Kerala SET 2019 Feb Question Paper Statistics

A
19231 120 MINUTES

1. lim →∞ 1 + , where is a finite number is equal to
A) B) C) 0 D) 1

2. Which of the following statement is not true?
A) A function which is uniformly continuous on a set is also continuous on
that set.
B) If a function is uniformly continuous on a bounded set, then the function
is bounded on the set.
C) The product of any two uniformly continuous functions is uniformly
continuous.
D) The function ( ) = , ∈ is not uniformly continuous on .
∞
3. ∫ =
A) 0 B) 1 C) D) does not exist

4. Which of the following statements are true?
1. The union of any collection of open sets is an open set.
2. The intersection of any collection of open sets is an open set.
3. The union of any collection of closed sets is a closed set.
4. The intersection of any collection of closed set is a closed set.
A) 1 and 3 only B) 1 and 4 only C) 2 and 3 only D) 2 and 4 only

5. Let and be two subspaces of ℜ . Then which of the following statement(s)
is/are true.
1. ∪ is always a subspace of ℜ .
2. ∩ is always a subspace of ℜ .
3. The sum + is the smallest subspace of ℜ that contains ∪ .
A) 1 only B) 2 only C) 1 and 3 only D) 2 and 3 only

6. Let be a vector space of dimension , and and be two subspaces of with
dimensions and respectively such that < . Then the maximum dimension
of ∩ is
A) B) C) D) + −

7. Assume that the sum of two idempotent matrices is again idempotent. Then product of
the matrices is
A) a non-zero idempotent matrix B) a zero matrix
C) an identity matrix D) none of these

8. If and are square matrices of order , then
A) ( )≥ ( )+ ( )−
B) ( )≤ ( )+ ( )−
C) ( )= ( )+ ( )−
D) None of the above

9. Let A + be the Moore-Penrose -inverse of a matrix A. Which of the following
statements are true:
1. A+ is unique 2. ( ) = 3. ( ) = ( )
A) 1 and 2 only B) 1 and 3 only
C) 2 and 3 only D) 1, 2 and 3

10. Which of the following statement is not true?
A) Similar matrices have identical characteristic polynomials
B) If and are × matrices then characteristic polynomial
of and are same
C) Characteristic polynomial of a matrix and its transpose are same.
D) Product of characteristic roots of a matrix is equal to its trace.

11. The matrix is a positive definite matrix. Then
A) | |>0
B) is positive definite
C) All the eigen values of are positive
D) All the above

12. Let { } be a nondecreasing sequence of sets, then lim is:
A) ⋃∞ B) ⋂∞ C) ∅ D) Does not exist

13. Let and be Lebesgue measurable sets, then which of the following
statement(s) is/are true?
1. ∪ is Lebesgue measurable
2. ∩ is Lebesgue meaurable
A) 1 only B) 2 only
C) Both 1 and 2 D) Neither 1 nor 2

14. Let , and be three events such that ( )= ( )= ( )= , ( ∩ )=
( ∩ ) = 0 and ( ∩ ) = . Then probability that at least one of the events
, and occurs is
A) B) C) D)

15. Let , and be three mutually exclusive and exhaustive events such that
( )= ( ) and ( ) = ( ). Then ( ) =
A) B) C) D)

16. The odds against an event is 4 ∶ 3 and odds in favour of another event is 7 ∶ 4, and
are independent. Then the probability that neither nor occurs is
A) B) C) D)
77 77 77

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17. The probability that a 3-card hand drawn at random and without replacement from
an ordinary deck consists entirely black cards is
A) B) C) D)
17 17

18. Among three urns, the first urn contains 7 white and 10 black balls, the second
contains 5 white and 12 black balls, and the third contains 17 white balls and no black
ball. An urn is chosen and a ball is drawn from the selected urn. It is found that the
ball is white. Then probability that the ball came from the second urn is
A) B) C) D)

19. Which of the following is not a probability density function?

( )= | |
A) , −∞ < <∞

( )
( )= , > , >0
B)
0, ℎ

C) ( )= , −∞ < <∞
( )

