QP-IES-ISS-25-STATISTICS-PAPER-I-230625

LE S/iS:S Examination - 2025 DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE TOLD TO DO SO T.B.C. : BGSP-A-STS Test Booklet Series Serial No MINN 1008053 TEST BOOKLET Se ay STATISTICS Paper I Time Allowed : Two Hours Maximum Marks : 200 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 Code 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. @ 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) Ifa 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 BGSP-A-STS (1-A) 2\\075 GANG. S08 - noiisnimsxa 1. X follows Beta distribution of first kind Beta (a, 8) with a > 1 and B > 1. What is the mode ? a-1 (a) a a bh) &) a+Bp atl (c) Gabe a+p @ a+B-1 2. ‘For Beta distribution of first kind Beta (a, B), consider the following statements : IL For a > 1, B > 1; density has unique mode. Il. For a +f = 2 such that a =f = 1, all points in [0, 2] are modes. Which of the statements given above is/are correct ? (a) Tonly (b) ILonly (c) BothIand II (d) Neither I nor II 3. Let X be a continuous random variable with the following pdf : 1 =; 2<x< fiz)x {g° 5 ** " 0; otherwise What is the distribution of Y = aX + b, provided a>0? (a) U(8a+b, 12a +b) (b) U(2a+b, 10a +b) (c) Poisson (ba!) (d) None of the above BGSP-A-STS 4, (2-A) A player rolls a fair dice and simultaneously tosses a fair coin. If the “head” appears on the coin, then the player wins twice the value that appears on the dice. However if the “tail” appears on the coin, then the player wins half of the value that appears on the dice. What is the player’s expected win ? (a) 10-250 (b) 8-625 (c) 7-625 (d) 4375 A miner is trapped in a mine containing 38 doors. The first door leads to a tunnel that will take him to safety after 3 hours of travel. The second door leads to a tunnel that will return him to the mine after 5 hours of travel. The third door leads to a tunnel that will return him to the mine after 7 hours of travel. Assuming that the miner is equally likely to choose any of the doors, what is the expected length of time until he reaches safety ? (a) 15 hours (b) 25 hours (c) 380 hours (d) 35 hours 6. " distribution function of standard Laplace distribution with the pdf f(x) = is: ! 50 (8) ) = ‎ 1- + e* =~, ± < 0 0) ) = ] 2 ail Ge x>0 ‎ اگوی‎ ৰ +: ৯৫৬০ Fy@)= x ‎ x>0 2 1-e* < @ / ` 1 x>0 7. Let X, and X, have the joint pdf given by _ 2 0૨૩૨-૨1 এ f{x,, 22) = | . What is the marginal distribution of X,? () () = 1, 0 < ±¡ < 1 ` £,(x,) = 2 ¬ 2], 0 < ±¡ < 1 66) () = 28), 0 < ±¡ < 1 ( f(x) = 3x?,0< x,<1 BGSP-A-STS (3- 10. A) Let k > 0 be a constant such that pdf of X is given by f(x) = . What is the value of E(X) ? (a) (1) © : (8) + Let (‎ and My(t) be the moment generating functions of two random variables X and Y respectively, for some teR If 800 = 2 , 900 = My(t) = (075 + 0:25e74)M,(t), where t € (¬ 1), h); h > 0, then what is Var(Y) equal to? - and (a) (b) () (d) Let X follow Chi-square distribution with n degrees of freedom. Which of the following statements is/are correct ? - distributed with mean 0 and variance 1. Ne ple ale i. normally is asymptotically 11. is asymptotically normal with mean n 1 and variance 4 i n Select the answer using the code given below : () [ IL only (6) BothIand II (d) Neither I nor II 11. The joint probability of the occurrence of two events A and B is given by the following contingency table : | 04 0: A | 02 08 Consider the following statements : A and B are independent events. . P(A|B)+P(A 1902 >. Which of the statements given above is/are correct ? () 107 (0) ۶۰۰۹۶۸۶ (€) 03176 II () Neither I nor 11 12. If k, and denote respectively the rth cumulant and the 7% moment about origin zero of a distribution, then which of the following statements are correct ? i 12] = 7 11. 1-12-12 111. “ + 2‎ Select the answer using the code given below : () Iand II only IL and III only () Land III only (0) 1, 11 ‎ 111 BGSP-A-STS (4- 13. 14. A) Let (X, Y) follow bivariate normal distribution with E(X) = E(Y) = 2, V(X) = 100, V(Y) = 144 and Corr(X, ) = 0:5. Which of the following statements is/are correct ? 