Indian Economic Service - Indian Statistical Service Examination
1. What is Var(X) equal to?
(a) 2-7
(b) 2-4
(c) 2-35
(d) 1-17
2. What is E(Y|X = 1) equal to?
(a) 1/7
(b) 3/7
(c) 4/7
(d) 6/7
3. What is P(|X - Y| ≤ 1) equal to?
(a) 0-06
(b) 0-7
(c) 0-12
(d) 0-41
For the next two (02) items that follow :
Let f_X(x) be p.d.f. of N(0,1) variates and
Φ(x) = ∫_{-∞}^{x} f_X(θ) dθ
4. What is Φ(-100) equal to?
(a) -Φ(100)
(b) 1-Φ(100)
(c) 1/2 Φ(100)
(d) 1/2 - Φ(100)
(a) 1/2
(b) 1/4
(c) 0
(d) 1/3
UPSC Previous year question paper Indian Economic Service - Indian Statistical Service Examination, 2026 exam
For the next two (02) items that follow :
The joint probability density function of X and Y is given as
6. What is the value of E[X|Y = 0.5]?
(a) 1/9
(b) 1/3
(c) 5/9
(d) 7/9
7. What is the value of conditional variance of X given Y = 0.5?
(a) 11/162
(b) 13/162
(c) 5/54
(d) 17/162
11. If X ~ N(0, 1) and Y ~ N(0, 1) are independent random variables, then the distribution of X / Y is
For the next two (02) items that follow :
Let the joint p.d.f. of (X, Y) be given by
8. The expression for conditional density of Y given X is
(a) (3x + y) / (2(3x + 1)); 0 < y < 2, 0 < x < 1
(b) x(3x + y) / (2x(3 + x)); 0 < y < 2, 0 < x < 1
(c) x(3 + y) / (3x + 1); 0 < y < 2, 0 < x < 1
(d) (3x + y) / (x(6 + y)); 0 < y < 2, 0 < x < 1
9. The expression for conditional density of X given Y is
(a) x(3x + y) / (2x(3 + x)); 0 < x < 1, 0 < y < 2
(b) (3x + y) / (x(6 + y)); 0 < x < 1, 0 < y < 2
(c) 2x(3x + y) / (2 + y); 0 < x < 1, 0 < y < 2
(d) x(6x + y) / (2 + xy); 0 < x < 1, 0 < y < 2
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 p, then E(X + Y - Z) = 0, if and only if p equals
(a) 1/3
(b) 3/4
(c) 2/5
(d) 3/5
(a) Chi-square with degrees of freedom 2
(b) Cauchy (0, 1)
(c) Cauchy (1, 1)
(d) 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?
(a) 35/139
(b) 200/417
(c) 112/417
(d) 105/139
13. Let (X, Y) have a bivariate normal distribution with parameters μ_x = 5, μ_y = 8, σ_x^2 = 16, σ_y^2 = 9 and ρ_xy = 0.6. What is the coefficient of regression of Y on X?
(a) 7/16
(b) 9/20
(c) 1/2
(d) 3/8
14. Let X be a continuous random variable whose density function is constant in a ≤ x ≤ b and zero elsewhere. What is the standard deviation of X?
(a) (2a + b) / (2√3)
(b) (b - a) / (2√3)
(c) (a + b) / 2
(d) (b - a) / 2
15. A random variable X has the following probability mass function p(x) = x / 15 where x = 1, 2, 3, 4 and 5. What is E(X|x > 1) equal to?
(a) 27/7
(b) 45/14
(c) 15/7
(d) 15/14
16. A random variable X has the following distribution :
What is P(5 < X < 7 | X > 6) equal to?
(a) 1/3
(b) 3/4
(c) 2/3
(d) 1/6
17. Let X have a binomial distribution with parameters n and p. If q = 1 - p, then what is P[X = 0 or even] equal to?
(a) 2(q - p)^n
(b) [1 + (q - p)^n] / 2
(c) [1 - (q - p)^n] / 2
(d) 1 + (q - p)^n
18. Let X_1, X_2, ... be a sequence of i.i.d. random variables and S_N = X_1 + X_2 + ... + X_N, where N is a random variable independent of X_i, i = 1, 2, 3, ... . If M_{X_1}(t) = E(e^{tX_1}), then the moment generating function of S_N is
(a) log(log M_{X_1}(t))
(b) M_N(log M_{X_1}(t))
(c) M_{X_1}(log M_N(t))
(d) M_N(e^{M_{X_1}(t)})
19. If the joint p.d.f. of two random variables X and Y is
then the value of P(X ≥ Y) is
(a) 0·7
(b) 0·3
(c) 0·12
(d) 0·88
20. A number X is chosen at random between 0 and 6, and another number Y is chosen at random between 0 and 9. What is the probability that the equation Z^2 - XZ + Y = 0 has real roots?