(2 − ), 0 < <2
D) ( )=
0, ℎ

0, <0
⎧ ( + 1)
20. Consider the function ( )= 2, 0 ≤ < 1. Then ( ) is
⎨
⎩ 1, 1≤

A) Distribution function of a discrete random variable
B) Distribution function of a continuous random variable
C) Distribution function of a mixed type random variable
D) Not a distribution function

21. Which measure is the most unreliable indicator of central tendency if the distribution is
skewed?
A) Mean B) Median C) Mode D) Range

0, ≤0
22. Let be a random variable with distribution function ( ) = .
1− , >0
Median of the distribution is
A) 0 B) C) 2log D)

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23. Which of the following statements are true for the variance of a random variable ?
1. > 0 for all non-degenerate random variables
2. If the distribution of is concentrated near ( ) then will be small.
3. A small value of means the probability is small that will deviate much
from its mean.
4. Large value of means the probability is large that will be far from the
mean.
A) 1and 2 only B) 1, 2 and 3 only
C) 1, 2 and 4 only D) 1, 2, 3 and 4

24. Let be integer valued random variable with probability generating function ( ),
| | ≤ 1. Then the second factorial moment of is given by
′ ′′
A) (1) B) (1)
′′ ( )
C) 1 − [ ′ (1)] D) ′′ (
1) − [ ′ (1)] + ′
(1)

25. Which of the following statement is false?
A) Two random variables and have the same moment generating function,
then and must have the same distribution.
B) If moment generating function of a random variable exist, then moments of
all orders exist.
C) If moments of all orders exist, then moment generating function exist in some
open neighborhood of zero.
D) All the above statements are true.

26. Let { } be a sequence of random variables for which ( ) < ∞ and let
= ∑ , ≥ 1. A necessary and sufficient condition for the sequence { }
to satisfy weak law of large numbers is

A) → 0, →∞ B) → 1, →∞

C) → 0, →∞ D) → 1, →∞

27. Let , , … be iid random variables with mean 0 and variance 1. Let Φ( ) denote the
cumulative distribution function of a standard normal random variable. Then for any
> 0, lim →∞ (|∑ | < ) equals

A) Φ( ) B) 1 − 2Φ( ) C) 2Φ( ) − 1 D) 1

28. Binomial distribution is positively skewed if the probability of success is

A) less than B) greater than C) equal to D) equal to 1

29. The mean of zero-truncated Poisson( ) variable is

A) B) C) D)

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30. Which of the following distribution possesses lack of memory property?
A) Poisson B) Binomial
C) Hypergeometric D) Geometric

31. The mean and variance of negative binomial distribution are related as:
A) mean is greater than variance B) mean is less than variance
C) mean is equal to variance D) None of the above

32. Suppose that the amount of time a customer spends at a fast food restaurant has an
exponential distribution with mean of six minutes. Then the probability that the customer
will spend more than 12 minutes in the restaurant given that she has been there for more
than 6 minutes is
A) B) C) D)

33. Which of the following statements are true?
1. If X and Y are iid Cauchy(1, 0) random variables,. Then X + Y follows Cauchy(2, 0)
distribution
2. If X and have the same distribution, then is ℎ (1, 0) random variable.
3. Let be a uniform random variable over − , . Then = tan is a Cauchy random
variable.
A) 1 and 2 only B) 1 and 3 only
C) 2 and 3 only D) 1 , 2 and 3

, > 0, , >0
34. Let has Pareto distribution with pdf ( ) = ( ) . Then
0, otherwise
( ) exists only when

A) >1 B) <1 C) >1 D) <1

35. Let the random variable has lognormal distribution with parameters and . Then
moment generating function of is

A) exp + B) +

C) log + D) does not exist

36. Let and be independent random variables with pdfs and , respectively. Then the
pdf of = is
∞ ∞
A) ( ) = ∫ ∞ ( ) ( )| | B) ( )=∫∞ ( ) ( )
| |

∞ ∞
C) ( )=∫∞ ( ) | | D) ( )=∫∞ ( ) | |

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37. Let , , … , be a random sample taken from discrete uniform distribution with pmf
( ) = , = 1, 2, … , , ≥ 1 . Then the pmf of the order statistic is
( )
A) ( )= , = 1, 2, … ,

