QP-IES-ISS-26-220626-STATISTICS-PAPER-I

LE.S./I.S.S. Examination, 2026 DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE TOLD TO DO SO T.B.C. : SNCS-A-STS Test Booklet Series . 23 TEST BOOKLET 7c ES STATISTICS : Paper—I Time Allowed : Two Hours Maximum Marks : 200 iy 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 /67 କା ୍‌ ଵଵ ପଯେ ಕಾಕಾ ee For the following three (03) items : Two random variables X and Y have a discrete joint distribution given by P(X=x,Y=y) | ಎ ഒഴി. for. x, శ్రి) டந்த 2 0; otherwise 1. What is Var(X) equal to? 2-7. (b) 24 6/ 2:35 (0 1:17 2. What is E(Y |X = 1) equal to? | (a) 1/7 (b) 3/7 (0 4/7 (0 6/7 3. What is P(|X-Y|< 1) equal to? (a) 0:06 (b) 0-7 (c) 0-12 (0 0-41 For the next two (02) items that follow : Let {> () be p.d.f. of (9 1) variates and (x) = [ fx (0) 0 4. What is ) (-100) equal to? (a) -%(100 () 1- (100) (8 13000) )4-100 SNCS-A-STS/67A 5. What is 2 | 2 F(x) equal to? (a) 1/2 (b) 1/4 ( 0 (0) 1/3 For the next two (02) items that follow : The joint probability density function of X and Y is given as 2(x+2y), (> ) = Bek? 0; elsewhere O<x <1. 01 6. What is the value of E[X|Y = 0-5]? (a) 1/9 (b) 1/3 (4 5/9 (0 7/9 7. What is the value of conditional variance of X given Y =0-5? (a) 11/162 (b) 13/162 (6) 5/54 (0) 17/162 For the next two (02) items that follow : Let the joint p.d.f. of (X, Y) be given by ঢেৰা, ; 3 = )9 (7 otherwise‏ ;0 02021 O<y<2 8. The expression for conditional density of Y given X is (a) ಚಾ ૦૬૪૬૦૦૬૪૨1 (b) 0૬૪૦૦૦૬૦૨1 (૦ ୪୮୪୬ ୦ ୧୮୦୭ । 3× + 1 ଏ 1‏ فقاو یا < 0 ೨೫೫;‏ رم x(6+y)‏ 9. The expression for conditional density of X given Y is (a) அன்றி 0૯૪૦૩0૨૬૪૦ 2x(3 + x) رم‎ ૪. 0૮૪૨10૨૭૨૦2 2(6- 9) (c) 6 YEO Q+y (‌) (62 0), ‎ 2+ xy 10. Let X, Y, Z be independent. If X and Y each has geometric distribution with parameter 1/2 and Z has a negative binomial distribution with parameters 3 and , then E(X+Y-Z)=0, if and only if p equals (a) 1/3 (b) 3/4 (© 2/5 (0 3/5 SNCS-A-STS/67A 11. If ~ 701) and ~ ?7(0 1) are independent random variables, then the distribution of X/Y is (a) Chi-square with freedom 2 degrees of (b) Cauchy (0, 1) (c) Cauchy (1, 1) () F-distribution with degrees of freedom (1, 1) 12. There are three identical urns containing white and black balls. The first urn contains 2 white and 3 black balls, the second urn contains 3 white and 5 black balls, and the third urn contains 5 white and 2 black balls. An urn is chosen at random and a ball is drawn from it. If the ball is white, then what is the probability that the second urn is chosen? (9) 35/139 (b) 200/417 (c) 112/417 (4) 105/139 13. Let (X,Y) have a bivariate normal distribution with parameters p, = 5, My = 8, 62 = 16, 2 = 9 and py =0-6. What is the coefficient of regression of Y on X? (a) 7/16 (b) 9/20 (c) 1/2 (0 3/8 [ P.T.O. | 14. Let X be a continuous random variable 18. Let Xj, 2, ... be sequence of whose density function is constant in iid. random variables and @ 5 × < ® and zero elsewhere. What is Sy =X, +Xqt...