1 E(Y| ® =x) = '2 + 0-6x . :: = 5005 Select the answer using the code given below : (a) Ionly ) Il only () 1 and II (d) Neither I nor 11 In a trivariate distribution, it is given that 12 = 0:8, 13 = ¬ 04, 723 = 0. The value of correlation coefficient between X, and X, after linear effect of X, on each of them is eliminated, is : (a) (1) 0) ¬ @ - 15. 16. BGSP-A-STS If nyx is the correlation ratio of Y on X, then which of the following statements is/are correct ? is independent of change of origin 2 "yx and scale. IL. Nyx ಗರ್ಗ 111. 72 where 7 is the correlation They te coefficient between X and Y. Select the answer using the code given below : () ] 880 11 only (0) 1110117 (6) Land III only () I, 11 7 The following is the joint probability mass function of two random variables and 7 : 3 | ° | 1 0 | 3/28 | 9/28 | 3/28 1 | 3/14 | 3/14 0 2 | 1/28 0 0 Which of the following statements is/are correct ? 1 = 3/4 11. = 80 Select the answer using the code given below : (8) 10017 (0) [ () ` 9011) 1 and II (d) Neither I nor II (5- 17. 18. A) Let (51, 22, . . . 100 be a random sample of size k from geometric distribution with parameter p with pmf पिंड) = एप; x = 1, 2, 3, ~ ~ ~ . Then 101002, Xo, . . . , X,) is distributed geometrically with parameter : (8) 1 0) ~ 1-01-7 ( (1-95 (0) None of the above If the random variables X and Y have a bivariate normal distribution with parameters Hy ಟ್ರೂ ता of and p, then the marginal distribution of X is : (a) , ) 0) அழுக pay ചള, मै) 2 ( , (1-02) (d) Nowy + , (1 = 2) 2 19. What would be the lowest value of r? in a sample of 27 pairs of observations which would be significant at 5% level of significance ? [1250-05) = 2:06] (8) 01451 ) 02451 (6) 51 (0) 0:4451 20. The male population in a state is 25 lakh. Among them the number of literates is 2 lakh, the number of criminals is 26000 and the number of literate criminals is 2000. What is the association between literacy and criminality ? (a) No association (b) Negatively associated (c) Positively associated (d) Association cannot be established BGSP-A-STS 21. 22. (6-4). 1 =) where What 15 (‌) equal 10, 02121? (take the interval of differencing to be unity) ab* 1 (a) (b) () ab* (6) 1-9 What is the value of A® 34521 x (take the interval of differencing to be unity) (a) 1 000: ತ್ತು () 1‎ 1 55 (4) 29 23. If the 9% divided difference of f(x) = 4 based 5 x on points x; = ih, 1 = 1 < 10 is a, then what is the 10‘ divided difference of f(x) based on points x; = 111, 1 = 1 = 11? a (a) = 118 ~ 101 a () tin இலவம்‌ = 10h 24. The Simpson’s one-third rule applied to 2 . 10) dx gives the value 2, where f(1) = 06. 0 If trapezoidal rule is applied, then what is the 2 value of | f(x) dx ? 0 (a) 8 (b) 4 ( 02‎ 25. Consider the following table : ‎ | 0 | 1/4 | 1/2 | 3⁄4 1 fix) 1 4/5 | 2/3 | 4/7 | 1/2 Simpson’s one-third rule with 4 equal intervals is used to approximate the area bounded by the curve f(x) and x-axis from x = 0 to x = 1. What is the approximate value of the area ? (a) 06951 (0) 06944 () 06982 () 0-70 BGSP-A-STS ( 7 26. 27. -A) Trapezoidal rule is used to find the absolute ath value of error in evaluating f f(x) dx. a If the of ‌ f"( , < <b, then what is the value of h? absolute value error is (a) 52 (b) 3 € 4 ) 8 Consider the following statements : b 1. Trapezoidal rule for evaluating | f(x) dx a gives exact result for linear polynomial. b Il. Trapezoidal rule for evaluating f f(x) dx a gives exact result for quadratic polynomial. III. Simpson’s one-third rule for evaluating b f(x) dx gives exact result for cubic a polynomial. IV. Simpson’s one-third rule for evaluating b ‌ f(x) dx gives exact result for quadratic a polynomial. Which of the statements given above are correct ? (a) I, 111 77 (0) 1800 1 only (€) Iand III only () , and IV 28. 