BPSC Exam 2025: Integrated 70th Combined Main (Written) Competitive Examination, Optional, Statistics
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Serial No.
Booklet Series
02/GO/CC/M-2025 – 27
I
Candidate’s Roll Number
Question Booklet
I
STATISTICS
Time Allowed : 2 Hours Maximum Marks : 100
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-1-
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1. Reliability function of Weibull life time 3. Let X be a discrete random variable with
x
the moment generating function
1
model f(x; , ) x e , x > 0, > 0,
(1 3et )3 (3 et )2
Mx (t ) , t R . Then
1024
> 0, will be
which of the following statements is
correct ?
x
(A) e
6
(A) P(X 0)
1024
1
(B) x 18
(B) E(x )
4
28
x (C) P(X 5)
(C) 1 e 1024
15
(D) V(x )
16
1
(D) x
4. The index numbers lie between
Laspeyre’s and Paasche’s index
2. Let x , x , . . ., x be a random sample numbers are
1 2 n
from Poisson distribution with mean (A) Fisher and Dorbish-Bowley
(B) Marshall Edgeworth and Fisher
> 0. Then variance of MLE of is
(C) Marshall Edgeworth, Dorbish-Bowley
and Fisher
2
(D) Marshall Edgeworth and
(A)
n Dorbish-Bowley
(B) 5. In a randomised block design with
4 blocks and 5 treatments having one
missing value, the error degrees of
2 freedom will be
(C)
n2 (A) 11
(B) 9
(C) 12
(D)
n (D) 10
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6. Hazard function of exponential life time 9. If r is the correlation
model with mean > 0 is given by sample of n pairs of observations, then
probable error of r is given by
x
(A) 1 e
(A) 0 . 6745 1 r 2 n
1
(B)
(B) 0 . 6745 (1 r 2 ) n
1 x
(C) e
(C) 0 . 6754 1 r2 / n
x
(D) e (D) 0.6754 (1 – r2)/n
7. Let X~ U(5, 5 + ),
10. e years values 15, 24,
T = k(x – c)2. The statistic T becomes
2 18, 33 and 42, the three years moving
unbiased estimate of if
averages will be
(A) k = 5, c = 0
(A) 29, 30, 31
(B) k = 5, c = 5
(B) 19, 22, 31
(C) k = 3, c = 0
(C) 29, 25, 31
(D) k = 3, c = 5
(D) 19, 25, 31
2
8. Consider the model y = i + , ~ N(0, i ),
i i i
11. Consider the following statements :
i = 1, 2, 3, where y y y , are uncorrelated.
1, 2, 3
T1 : R12. 23 2
r12 2 if r = 0
r13
y1 y 2 y 3 23
e T1 and
6 T : If R = 0 then r =r =0
3y1 2y 2 y3 . Which of the 2 1.23 13 12
T2
10 Which of the following statements is
following statements is true ?
correct ?
(A) E(T ) = E(T ) = and V(T ) < V(T )
1 2 1 2 (A) Only T is true
2
(B) E(T ) = , E(T ) and V(T ) < V(T ) (B) T and T both are true
1 2 1 2
1 2
(C) E(T ) E(T ) and V(T ) (C) Both T and T are false
1 2 1 2 1 2
(D) E(T ) = , E(T ) and V(T ) < V(T ) (D) Only T is true
2 1 2 1 1
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12. Let X follows U(0, 1) uniform distribution. 15. The probability of accepting a lot of
quality LTPD is called
1
Then P X2
2 (A) 1 – producer’s risk
1
(A) (B) producer’s risk
3
1 (C) 1 – consumer’s risk
(B)
4
2 (D) consumer’s risk
(C) 3
1
(D) 16. Let random variable X follows
6
1
exponential distribution with mean 0.
