i Ped ae = af (x0) + BFlxo +h) + Wf Xo +2h) 40 m wide : is exact for polynomial of as high order Distance from one as possible for any h> 0, then what are of the sides (x) : O 10 20 30 40 the values of a, B, y respectively? Depth (y) at a (a) he 4h ch distance (x) in metre: 0 4 7 9 12 BF 232153 A Ueoh What is the approximate area of cross- (b) Be Bg section of the river? (Use Simpson’s one-third rule) (o) h, 3, h (a) 780 square metre h h (a) 2’ - 2 (b) 390 square metre A (c) 260 square metre 78. For i. f(x)dx, where f(x) is cubic polynomial, consider the following (a) 195 square metre statements : I. Trapezoidal rule will give the exact value of the integral. 80. Consider the initial value problem Il. Simpson’s one-third rule will give gy =- 8y; y(0)=1. What is the largest the exact value of the integral. a ; interval for step size h so that the forward Euler method to solve the initial Ill. Simpson’s three-eighths rule will value problem is absolutely stable? give the exact value of the integral. Which of the statements given above are (a) (0, 1/8) correct? (a) 1 and II only (b) (0, 1/4) (b) II and III only (c) ( 1/3) (c) 1 and III only (a) 1, Il and (a) (0 1/2) SGSE-E-STS/62B 16 SPACE FOR ROUGH WORK SGSE-E-STS/62B 17 [ P.T.O. SPACE FOR ROUGH WORK SGSE-E-STS/62B 18 SPACE FOR ROUGH WORK SGSE-E-STS/62B 19 (1.. SPACE FOR ROUGH WORK kkk SGSE-E-STS/62B 20 SB24—4896