(a) 1/2
(b) 1/3
(c) 2/3
(d) 1/4
21. The random variables X and Y have joint p.d.f.
f(x,y) = 1/3 (x + y); 0 ≤ x ≤ 1, 0 ≤ y ≤ 2
What is the value of E[Y|X = 2]?
(a) 8/9
(b) 10/9
(c) 11/9
(d) 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
(c) 13 < n < 15
(d) n > 15
23. If b_{13.2} is the regression coefficient of X_1 and X_3 on X_2, and b_{31.2} is the regression coefficient of X_3 and X_1 on X_2, then what is b_{13.2} × b_{31.2} equal to?
(a) r_{12.3}^2
(b) r_{13.2}^2
(c) r_{23.1}^2
(d) r^2
24. Let (X_1, X_2, ..., X_n) be a random sample from N(μ, σ^2), then the distribution of n s^2 / σ^2, where s^2 = 1/n ∑_{i=1}^n (X_i - X̄)^2, is
(a) χ_{(n)}^2
(b) χ_{(2n)}^2
(c) χ_{(n - 1)}^2
(d) χ_{(2n - 2)}^2
25. If the coefficient of correlation r(2x, y) = 0.25, then what is
r(4x - 3, (7 - 4y)/8)
equal to?
(a) 1/4
(b) 1/8
(c) -1/4
(d) -1/8
26. Let Y_1 ≤ Y_2 ≤ Y_3 ≤ Y_4 ≤ Y_5 ≤ Y_6 be the order statistics from a c.d.f. F(x) = 1 - e^{-λx}, where λ = 0.5. What is the c.d.f. of Y_1?
(a) 1 - e^{y/2}
(b) 1 - e^{y/3}
(c) 1 - e^{3y}
(d) 1 - e^{-3y}
27. If R_{i,j,k} represents multiple correlation coefficient, r_{ij} represents simple correlation coefficient and σ_i represents standard deviation of variate X_i for a multivariate data set, then consider the following statements :
R_{1.23}^2 = σ_1^2 σ_2^2 (1 - r_{12}^2)(1 - r_{13})^2
R_{1.23}^2 ≥ r_{12}^2
R_{1.23}^2 ≥ r_{13.2}^2
Which of the statements given above are correct?
(a) I and II only
(b) II and III only
(c) I and III only
(d) I, II and III
28. The first of the two samples has 100 items with mean 15 and standard deviation 3. If the whole group has 250 items with mean 15-6 and standard deviation √13.44, then what is the variance of the second group?
(a) 16
(b) 12
(c) 4
(d) 2
29. Let (X,Y) ~ BVN(μ_1, μ_2, σ_1^2, σ_2^2, ρ). If σ_1^2 = 2, σ_2^2 = 3 and ρ = 0, then consider the following :
I. E(X) = E(Y)
II. E(XY) = 2
III. X and Y are independent
Which of the above is/are correct?
(a) I only
(b) II only
(c) III
(d) I and II
30. The joint p.d.f. of X and Y is given by
f(x,y) = y / (1 + x)^4 e^{-y/(1 + x)}; x,y ≥ 0
Then the regression of Y on X is
(a) exponential
(b) linear
(c) polynomial of degree 2
(d) polynomial of degree 3
31. Two normal populations π_1 and π_2 have their means equal, but standard deviation of π_2 is twice that of π_1 having standard deviation σ. Then in the samples of size 2000 from each population, the difference of sample means in all probability is
(a) less than 0.05σ
(b) less than 0.15σ
(c) greater than 0.05σ
(d) greater than 0.15σ
32. The mean height of 50 male students who showed above average participation in college athletics was 68:2 inches with a standard deviation of 2:5 inches, while 50 male students who showed no interest in such participation had a mean height of 67:5 inches with a standard deviation of 2:8 inches. In order that the observed difference in the mean height to be significant at 5% level of significance, the sample size of each of the two groups has to be increased by (the critical value of Z at 5% level of significance is 1:645)
(a) at least 78
(b) at most 78
(c) at least 28
(d) at most 28
33. A normal population has a mean of 0:1 and standard deviation of 2:1. For a random sample of size 900, what would be the value of P(x̄ < 0)?