13. If l = 25000 and l = 20000, then the Then the median of the distribution is
30 31
probability that a person of age 30 years
will survive till his next birthday is
(A) n2
3 2
(A)
5
1 (B) ln2
(B)
5
4
(C) 5 (C) e 2
2
(D)
5 1
(D) n2
14. Suppose {Y , Y , Y , Y } be a
13 8 2 7
SRSWOR from the population of size
101 with variance 10. Then covariance 17. In a life table generally the value of
between Y and Y will be radix (l ) is taken to be
8 2
0
1
(A)
10 (A) 10000
6
(B)
101 (B) 100
2
(C)
101 (C) 100000
1
(D) (D) 1000
10
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18. If the annual trend equation is 21. If f (y) is the probability density function
y
of random variable y, then probability
y 112.8 43.2 X , then the monthly
density function of |y| will be
trend equation will be
(A) y 112.8 3.6X (A) 2f (–y)
y
(B) y 9.4 3.6X (B) f (y) + f (–y)
|y| |y|
(C) y 9.4 0.3X
(C) f (y) + f (–y)
y y
(D) y 112.8 0.3X
(D) 2f (y)
y
19. Let x , x , ....., x be a random sample
1 2 n
–( H)
from the distribution f(x, ) = 2 x , 22. Suppose that a population consists of a
x > 2, > 2. The CRLB of the variance of linear trend Y = a + ib, for i = 1, 2, . . . . N,
i
where a and b are constants. If variance of
an unbiased estimate of ln is
the sample mean of n observations drawn
(A) n
2
(B) sampling and simple random sampling
from the given population are respectively
1
(C) – – –
n denoted by V(Y ), V(Y ) and V(Y ),
st sy ran
– – –
1 then V(Y ) : V(Y ) : V(Y ) equals to
(D) st sy ran
2
(A) 1
:n:1
20. The relation between NRR and GRR is n
1
(A) NRR GRR (B) n : :1
n
(B) NRR GRR
1
(C) : 1: n
n
(C) NRR = GRR
(D) None of the above 1
(D) n : 1:
n
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23. In usual notations of life table, if 26. In SPRT, an approximate formula to
l = 750000 and l = 740000, then q calculate ASN is
20 21 20
will be
2 L( )log B (1 L( ))log A
(A)
(A) 75 E(Z)
7 L( )log A (1 L( ))log B
(B) (B)
75 E(Z)
1
(C)
75 L( )(1 log B) (1 L( ))log A
(C)
4 E(Z)
(D)
75
L( )(1 log A ) (1 L( ))log B
E(Z)
24. The probability density function
of a random variable X is Let {1, 0, 1, 1, 0, 0} be a random sample
3 from Bernoulli distribution with
f(x, ) , x 3, 0 . Then the
x x mean 0< < 1. Then UMVUE of
most powerful critical region of size 0.027 (1 – ) is
to test H : = 3 versus H : = 2 is
0 1
(A) 0.60
(A) x > 10
(B) x > 8 (B) 0.96
(C) x > 11 (C) 0.50
(D) x > 9
(D) 0.70
25. The relation between the central
mortality rate (m ) and force of mortality 28. How many parameters of a bivariate
x
( ) is
x
x x 1 simple hypothesis ?
(A) mx
2
(A) 3
(B) mx 1
x
2
(B) 5
1
(C) mx x
2 (C) 2
x 1
(D) mx (D) 4
2
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29. If two regression lines are X + 2Y – 5 = 0 32. Purchasing power of money equal to
and 2X + 3Y – 8 = 0, then for X = 1, the
estimate of Y will be
1
(A) 1 (A)
Fisher’s index number
number
(B) 2
1
(C) –1 (B) Money wages
(D) –2
1
(C) Cost of living index number
30. For the fundamental EOQ problem with
no shortages, D = demand per unit time,
1
C = set up cost and C = holding cost, then (D) Wholesale price index number
s 1
optimum lot size due to R.H. Willson is
(A) 2DC1 / Cs
33. n of the two variates
(B) 2DCs / C1
X and Y are equal, then ratio estimator of
(C) 2C1Cs / D
the mean of an equivalent simple random
(D) 2Cs / (DC1)
sample drawn without replacement, when
equals to
31. Let random variable X has probability XY
–x
distribution P(x) = Ce , x = 1, 2, 3, . . .