30. BGSP-A-STS Forward Euler method is used to find numerical solution ] at x = 0:2 with h = 2 dy of the initial value problem — = 25, y(0) = 1. What is | y(0-2) - y, | approximately equal to ? (a) 00215 ) 00385 () 5 1 Picard method is used to find approximate solution yx) {first iterate of Picard’s approximation} of initial value problem dy _ dx [ - |at x = 0-2? - 22, y(0) = 1. What is the value of (a) 09915 (0) 09615 (€) 00015 (1) 00005 If y(x) is the solution of =1l+x+y,y(0)=1 for step h = 0-1 by Euler’s method, then y(0-3) is equal to: (a) 0968 5 (6) 1-698 (1) 1968 (8- 81. 32. 33. 34. A) In which of the following topology, all the nodes are connected to a single backbone cable ? (a) Mesh (b) 58 (c) Hybrid () Bus Which device is used for connecting the computers in a star topology ? (a) Router (b) Hub (c) Repeater () Bridge In which type of scheduling is the process allocated to CPU for a specific time period or time slice ? (a) FCFS (b) Shortest Job First (©: Priority | (0) Round Robin Consider the following : Better performance II. 1.077 price 111. Low power and heat IV. High reliability Which of the above are advantages of solid-state storage devices as compared to hard disk drives ? (a) 1, and III (0) I, I and IV (€ I, ]]] andIV () II, 111 and IV 35. Consider the following statements : 4 The hexadecimal equivalent of octal number 366 is OF6. II. The decimal equivalent of hexadecimal number A53 is 2643. Which of the statements given above is/are correct ? (a) Ionly (b) ILonly (c) BothI and II (d) Neither I nor II 36. Consider the following : ib Multi-user II. Multitasking III. Multiprocessor Which of the above are correct in respect of Linux operating system ? (a) Iand II only (b) IL and III only (c) Land III only _@) I, Mand Il 37. Which one of the following terms specifies the amount of memory required by an algorithm for performing the desired task ? (a) Time complexity (b) Space complexity (c) Hybrid complexity (d) Processing complexity BGSP-A-STS 39. 40. (9-A) In high-level programming languages, what role does the runtime environment play ? (a) It provides libraries and frameworks for application development. (b) It executes the compiled machine code directly on the CPU. (c) It manages memory allocation, garbage collection, and other runtime tasks. (d) It translates source code into machine code during compilation process. What is the purpose of using breakpoints in debugging ? (a) To mark a point in the code where execution should pause for inspection. (b) To measure the time elapsed to run a block of code. (c) To skip a block of code during execution. (d) To program. identify memory leaks in the In a client-server Database Management which one of the primarily responsible for System architecture, following is managing database access, security, and data integrity ? (a) Query processor (b) Client (c) Server (d) Database administrator Consider the following for the next four (04) items : Let (X,, 25, 303) be a random vector with the joint pmf given by : PUK, = ], Xp = 9, 23 = 3) = 50! வதிய کک‎ ±] 3912, 160-௨௧௨) -25 -ಸ್ಟಾ)!(4) (4) (4) (4 if xj + ± + ৯৪3 < 0 0; otherwise 41. The distribution of (50 - X,) is: (a) Binomial (50, 1/4) (b) Binomial (50, 3/4) (6) Poisson (3/4) () Poisson (1/4) 42. The correlation between X, and X, 15 : (8) -1/4 (0) -1/3 (c) 1/4 () 18 43. What is E(X, |X, = 5) equal to? (a) 8 (b) 10 (€ 12 (0) 15 44. What is V(X, |X, = 5) equal to? (a) 8 (b) 10 (c) 12 () 15 BGSP-A-STS (10-A) Consider the following for the next three (03) items: Let random variables X and Y be independently ‘distributed as Poisson (1) and their moment generating functions together with random satisfy the M(t) = 04008 M,(t)) for some (-h,h);h>0. variable Z following relation 45. What is the value of E(Z) 7 (a) 241 0) 07 (60 344). ( 141 46. What is the value of Var(Z) ? (a) 117,473 01 ର ଙ୍କୁ ( 12 4:20 () 27423425 47. What is the moment generating function of random variable 2 ? (a) 2 (-1) 7 0) ९५०५९ -1) _ 1] () ere 1) (को ` ୬୩୨୯ -9-3 BGSP-A-STS Use Poisson distribution to compute the probability of getting the ace of spades at least twice in 104 trials of drawing a single card from a pack of well-shuffled cards