[Given that P(0 < Z < 1.43) = 0.4236]
(a) 0.0764
(b) 0.4236
(c) 0.5764
(d) 0.9236
34. When sampling is without replacement, then V(x̄) = σ^2/n ((N - n)/(N - 1)). The factor (N - n)/(N - 1) is ignored under which of the following conditions?
I. n is small compared to N
II. n = N
III. Population is infinite
Select the correct answer using the code given below.
(a) I only
(b) III only
(c) I and III only
(d) I, II and III
35. A random sample of size 5 is taken from a normal population with mean 2·5 and variance 36. The probability that the sample variance
S^2 = 1/(n - 1) ∑_{i=1}^n (X_i - X̄)^2
lies between 27 and 45 is
[Given, P(χ_4^2 ≥ 3) = 0.55783 and
P(χ_4^2 ≤ 5) = 0.71270]
(a) 0·15487
(b) 0·84513
(c) 0·27053
(d) 0·34513
36. The t-statistic for a given sample is calculated to be 3. If the sample variance is doubled, then the new t-statistic will become
(a) 6
(b) 3/2
(c) 3/√2
(d) 1/√2
37. Kruskal-Wallis test is exactly equivalent to one-way ANOVA when
(a) data are normal
(b) population distributions are identical and normal
(c) sample sizes are equal
(d) all observations are replaced by their ranks
38. If X ~ U(0, 1), then P(X_{(1)} > x) is
(a) 1 - x^n
(b) (1 - x)^n
(c) n x^{n - 1}
(d) x^n
39. Consider a sample of odd size n from U(0, 1) population. Then the mean and variance of distribution of median are respectively
(a) 1/2 and 1/4
(b) 1/2 and 1/(4n + 8)
(c) 1/4 and (n - 3)/(2n + 4)
(d) Both mean and variance do not exist
40. What is the mean of the numbers 1·n^2, 2·(n - 1)^2, 3·(n - 2)^2, ..., n·1^2?
(a) (n + 1)^2 (n + 2) / 12
(b) (n + 1)^2 / 4
(c) n(n + 1)^2 / 12
(d) (n + 2)^2 (n + 4) / 12
41. If the interval of difference is h, then what is Δ[log f(x)] equal to?
(a) log[1 + E f(x)/f(x)]
(b) log[1 - Δ f(x)/f(x)]
(c) log[1 + Δ f(x)/f(x)]
(d) log[1 - E f(x)/f(x)]
42. A cubic polynomial f(x) takes values as in the following table :
x: 0 1 2 3
f(x): 1 0 1 10
What is f(4) equal to?
(a) 35
(b) 33
(c) 31
(d) 29
43. Consider the following statements :
I. The (n + 1) th divided differences of a polynomial of nth degree are zero, where n is a natural number.
II. 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) I only
(b) II only
(c) Both I and II
(d) Neither I nor II
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
(c) 5
(d) 6
45. If the interval of difference is h, then what is
(Δ^2/E) e^x × (E e^x)/(Δ^2 e^x)
equal to?
(a) 1
(b) e^{-x}
(c) e^{x + h}
(d) e^x
46. The third divided difference of the function (x^3 - 2x) with arguments 2,4,9,10 is
(a) 1
(b) 2
(c) 3
(d) 4
47. What is μ[f(x)g(x)] equal to?
(a) δ f(x) δ g(x) + 1/4 μ f(x) μ g(x)
(b) μ f(x) μ g(x) + 1/4 δ f(x) δ g(x)
(c) μ f(x) δ g(x) + δ f(x) μ g(x)
(d) 1/4 [μ f(x) δ g(x) + δ f(x) μ g(x)]
48. Which one of the following is the second iterate of the Picard's approximation of the initial value problem given below?
dy/dx = 1 + x + y, y(0) = 1
(a) 1 + 3x + x^2 + x^3/6
(b) 1 + 2x + 3/2 x^2 + x^3/6
(c) 1 + x + 3/2 x^2 + x^3/6
(d) 1 + 2x - 3/2 x^2 + x^3/6
49. If the interval of difference is 1, then what is the value of the following?
Δ^4[(x - 1)(1 - 2x)(1 - 3x)(1 - 4x)]
(a) 576
(b) 24
(c) -24
(d) -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) 1547
(b) 1650
(c) 1747
(d) 1850
51. It is given that f(1) = 1, f(2) = 0.5, f(3) = 0.3, f(4) = 0.25, f(5) = 0.20. What is the estimated value of the integral ∫_1^5 f(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) = 1, f(2) = 0.5, f(3) = 0.3, f(4) = 0.25, f(5) = 0.20. What is the estimated value of the integral ∫_1^5 f(x) dx when estimated by Simpson's one-third rule?