(A) 0.5
Then the value of the constant C is
(A) e – 1 (B) 0.3
(B) 1
(C) 0.6
1
(C)
e
(D) 0.4
(D) e
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34. Let x , x , . . ., x be a random 36. For M/M/1 queuing model with = 30
1 2 n
1
sample from gamma distribution customers per day (24-hours) and
x 36
1
f(x; , ) e x 1; x 0, 0 customers per minute, then the expected
number of customers in the system will be
is unknown. The most powerful critical (A) 3
region for testing H : = 3 versus (B) 5
0
H : = 2 will be (C) 1
1
n (D) 4
(A) xi C
i 1
37. If X and X – Y are uncorrelated random
n
xi C variables, then V(X – Y) equals to
(B)
i 1
(A) V(X) + V(Y)
n
(C) xi C (B) V(Y) – V(X)
i 1
(C) V(X) + V(Y) – Cov (X, Y)
n
(D) xi C
(D) V(X) – V(Y)
i 1
38. If interaction AB is compounded in a
3
35. For k-variate general linear 2 factorial experiment, the entries of two
blocks in a replicate will be
model y x u , where
2 abc ab ac c
yn 1, xn k , k 1
and un 1 u ~ N(0, In ) . (A)
(1) a b bc
Then the distribution of is
b ac bc a
(B)
(A) N ( , ( x1x) 1) (1) ab c abc
(B) N ( , 2
( x1x) 1) (1) abc a c
(C)
ab bc ac b
2
(C) N ( , In )
(1) ab a b
(D)
(D) None of the above abc c bc ac
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39. Below given is a transportation problem 42. If a.P(A B) = 3.25, P(B) = P(A) = 0.25
model. 4
and P(A B) , then the value of
Supply : a = 9, a = 6, a = 15, a = 15 13
1 2 3 4 constant ‘a’ is
Demand : b = 9, b = 14, b = 10, b = 5,
1 2 3 4
b =4 (A) 12
5
To balance it, we need to add
(B) 10
(A) both a dummy origin and dummy
destination (C) 13
(B) a dummy origin
(D) 11
(C) neither dummy origin nor dummy
destination
(D) a dummy destination
43. In replacement problem if n and n
1 2
are optimum replacement age for
40. In a test of randomness, if U denotes the equipments A and B respectively and
total number of runs in n observations, W(n) is weighted average cost for each
then expected number of runs will be
equipment at age n, then purchase
n 2 equipment B in place of equipment A, if
(A)
2
(A) W(n ) = 2W(n )
n 2 1
(B)
2 (B) W(n ) < W(n )
1 2
n 1
(C) (C) W(n ) > W(n )
2 1 2
n 1 (D) W(n ) = W(n )
1 2
(D)
2
44. used when
41. If the index number for 1990 at the base
1980 is 250, the index number for 1980 (A) Xmax Xmin 6
at the base 1990 will be
6
(B) Xmax Xmin
(A) 150 n
(B) 4 (C) Xmax Xmin 3
(C) 400
6
(D) Xmax Xmin
(D) 40 n
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3
45. In a 2 factorial experiment with 48.
10 replicates, the error degrees of 50 items is 0.60. Then the reliability of a
test having 125 comparable items will be
freedom is
(A) 0.89
(A) 63
(B) 0.69
(B) 19 (C) 0.99
(C) 79 (D) 0.79
(D) 59
49. Consider the problem of testing
H : f (x) = f (x) versus H : f (x) = f (x)
0 X 0 1 X 1
based on a single observation, where
46. If two regression lines are X + 2Y – 5 = 0 x: 0 1 2 3 4
and 2X + 3Y – 8 = 0, then mean of X and f (x) : 0.1 0.2 0.1 0.4 0.2
0
f (x) : 0.2 0.2 0.2 0.2 0.2
Y are respectively 1
If the test function (x) is given as
(A) 5 and 8 (0) = (1) = (2) = 1 and (3) = (4) = 0,
then probability of type II error will be
(B) 1 and 2 (A) 0.5
(B) 0.3
(C) 8 and 5
(C) 0.6
(D) 2 and 1 (D) 0.4
50. In R-chart, estimate of unknown process
47. Let {15, 4, 6, 5, 10} be a random sample
standard deviation is
–(x – )
from the distribution f(x, ) = e ,
, (– 3], is unknown MLE of R
(A) D1
is
R
(A) 6 (B) d1
(B) 3 R
(C) D2
(C) 7
R
(D) d2
(D) 4
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51. Below is the layout of Direction (Q. No. 54 to 56) :
A C A B Read the following data and answer the questions
D D B C given below it :
C B C D 25 families live in 4 blocks A, B, C and D and the
B A D A number of individuals per block is shown below.