with replacement (Given e~? = 01353). What is the probability ? (a) 0:6285 00) 05941 (6) 05419 (d) None of the above 49. Let 2], 29, be iid sequence of 7876080 variables with common ୮୦୯୮ = _ 1712 £ = 1 ~ = ~ 0 50 then x converges in probability to : (a) 1-9 1+6 ( 1-6? (1) 1+6 The characteristic function of a random variable X is given by ®,(t) = 8-0 || = 1. What is the fourth order central moment of the distribution ? (a) 12 (0) 24 (c) 36 (d) 48 (11-A) 51. Let 0, ‎ . . . , X, be a random sample from uniform [0, 1] distribution. The asymptotic distribution of empirical cdf distribution function F(x) is : (a) Normal with mean zero and variance 1 (b) Normal with mean zero and variance x(1-x)/n (c) Normal with mean x and variance 21-30 (d) None of the above 52. A two dimensional non-negative continuous random variable (X, Y) has the joint density ‎ 4256 ± +>); x>0,y>0 0; otherwise Let T? = X? + Y? such that 0 = (x? + y?) < . The density function of T is given by : 2 (8):: 986." 3 _+8 (8) 22 7 2 386-7 3 () BGSP-A-STS 53. 55. (12-A) Given that for three distinct attributes A, B and C with (A) = () = () = (0) = () = ¢) = N/2 such that (ABC) = (001). Which of the following statements is/are correct ? 1. () = (00), (AB) = (8) TI. 2486) = (AB) + (AC) + (BC) ¬ N/2 Select the answer using the code given below : (8) 10017 (b) IL only (6) BothI and II (d) Neither I nor II If X is uniformly distributed with mean 1 and variance 4/3, then P[X < 0] equals : (a) 1/2 (0) 18 (c) 1⁄4 () 1/8 If ~ N(u, 2), then Pla 2 0] in terms of cdf of the standard normal distribution ®(.) for some constants ‘a’ and ‘b’ 18 : இதன்‌ હાટ 0 0 5 0 0 ‌ ‎ 0 0 ‎ 9 0 0 56. 57. Let (X,, . 253) be a random sample from N(0, 1) distribution. If for some 00, 95 7 53 and 1/5, : 5 0, = 0 and 58 ene ‎ 51 1 18: = 0, then the distribution of (a) ¦ () . Let M be the minimum observation in a random sample of size n from a distribution —(x—6) > e 6 << that has the pdf f(x) = 0, otherwise What is the distribution of 2n(M - 0) ? (b) Exponential distribution with mean equal to 2 (c) Exponential distribution with mean equal to 0-5 @ 58. Let ® ~ Niu, 2). If 2 = 2, > 0, then the expression of P[(X < - ) |(X < ] in terms of the cdf of N(0, 1) distribution ®(.) is : C2) 00) (2) () 1-(2) (1) 21-02) BGSP-A-STS 59. 60. (13-A) Suppose that the random variables X,, 302, 3, , X; are independent and each has a standard normal distribution. A constant C such that the (24 425) random variable will have a t-distribution, 00 ‏ has value : (d) Consider the following 2 x 2 table for two attributes A and : B not B A a b notA | d Given that Q = Yule’s Coefficient of Association and Y = Coefficient of Colligation. Which correct ? 1585 ... 11. Q=-1lifad=0 111. & = 08 = Select the answer using the code given below : () Iand 11 only (2) and III only (c) Land 111 only (0) 1,118 7 of the following statements are 61. BGSP-A-STS What is the function whose first difference is 2‎ + ± + 4? (take interval of differencing to be unity) (8) 4x24 7± + 7. 0) ‎ + ±2 +} 2x8 + ±2 + 8± + 7. () ` ±3 _ ±2 + 4± + 7. What is forward difference approximation for the second order derivative 17%) at x; 7 ¬ 20 +1 + fi a ‎ ' 6-1-25 + 84] (b) en (0 +2; ~ 29-22-71 (d)‏ Suppose g 15 நமனை such 1108 y = f(x) x = ) and ffx) is given by _ 6-) -) (x—a)(x—c) ) = |. where a<b<canda<B<y are real numbers. What 15 (3) equal to ? (a) 3 a Ch (6); Suppose f(x) = x + x? + , If g(x) is a quadratic polynomial determined using Lagrange’s interpolation formula using the fact that g(f(0)) = 0, g(f(1)) = 1 and g(f(2)) = 2, then what is g(13) equal to ? (a) 70/19 71/21 () 71/20 8) 70/21 65. 66. 67. (14 - ^ ) If y,(x) is the 40 iterate of Picard’s approximation for the solution of the initial value problem = 1 + ±2 + 72; y(1) = 2, then what 15 (1) equal to? (a) 1 (b) 2 ( (1) 4 Consider the following statements in respect of forward and _ backward (A and V): operators i: A-V=AV 11. V-A=VA 111. A+V=AV 17. A+V లా దీది Which of the statements given above is/are correct ? (8) 10017 Il only () 1874111 () IlandIV A100 What is the value of (® (take interval of differencing to be unity) (a) 100! ! - !100 )0 ( 100! -1 1 + ! - !100 )0( 68. 