(a) 1·60
(b) 1·62
(c) 1·65
(d) 1·68
53. The estimation of the global truncation error for trapezoidal rule for ∫_a^b f(x) dx by partitioning into n equal subintervals with h = (b - a)/n and ξ ∈ [a, b] is
(a) -1/2 n h^2 f'(ξ)
(b) -1/12 n h^3 f''(ξ)
(c) -3/80 n h^5 f^{iv}(ξ)
(d) 1/2 n h^2 f''(ξ)
54. Let
dy/dx = (y - x)/(y + x); y(0) = 1
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 dy/dx = -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
(d) For any step size h < 1 only, the method is absolute stable
56. An estimate of the truncation error e(x) in the Lagrange interpolation polynomial is given by
(a) ((x - x_0)(x - x_1)...(x - x_n))/((n + 1)!) y^n(ξ)
(b) ((x - x_0)(x - x_1)...(x - x_n))/((n + 1)!) y^{n + 1}(ξ)
(c) ((x - x_0)(x - x_1)...(x - x_n))/(n!) y^{n - 1}(ξ)
(d) ((x - x_0)(x - x_1)...(x - x_n))/(n!) y^n(ξ)
where x_0 ≤ ξ ≤ x_n.
57. Let du/dt = -2t u^2; 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
(c) 0·92
(d) 0·96
58. If -1 < x < 1 then what is the product (1 + x + x^2 + x^3 + ...)(1 - x + x^2 - x^3 + ...) equal to?
(a) 1 + x^2 + x^4 + x^6 + ...
(b) 1 - x^2 + x^4 - x^6 + ...
(c) 1 - x^2 + x^6 - x^{10} + ...
(d) 1 + x^2 + x^6 + x^{10} + ...
59. Let
P = (C(n,1) + C(n,2)Δ + C(n,3)Δ^2 + ... + C(n,n)Δ^{n - 1}) y_1
and Q = y_1 + y_2 + ... + y_n. If the interval of difference is h = 1, then what is P/Q equal to?
(a) 1
(b) n
(c) 2n + 1
(d) n^2
60. Forward Euler method is used to solve the equation dy/dx = -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
(d) 0·35
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
(d) Device driver
62. Which one of the following IEEE wireless standards has the maximum data rate (Mbps)?
(a) 802.11a
(b) 802.11ad
(c) 802.11g
(d) 802.11b
63. Consider the following statements :
I. The equivalent octal number of the decimal number 45796 is 131344.
II. The equivalent hexadecimal number of the octal number 365 is OF5.
Which of the statements given above is/are correct?
(a) I only
(b) II only
(c) Both I and II
(d) Neither I nor II
64. Which one of the following is not a process state?
(a) Ready
(b) Executing
(c) Deadlock
(d) Terminated
65. Paging with swapping is also known as
(a) demand paging
(b) segmentation
(c) swapping
(d) None of the above
66. What is the fixed length of the instruction format used in RISC processors?
(a) 30 bits
(b) 16 bits
(c) 32 bits
(d) 24 bits
67. Consider the following statements in respect of Python programming language :
I. It can be used to create web applications.
II. It is interpreted language.
III. 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
(d) I, II and III
68. Which one of the following is an International Standard for Information Security Management?
(a) ISO/IEC 27001
(b) ISO 9001
(c) ISO 14001
(d) 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
(d) SMTP
70. UNIX operating system is written in which language?
(a) C
(b) COBOL
(c) JAVA
(d) FORTRAN
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
(d) 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
(d) 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
(d) 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
(d) Interrupt-driven data transfer
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
(d) 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
(d) refers to an error specific page
78. A laser printer has a resolution of 1200 dpi. If a page measures 8 × 10 inches, how many total dots can theoretically be printed on one page?
(a) 0-096 million
(b) 96 million
(c) 115-2 million
(d) 120 million
(b) 96 million
(c) 115-2 million
(d) 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
Indian Economic Service - Indian Statistical Service Examination STATISTICS I
Indian Economic Service - Indian Statistical Service Examination-INDIAN ECONOMICS
Indian Economic Service - Indian Statistical Service Examination-GENERAL STUDIES
Indian Economic Service - Indian Statistical Service Examination-GENERAL ENGLISH