(A) Randomised block design Block : A B C D
(B) Latin square design
Number of
(C) Split plot design individuals : 20 50 20 10
(D) None of the above
54. Average number of individuals per family is
52. The cumulative distribution function of a
random variable X is (A) 5
0, if x 0
(B) 3
1 1 2
F(x) (4 x x ), if 0 x 1
4 6 (C) 6
1, if x 1
(D) 4
1
(A) 55. Average number of individuals per block is
4
1 (A) 15
(B)
6
(B) 20
1
(C)
3 (C) 23
1
(D) (D) 25
5
53. Let {10, 5, 6.6, 8, 4.7, 4} be a random 56. Average number of families per block is
sample from uniform U( – 5, + 5)
(A) 5.25
distribution. Then MLE of is
(A) 7 (B) 3.25
(B) 2
(C) 6.25
(C) 10
(D) 4.25
(D) 4
02/GO/CC/M-2025 – 27/I -20-
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57. Let x , x , . . ., x be a random sample The characteristic function of uniform
1 2 n
from Bernoulli distribution with mean , distribution over (–a, a) is
0<
(A) ta.sin(ta)
C, if x1 x 2 1
T(x1, x 2 )
0, otherwise
(B) sin(ta)
(1 )
Then T becomes unbiased for , if ta
n
C equals to
(C) ta.cos(ta)
(A) 2n
cos( ta)
(B) 2 (D)
ta
1
(C)
2n
1 60. Let 3 components be connected as
(D)
2 depicted in the following system.
c1
58. Let _y = (y , y , y ) is N ( , ), where c3
1 2 3 3
4 0 2 c2
3
1 and 0 1 1 then which
2 2 1 3 Each of the component has independent
one of the following statements is false ? and identically distributed exponential life
time model with mean 1. Then the reliability
(A) y – 2y + y ~ N(3, 29)
1 2 3
function of the given system will be
(B) y and y are independent –x –2x
1 2
(A) e (1 – e )
1 –3x
(C) y1y3 (B) e
3
–x –x 2
(C) e (1 – (1 – e ) )
y1 3 4 2
(D) ~ N2 ,
y 2 2 3 –x 2
3 (D) (1 – e )
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61. For uniformly bounded random variables, 64. Consider the following two transition
probability matrices
n
V Xi 1 1
i 1 0 0
the condition lim 0 to hold 2 2
2 0 1 0
n n 1 1
0 0 1 1
weak law of large numbers is P1 2 2 and P2 0
2 2
1 1 1 1
0 1 0
4 4 4 4
0 0 0 1
Then Markov chain of
(A) P and P both are reducible
1 2
62. Average sample number of double
(B) P is reducible but P is irreducible
sampling plan is given by 1 2
(A) n + n (P ) (C) P and P both are irreducible
1 2 2 1 2
(B) n + n (P )
1 2 1 (D) P is reducible but P is irreducible
2 1
(C) n + n (1 – P )
1 2 1
(D) n + n (1 – P )
1 2 2
65. If X is a standard normal variate, then
63. In SPRT, let B and A (B < A) are
stopping bounds. The null hypothesis is
point x, when
rejected and the process will terminate at
th
m stage when (A) x > 1
(A) m B
1 (B) x = 1
1
(B) m A
(C) x = 0
1
(C) m B
(D) x < 0
(D) m A
1
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66. Consider the following linear 68. Let _
X = (x , x , x ) be distributed
1 2 3
programming problem as trivariate normal distribution with
parameters _ = (2, 1, 2) and
maximize z = 3x + 7x
1 2
2 1 1
subject to x + x
1 2
3 1 3 0 . Then Y
_ = (x + x + x , x –x )
1 2 3 1 2
1 0 1
x + 2x 7
1 2
will be N(_, ), where _ and are
x > 0, x > 0 respectively
1 2
Then an optimal solution is given by 10 1
(A) (1, 5) and
1 3
(A) (3, 0)
6 1
(B) (0, 3.5) (B) (5, 1) and
1 5
(C) (0, 3)
10 0
(C) (5, 1) and
(D) (7, 0) 0 3
6 0
(D) (1, 5) and
0 5
67. Let {1, 1, 0, 0, 3} be a random sample
from exponential distribution with 69. Consider the following statements :
1 T
mean 0 . MLE of the median of the 1
increases as the mean square
distribution is within the cluster increases.