70. BGSP-A-STS Consider the following data : x 6-1 6-2 f(x) 6:3 6-4 — 0:1998 | — 0-2223 |— 0-2422 |— 0-2596 What is the approximate value of f”(6-3) with error O(h?) obtained using central difference operator ? (a) (b) (c) 0-23 (d) 0:27 0-25 0-21 What is the approximate value of the integral 1 al x3 dx, if it is obtained using the 0 1 quadrature formula f f(x) dx = af{0) + pflx,) 0 which is exact for polynomials of all small degrees as high as possible ? (a) (b) (c) 2/3 (d) 1/4 V9 2/9 It is given that f(0) = 8, f(1) = 11, (4) = 68 and f(5) = 120. What is the approximate value of f(2) if it is obtained by applying Lagrange’s interpolation formula ? (a) (b) (c) 186 16:6 176 (d) 196 71. 72. 73. (15-A) Which one of the following statements best describes the use of flowcharts in problem solving ? (a) Flowcharts are used to visualize data structure. (b) Flowcharts are used to create graphical user interface. (c) Flowcharts consist of short, readable and formally-styled English language statements. (d) Flowcharts are diagrammatic representations of the logic for solving a task. What is the main purpose of the linker in the compilation process of the programming languages ? (a) To perform optimization of compiled code (b) To translate source code into machine code (c) To check syntax of the program (d) To combine object files and libraries into an executable program Which of the following are the characteristics of virtual reality ? di; It is created using multimedia. II. It enables users to move and react in a computer-simulated environment. Ill. It stimulates senses like sight, hearing, and touch. Select the answer using the code given below : (a) (b) IL and III only (c) Tand III only (ad) I, IWand Il Tand II only i 74, 75. 76. BGSP-A-STS Multimedia refers to combination of different types of media in one application. How many of the following are part of multimedia ? I Text II. Video III. Audio IV. Graphics and Images V. Animation Select the answer using the code given below : (a) Only two (b) Only three (c) Only four (d) All five Consider the following in respect of working of router in computer networks ? L It is the purpose of interconnecting computer networks. used for II. It determines on which outgoing link an IP packet is to be forwarded. Ill. The destination address of the IP packet is examined, and it is routed to the next router. Which of the statements given above are correct ? (a) Iand II only (b) Iland III only (c) Land III only (d) I, Mand III In which one of the following the same set of statements are executed several times on the basis of the condition specified ? (a) If-Then (b) If- Then - Else (c) Do- While (d) Case Type 77. 78. 79. (16-A) Consider the following statements about the virtual memory in the computer system : i It provides large addressable space without worrying about size limitations of physical memory. II. It provides efficient sharing of main memory among different users in multiprogrammed operating system. Ill. It provides a large secondary memory. Which of the statements given above are correct ? (a) Land II only (b) ILand III only (c) Land III only (d) 1, IWandIIl A packet filter firewall can : (a) filter the specific users from accessing the services of the network. (b) block all IP packets from entering the networks. (c) check all IP packets and if found valid, then it is allowed to enter or exit the networks. (d) block any kind of viruses from entering the networks. A Wi-Fi is often used as a synonym for which IEEE technology ? (a) 802-1 (b) 802:5 (c) 802-9 (d) 802-11 AND and OR operations are part of which unit of CPU ? (a) Arithmetic Unit (b) Logic Unit (c) Control Unit (d) Main Memory Unit SPACE FOR ROUGH WORK BGSP-A-STS (17=A) SPACE FOR ROUGH WORK BGSP-A-STS (18-A) SPACE FOR ROUGH WORK BGSP-A-STS (19-A) 1008053 SPACE FOR ROUGH WORK BGSP-A-STS (20-A)