+Xy, where 27 is the standard deviation of X? a random variable independent of (a) (57) /(2V3) 1123... If My,=E(e%), ) (b-a)/(2V3) er the moment generating function © 9 ‌ ‎ (0 (b-a)/2 (a) log(log Mx, () | 15. A random variable X has the following (b) My (log Mx, (0) | probability mass function p(x) = 2/15, where x=1 2 3 4 and 5. What is (0) M x, (log M ‎ 9) E(X|x > 1) equal to? Mx, (0) 4 (b) 45/14 ( 15/7 19. If the joint p.d.f. of two random variables (0 15/14 X and Y is 16. A random variable X has the following Fix he 2:28; 22 < ‰, O<x<1 distribution : 0; ‘otherwise 9 <~ < 315 F(x) =112 then the value of P(X > ) is 0; otherwise (a) 07 What is 2(5 < X < 7|X > 6) equal to? (b) 03 (4) 3 (9 0-12 ) 3/4 (2, > () 8 (0 2/3 (0 1/6 20. A number X is chosen at random between 0 and 6, and another number 17. Let X have a binomial distribution with Y is chosen at random between 0 and 9. parameters n and , If 1- 5, then What is the probability that the what is P[X = 0 or even] equal to? equation 2 2 - XZ + Y = Ohas real roots? (a) 2(q- 2)" (a) 1/2 ) [1+(- 492‎ 1/3 (© [1-(q- p)"]/2 (© 2/3 (0 1+(q- 2)" (0 1/4 SNCS-A-STS/67A 4 : . 21. The random variables X and Y have joint p.d.f. f(x, 9) = (57 9); 021 O<y<2 What is the value of E[Y|X =2]? (a) 8/9 (b) 10/9 11/9 () 13/9 22. How many pairs of observations (n) must be included in a sample in order that an observed correlation coefficient of 0:6 shall have a calculated value of t greater than 2.72 ? (a) n<10 (b) 10<n<13 ( 13 < 71 < 15 (0 n>15 23. If 93.2 is the regression coefficient of 2 and 23 on 22, and 931.2 is the regression coefficient of 23 and 20 on 202, then what is 213.2 × ‎ equal to? ( 2.3 (b) 72.2 © ‎ (0 72 24. Let (20. 22, ..., 201) be a random sample from N(p, 02}, then the distribution of 2 ns 1 2 = 2 ” where s = ~ >+-1 22), is fey ‌) 2 (b) Xan) 2 9 7 2 (0 X2n-2) SNCS-A-STS/67A 25. If the coefficient of correlation r(2x, 9) = 0-25, then what 15 7-4 (3 88) equal to? (a) 1/4 (b) 1/8 © -1/4 (0 -1/8 26. Let < ‎ < 575 5 be the order statistics from a 6.6.8 F(x) = 17, where 2. = 0-5. What is the c.d.f. of Y,? (a) 1-e¥/? () 1-/3 6) 1-63 (0 1--3 27. If jk represents multiple correlation coefficient, 7, represents simple correlation coe: ficient and 6; represents standard deviation of variate X; for a multivariate data set, then consider the following statements : 200 1 21 _ ^ )2 1, ووتھ‎ 290850 (10556) (2713) 2 2 1. 123 = 2 ‎ . , = 53.2 Which of the statements given above are correct? (a) 1 and II only (b) Il and III only ( 1 and 111 only (0 1, Il and 111 (.7.. 3. . . --: - 28. The first of the two samples has 100 31. Two normal populations m, and 7, have items with mean 15 and standard their means equal, but standard deviation 3. If the whole group has deviation of m4 is twice that of m, having 250 items with mean 15-6 and standard standard deviation o. Then in the deviation /13-44, then what is the samples of size 2000 from each variance of the second group? population, the difference of sample (a) 16 means in all probability is (a) less than 0-050 (b) 12 (b) less than 0-150 4 (<) (c) greater than 0-050 ig) 2 (ad) greater than 0-150 29. Let (X, Y) ~ BVN(u), Hg, 07, 02, p). If 32. The mean height of 50 male students o? 29. o3 = 3 and p =O, then consider who showed above average participation in college athletics was 68:2 inches the following : with a standard deviation of 2:5 inches, I. E(X)=E() while 50 male students who showed no interest in such participation had Il. E(XY)=2 a mean height of 67:5 inches with : a standard deviation of 2-8 inches. Il. X and Y are independent In order that the observed difference in the mean height to be significant at Which of the above is/are correct? 