T : In usual notations, if n = N,
2
1 two stage sampling reduces to
(A) log 2
5
sampling.
1 Choose the correct answer from the
(B) log 2
2 following :
(A) Only T is true
1 2
(C) 5 log
2 (B) Both T and T are true
1 2
(C) Both T and T are false
(D) log2 1 2
(D) Only T is true
1
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70. Suppose E and F are events with 73. Let simultaneous equation model be
P(E) = 0.25 and P(F) = 0.80. Then which Y = + Y + X +u
1 10 12 2 13 1 1
of the following statements is correct ?
Y = + Y + X +u
2 20 21 1 21 1 2
(A) P(E Here Y and X are
1 1
(A) both endogenous variables
F)
(B) endogenous variable and exogenous
variable respectively
(C) both exogenous variables
(D) exogenous variable and endogenous
variable respectively
71. If V ,V and V denote the
rand prop opt
variances of the sample mean under 74. Let X and X be independent random
1 2
SRSWOR, proportional and optimum variables, each having probability
x
distribution p(x) = pq , q = 1 – p, 0 < p < 1,
x = 0, 1, 2, . . . The distribution of
respectively, then correct relation is
(X |X + X = 5) is
1 1 2
(A) V V V
prop opt rand (A) negative binomial distribution with
parameters 5 and q
(B) V V V
opt prop rand
(B) discrete uniform distribution over
(C) V V V 0, 1, 2, .... , 5
prop rand opt
(D) V V V (C) binomial distribution with parameters
opt rand prop 5 and q
(D) negative binomial distribution with
72. Let x , x , . . ., x be a random sample parameters 5 and p
1 2 n
2
from N( , ne
n
( xi x)2 75. Average amount of total inspection in a
i 1 2
Sn2 , Sn2 is single sampling plan is given by
n
(A) n + (N – n) (1 – L(p))
(A) Consistent and unbiased both
(B) Consistent only (B) n + (N – 1) (1 – L(p))
(C) Neither consistent nor unbiased (C) n + N – n(L(p))
(D) Unbiased only (D) N – n + N(1 – L(p))
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76. In simple random sampling without 79. The following observations were recorded
from two populations with distribution
replacement of size N, the probability of
functions F (x) and F (x) population I : {3, 7},
th 1 2
selecting a particular unit at r draw will be
population II : {2, 5, 8, 9}. The Mann-Whitney
1 test statistic for testing H : F (x) = F (x)
0 1 2
(A) (N r ) will be
r (A) 6
(B)
N (B) 3
1 (C) 8
(C)
(N r 1)
(D) 4
1
(D)
N
80. If L(p) and P(q) represent respectively
77. X is a Poisson random variable with Laspeyre’s price index number and
mean 2.5. Which of the following Paasche’s quantity index number, then
statements is false ?
(A) L(p).P(p) = L(q).P(q)
(B) L(p).L(q) = P(p).P(q)
2
(B) E(X ) = 8.75
(C) Both (A) and (B) are true
P(X 1)
(C) 2.5 (D) Both (A) and (B) are false
P(X 0)
1
2.5
81. Consider the following data :
78. Consider the following linear
programming problem X : –3 –2 –1 1 2 3
maximize z = x + x Y: 9 4 1 1 4 9
1 2
subject to x + x 1
1 2
–3x + x 3
1 2
and Y will be
x 0, x
1 2
The given problem has (A) 0.5
(A) Infeasible solution (B) 1
(B) Unique solution
(C) 0
(C) Exactly two solutions
(D) –1
(D) Unbounded solution
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82. Total number of possible Latin squares 85. In an examination with mean = 60 and
of order 4 × 4 is standard deviation = 10 marks, if a score
of a student is 55, then his standard
(A) 144
score in a distribution with mean = 100
(B) 4 and standard deviation = 20 will be
(C) 576 (A) 80
(D) 64 (B) 60
(C) 90
83. ne segments of length (D) 70
1, 2, 4, 7 and 10 units. Three of them are
chosen at random. The probability that
86. Let x , x , . . ., x be a random sample
1 2 n
a triangle can be formed by using these 2
from N( , ) normal distribution. Then
three chosen line segments is
respect to sample mean is
1
(A)
10
(A)
2
1
(B)
2
(B)
6
3
(C) 2
10 (C)
1
(D) 6
5 (D)