( )
B) ( )= , = 1, 2, … ,

( ) ( )
C) ( )= , = 1, 2, … ,

( ) ( )
D) ( )= , = 1, 2, … ,

38. Consider a system of identical batteries operating independently in a system,
and the batteries operate in series. Suppose the length of life of the batteries have
the common distribution function ( ) and pdf ( ). Then the pdf of length of
life of the system has the form
A) [1 − ( )] ( ) B) [ ( )] ( )

C) {1 − [1 − ( )] } ( ) D) None of the above

39. Let : , : , : be the order statistics corresponding to the random sample , ,
taken from a population with pdf ( ) = , > 0, > 0 . Define = 3 : ,
= 2( : − : ) and = ( : − : ). Then which of the following statements are
true?
1. ( , , ) and ( , , ) are identically distributed.
2. , and are independent.
3. , = 1, 2, 3 has the pdf ( ).

A) 1 and 2 only B) 1 and 3 only C) 2 and 3 only D) 1, 2 and 3

40. The mean of a non-central chi-square random variable with degrees of freedom and
noncentrality parameter is
A) B) + C) +2 D) 2 +

41. Which of the following statement is false for t distribution with degrees of freedom?
A) Mean of the distribution is zero for all ≥ 1.
B) Pdf of the distribution is symmetric about zero.
C) Pdf of the distribution can be approximated by a standard normal density
for large .
D) For small , t-distribution assigns more probability to its tails compared
with standard normal distribution.

42. If ~ ℎ (1, 0) then ~
A) ℎ (1, 0) B) (1) C) (2) D) (1, 1)

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43. Let ( , ) be bivariate random vector. Then the estimate ( ) of based on ,
which minimizes the MSE ( − ( )) is
A) ( ) B) ( ) C) ( | ) D) ( | )

44. Let ( , )~ , , , , . Then ( | ) =

A) B) (1 − ) C) (1 − ) D) ( − 1)

45. Let , , … . , be a random sample from a continuous population with
distribution function (. ). Define
Number of ′ ≤
( )= , ∈
Then
A) ( ) is unbiased but not consistent for ( )
B) ( ) is consistent but not unbiased for ( )
C) ( ) is unbiased and consistent for ( )
D) ( ) is neither unbiased nor consistent for ( )

46. Let be an unbiased estimator of the parameter of a family of distribution { , ∈ Θ}
such that ( ) < ∞, and be a sufficient statistic for { , ∈ Θ} . Also let =
( | ). Then
A) is unbiased estimator with ( ) ≥ ( )
B) is not unbiased estimator but ( ) ≥ ( )
C) is unbiased estimator with ( ) ≤ ( )
D) is the UMVUE

47. Let , , … . , be a random sample from the Poisson distribution with mean .
Then the Cramer-Rao lower bound of unbiased estimator of is
A) B) C) D)

48. The MLE of  based on a random sample , ,…., taken from the PDF
2
f ( x )   x I ( ,  ) ( x ),   0 ,
where (. ) is the indicator function defined on the set A is
∑ ∏
A) B) C) min D) max

49. State whether the following statements are true (T) or false (F)
1. Maximum likelihood estimate is always unique.
2. Maximum likelihood estimate is unbiased if it is unique.
3. Maximum likelihood estimate itself is a sufficient statistic.
4. Asymptotic distribution of maximum likelihood estimate is always normal.
A) 2 & 3 true, 1 & 4 false B) 1 & 2 True, 3 & 4 false
C) All the statements are true D) All the statements are false

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50. Let , ,…., be a random sample taken from population with pdf
( )= , 0 < < 1, > 0.
0, ℎ
∑
If = then the moment estimator of is

A) B) C) D)

51. An urn contains 10 balls, of which are red and 10 − are black. To test : = 5
against the alternative : = 6, one draws 3 balls from the urn without replacement.
The null hypothesis is rejected if the sample contains 2 or 3 red balls; otherwise it is
accepted. Then the power of the test is

A) B) C) D)