5% level of significance, the sample size of each of the two groups has to be (a) I only increased by (the critical value of Z at 9, j 7 i Pr (b) II only 5% level of significance is 1-645) t least 78 () (a) at leas (b) at most 78 (d) 1 and II (c) at least 28 30. The joint p.d.f. of X and Y is given by (d) “at most. 28 y ov 33. A normal population has a mean of F(x% y= (1+.x)4 evs; x y20 0-1 and standard deviation of 2:1. For a random sample of size 900, what would be the value of P(x < 0)? Then the regression of Y on X is [Given that P(0 < Z < 1-43) = 0-4236] (a) exponential (a) 0-0764 (b) linear (b) 0-4236 (c) polynomial of degree 2 (c) 0-5764 (ad) polynomial of degree 3 (a) 0-9236 SNCS-A-STS/67A 6 34. When sampling is without replacement, 37. Kruskal-Wallis test is exactly equivalent 2 - - ‌ 5 0‎ then V(x) = 9 3 The factor ~ 02 5 2 (a) data are normal is ignored under which of the following conditions? (b) population distributions are 1. nis small compared 10 N identical and normal Il. ce N (c) sample sizes are equal 111. Population is infinite () all observations are replaced by Select the correct answer using the code their ranks given below. (a) I only 8. If X - 0 (9 1), then P(X) > x) is (b) 111 only (a) 1-2” ( 1 and 111 only / 1, Wand 111 ) (1-2) -1 35. A random sample of size 5 is taken from ( nx” a normal population with mean 2-5 and variance 36. The probability that the (0) sample variance 52 = 1 se 3 39. Consider a sample of odd size n from n-1 (0 1) population. Then the mean and variance of distribution of median are respectively + 2 [Given, 2004 > 3) = 0-55783 and (a) 1/2 and 1/4 204 < 5 = 0-71270 () 1/2 and 1/(7+ 8 lies between 27 and 45 is > © 1/4 and ( _ 3) /(2# + 4) (2 0:27053 (0 Both mean and variance do not () 0-34513 7 36. The t-statistic for a given sample is 40. What is the mean of the numbers 1. 72, calculated to be 3. If the sample 2 2 2 variance is doubled, then the new 2. -0, 3. (-2), ..... 707 t-statistic will become + 0260+ 19 6‎ (a) (n+1)*(n+2)/ 3 2 (b) 2 (b) (n+1)* /4 3 (6) Va ( n(n+1)? /12 1 (8) — 42 (0 (n+2)?(n+4)/12 SNCS-A-STS/67A 7 [.110 41. Ifthe interval of difference is h, then what 16 ^ [108 f(x)] equal to? | Ef (x) 1 1+ ‌ (a) log | + 7 | | 8 109 ) 1 1- ‌ ced (c) log + 79 ] | Ef (x) ) 1 1- 88 a 42. A cubic polynomial f(x) takes values as in the following table : x | 0 | 1 f(x) | 1 10 What is f(4) equal to? (a) 35 (b) 33 () 31 (d) 29 SNCS-A-STS/67A 43. Consider the following statements : 1. The (851) divided differences of a polynomial of nth degree are zero, where n is a natural number. 11. The nth divided difference for a polynomial of degree n can be expressed as the quotient of two determinants each of order n, where n is a natural number. Which of the statements given above is/are correct? (a) 1 only (b) 11 only (c) Both I and II (‌) Neither I nor 11 44. The lowest possible degree of the polynomial p(x) using Newton’s divided difference formula which assumes the values 48, 100, 294, 900, 1210 and 2028, when x takes the values 4, 5, 7, 10, 11 and 13 respectively, is (a) 3 (b) 4 ( 5 (0 6 45. If the interval of difference is h, then what is 42]... 