–x(y + 1) 87. Which of the following relations, given
84. Given f(x, y) = x e , x > 0, y > 0.
in standard notations for M/M/1 model is
The regression curve of y on x will be not true ?
(A) x – y = 1 (A) L = .W
q q
(B) x + y = 1 (B) L = .W
s s
(C) xy = 1 (C) L s Lq
1
(D) x – y = 0 (D) Ws Wq
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88. Let X ~N (0, 1) and Y ~ N (0, 1) be 92. The two regressio
independent random variables. Then the variables X and Y are b = – 0.9 and
yx
X b = – 0.4, then r equals to
xy xy
distribution of is
Y (A) – 0.36
(A) Beta type – I (1, 1) distribution
1
(B) 0.36
(B) Standard Laplace distribution
(C) Beta type – II (1, 1) distribution (C) – 0.6
2
(D) Standard Cauchy distribution (D) 0.6
89. Which of the following relation is false ? 93. Consider a linear programming problem
with constraints 3x + 5x 17 and
(A) Tx 1 Tx Lx 1 3
x + 2x 8.
1 2
dx We rewrite the above constraints as
(B) qx
lx 3x + 5x + y 17 and x + 2x – z 8.
1 3 1 2
1 Then which of the following statements is
(C) e x e0x
2 true ?
lx 1 (A) Both y and z are called slack
(D) px variables
lx
(B) y is called slack variable and z is
90. Let X follows U(0, ), > 0 uniform called surplus variable
distribution. To test H : = 1 against
0 (C) Both y and z are called surplus
H : = 2, the critical region is given as variables
1
(D) y is called surplus variable and z is
(A) 0.10 called slack variable
(B) 0
(C) 0.25
94. For a moderately asymmetrical
(D) 0.05
distribution, which of the following
relationship is true ?
91. In T-score given by T = + where
1
is normalised score, then the value of (A) Mean – Median = (Mean – Mode)
3
( , ) is
1
(A) (50, 10) (B) Mean – Mode = (Mean – Median)
3
(B) (0, 1) (C) Median = 3(Mode) – 2(Mean)
(C) (5, 1) 2
(D) Median – Mode = (Mean – Median)
(D) (100, 10) 3
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95. If process mean and process standard 98. Let {2.5, 1.5, 0.5, –1.0} be a
deviation and random sample from the distribution
1
1 |x 2| 3 (x )2
respectively, then control limits for f(x, ) e e 2 ,
– 8 4 2
X-chart will be
x R, R, is unknown. The moment
(A) A estimate of belongs to the interval
1
(B) A (A) (0.25, 0.45)
2
(C) A (B) (0.48, 0.52)
(D) None of these (C) (0.45, 0.49)
(D) (0.60, 0.72)
96. To test whether the observations have
uniform U(0, 1) distribution or not, if the 99. The maximum limit of percentage
observations are 0.85 and 0.30, then
which of the following is the value of the called
Kolmogorov-Smirnov statistic ?
(A) Average outgoing quality limit
(A) 0.30
(B) Lot tolerance percentage defective
(B) 0.15
(C) Acceptance quality level
(C) 0.85
(D) None of the above
(D) 0.20
100. The number of persons living above the
20
97. If the demand curve is x , then the age 10 is 5000000 and the number of
p 1
price elasticity of demand at p = 3 will be persons living at the age 10 is 75000,
then expectation of life at the age 10 is
3 250
(A) (A)
4 3
5
(B) 100
4 (B)
3
4 275
(C) (C)
5 3
3 200
(D) (D)
5 3
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Source: Bihar Public Service Commission. Question paper published by Bihar Public Service Commission.