867 —|e*x E 46 equal to? (a) 1 ) e* (0) exth (0 €> 46. The third divided the function (x3 - 22) with arguments 2, 4, 9, 10 is (a) (0) (0 (0) 1 2 3 4 47. What is [79 00] equal to? () (2) (6) 808000 + auf (0190 uf (x)ug(x) + 8) 894 uf (x) 6960 + 87292) 2 [ 991904] 48. Which one of the following is the second iterate of the Picard’s approximation of the initial value problem given below? (a) (8) (c) (0) ‌ ~= = 1+ × + , 01 de x+y, 0 3 x‏ 2 17 1 + 2× + = ×5= +— 2 2 3 1 + × + -- × 2 6 14:25 ~ 2 x? + 5 6 SNCS-A-STS/67A difference of 49. If the interval of difference is 1, then what is the value of the following? ^4[(× - 1) (1- 2x) (1- 3901-43] (a) 576 (b) 24 (© -24 (0) -576 50. Consider the following table : x | 0 | 1/4 | 1/2 | 3/4 1 F(x) 1 4/5 | 2/3 | 4/7 1/2 Apply Simpson’s one-third rule with 4 equal subintervals to estimate the area bounded by the curve and the x-axis from x = 0 to x = 1. If the area is A, then what is 2520A equal to? (a) 7 (b) 1650 (© 1747 (0) 1850 (2.1.0. 51. It is given that f(l)=1, f(2)=0-5, Ff (3) = 0-3, f(4) = 0-25, f(5) = 0-20. What is the estimated value of the integral ey (x) dx when estimated by trapezoidal rule? (a) 1:45 (b) 1:55 (c) 1:65 (d) 1:75 52., It “is. given that /f()=1, f(2)=0-5, Ff (3) = 0-3, f(4) = 0-25, f(5) = 0-20. What is the estimated value of the integral fr (x) dx when estimated by Simpson’s one-third rule? (a) 1-60 (b) 1-62 (e} $1465 (d) 1-68 53. The estimation of the global truncation error for trapezoidal rule for Sf (x) dx by partitioning into n equal subintervals with h= 2= and Ee ja, By is n (a) -Snh?f" (b) -+nh?7"®) 12 S85 civ (o) ->nnd5¥(e (a Sn? 5") SNCS-A-STS/67A 10 54. Let di = FX y=1 dx y+x’ with step size h=0-05. Apply forward Euler method to estimate the value of y(0-1). What is the approximate value of y(0-1)? (a) 1-005 (b) 1-05 (c) 1:10 (d) 1:50 55. Backward Euler method is used to solve the initial value problem Pe =-8y; y(0)=1. Which one of the following is correct? (a) For any step size h> 0, the method is absolute stable (b) For any step size h< 1/8 only, the method is absolute stable (c) For any step size h< 1/4 only, the method is absolute stable (a) For any step size h<1 only, the method is absolute stable 56. An 3 of the truncation error e(x) in the Lagrange interpolation polynomial is given by (x ¬ 0) (x - x1)... - Xn) 7 (n+ 1)! ) (a) (४-0) रिया: 067 Xn) பர (b) n+ DI प्री (8 6 (a) mee 0102 || n! (© where < < . 57. Let 22 ‎ ; u(0)=1 with step size h=0-2. Apply forward Euler method to estimate the value of u(0-4). What is that value? (a) 0:84 (b) 0.86 () 0-92 (d) 0-96 SNCS-A-STS/67A 58. If ¬1 < × < 1, then what is the product (1+ × + ×2 423 +...) (1-21-24 -x3 + sa) equal to? ஐயிரு? ப்ட்‌ () 1— x2 + 26 ~ 310 +... (‌) + 0... 59. Let and +...+ + 9 = ‎ . If the interval of difference 18 h= 1, then what is P/Q equal to? (a) 1 (b) n (c) 2841 (0) 72 60. Forward Euler method is used to solve the equation =-20y; y(0)=1 with step size h. For which one of the following values of h, the method is absolutely stable? (a) 0-05 (b) 0-15 (c) 0-25 (9) 0-35 (2.1.0. 61. Which one of the following is the central part of UNIX operating system that manages and controls’ the communication between various hardware and software components of a computer system? (a) Shell (b) Kernel (c) Compiler (a) Device driver 62. Which one of the following IEEE wireless standards has the maximum data rate (Mbps)? (a) 802.1la (b) 802.11ad (c) _802.11g (da) 802.11b 63. Consider the following statements : I. The equivalent octal number of the decimal number 45796 is 131344. ll. “The equivalent hexadecimal number of the octal number 365 is OF5. Which of the statements given above is/are correct? (a) I only (b) Il only (c) Both I and II (a) Neither I nor II 64. Which one of the following is not a process state? (a) Ready (b) Executing (c) Deadlock (ad) Terminated 65. Paging with swapping is also known as (a) demand paging (b) segmentation (c) swapping (da) None of the above SNCS-A-STS/67A 12 66. What is the fixed length of the instruction format used in RISC processors? (a) 30 bits (b) 16 bits (c) 32 bits (ad) 24 bits 67. Consider the following statements in respect of Python programming language : I. It can be used to create web applications. Il. It is interpreted language. Ill. It is platform-independent language. Which of the statements given above are correct? (a) I and II only (b) II and III only (c) I and III only (a) 1, Il and Il 68. Which one of the following is an International Standard for Information Security Management? (a) ISO/IEC 27001 (b) ISO 9001 (c) ISO 14001 (a) ISO 45001 69. Which one of the following protocols is responsible for determining the Media Access Control (MAC) address corresponding to an IP address? (a) TCP (b) FTP (c) ARP (dq) SMTP 70. UNIX operating system is written in which language? (a) C (b) COBOL (c) JAVA (a) FORTRAN Sa eee ee ee ee ees oe 71. What is the medium used in optical storage systems for reading and recording data? (a) Ultraviolet light (b) High-energy visible light (c) Laser light (ad) Infrared light 72. A flatbed plotter moves its pin in X-Y direction. If the pen speed is increased too much, what will be the effect on drawing quality? (a) Better resolution (b) No effect (c) Better accuracy (da) Lines may be distorted 73. In an absolute loading scheme, which function is normally performed by the assembler instead of the loader? (a) Combining object modules (b) Allocation (c) Linking (ad) Loading 74. A virus that modifies its code each time it spreads to evade detection by antivirus software is known as (a) stealth virus (b) multipartite virus (c) polymorphic virus (d) boot sector virus 75. Which one of the following techniques is being used by a peripheral device to transfer data continuously without constant intervention of CPU? (a) Program-controlled data transfer (b) Direct memory access (c) Polling (a) Interrupt-driven data transfer SNCS-A-STS/67A 13 76. Rohit receives an e-mail that states that his credit card is going to be cancelled. In order to prevent this, the e-mail advises Rohit to click on the provided link and sign into his online banking account, which Rohit does. A week later, Rohit notices that ~10,000 has been withdrawn from his bank account. What has Rohit fallen victim to? (a) Trojan Horse (b) Online blackmail (c) Phishing scam (ad) Virus 77. A page fault in operating system (a) refers to the page used in the previous page reference (b) occurs when a page program occurs in a page memory (c) refers to an access to the page not currently in the memory (ad) refers to an error specific page 78. A laser printer has a resolution of 1200 dpi. If a page measures 8x10 inches, how many total dots can theoretically be printed on one page? (a) 0-096 million (b) 96 million (c) 115-2 million (ad) 120 million 79. A malware that is capable of transferring from one computer to another without any user interaction is known as (a) Trojan (b) worm (c) boot sector virus (d) spyware 80. Which one of the following is used for storing the results produced by arithmetic and logic unit? (a) Program counter (b) Accumulator (c) Memory address register (d) Instruction register [ P.T.O. —— ———————E———————— EE SNCS-A-STS/67A SPACE FOR ROUGH WORK 14 SPACE FOR ROUGH WORK SNCS-A-STS/67A 15 [ SPACE FOR ROUGH WORK kkk SNCS-A-STS/67A 16 26BS—3072