52. Which of the following distribution does not have monotone likelihood ratio
property?
A) Uniform over [0, ] B) Poisson
C) Binomial D) Cauchy (1, )

53. The critical region of the UMP test for testing : ≥ against : < , based
on random sample , , … . , taken from the pdf ( ) = ( )! , > 0,
> 0 has the form

A) ∑ > B) ∑ <
C) ∏ > D) ∏ <

54. Which test is the nonparametric analogue of the analysis of variance test for the
two way classification?
A) Kruskal-Wallis test B) Friedman test
C) Shapiro-Wilk test D) Freund-Ansari test

55. The decision in a sequential probability ratio test depends on
A) (type I error)
B) (type II error)
C) (type I error) and (type II error)
D) None of the above probabilities

56. Let denote the region of acceptance of an -level UMP test of : = against
: ≠ . For each observation,
= ( , , … , ) , let ( ) is the set ( ) = { : ∈ }. Then

A) ( ) is UMA confidence interval with confidence level 1 − .
B) ( ) is confidence interval with confidence level 1 − but not UMA
C) ( ) is confidence interval with a different confidence level
D) ( ) is not a confidence interval

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57. Choose the conjugate priors for the distributions in List I from List II and select
the correct answer using the codes given below.
List I List II
a. Poisson distribution with unknown mean 1. Normal distribution
b. Binomial distribution with unknown 2. Gamma distribution
probability of success
c. Uniform over [0,θ], θ>0 is unknown 3. Beta distribution
d. Normal distribution with unknown mean 4. Pareto distribution
and known variance

A) a-2, b-3, c-4, d-1 B) a-4, b-2, c-3, d-1
C) a-4, b-1, c-3, d-2 D) a-2, b-4, c-3, d-1

58. In a hypothesis testing about a population mean, the p-value is found to be 0.04. Assume
that the population mean given the null hypothesis is . Which of the following is/are
true about the population mean?
A) The 95% confidence interval includes
B) The 99% confidence interval includes
C) The 90% confidence interval includes
D) All the above are true

59. Suppose a finite population contains 7 items and 3 items are selected at random without
replacement, then the number of all possible samples will be
A) 21 B) 35 C) 14 D) 7

60. Consider a population of units, in which the proportion of units possessing a
given characteristic is . A random sample of size is taken from the population
using simple random sampling without replacement. Then the variance of the
unbiased estimate of is equal to

( ) ( ) ( ) ( )
A) B)
( )

( ) ( ) ( )
C) D)
( )

61. The manager of the customer service division of a major consumer electronics company
is interested in determining whether the customers who have purchased a LED television
made by the company over the past 12 months are satisfied with their products. If there
are 5 different types of LED televisions made by the company, the best strategy would be
to use a

A) simple random sample
B) stratified random sample
C) cluster sample
D) systematic sample

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62. Which of the following statement(s) is/are true for Horvitz-Thomson estimator?
1. Horvitz-Thomson estimator is the best estimator in the class of linear unbiased
estimators of the population mean.
2. Horvitz- Thomson estimator is the UMV in the class of all unbiased estimators

A) 1 only B) 2 only
C) Both 1 and 2 D) Neither 1 nor 2

63. The ratio estimator of population mean is unbiased if
A) The variable under study and the auxiliary variable have a positive correlation
B) The variable under study and the auxiliary variable have a negative correlation
C) The variable under study and the auxiliary variable are linearly related
D) None of the above

64. Which of the following statement is/are true for a connected design?
1. All treatment contrasts of a connected design are estimable.
2. The rank of the C-matrix of a connected incomplete block design with treatments
is − 1.
3. Latin square designs are connected.

A) 1 and 2 only B) 1 and 3 only
C) 2 and 3 only D) 1, 2 and 3

65. Assuming no bias, the total variation in a response variable is due to error (unexplained
variation) plus differences due to treatments (known variation). If known variation is
large compared to unexplained variation, which of the following conclusions is the best

A) There is no evidence for a difference in response due to treatments.
B) There is evidence for a difference in response due to treatments.
C) There is significant evidence for a difference in response due to treatments
D) The treatments are not comparable.

66. In × Latin square design, the degrees of freedom of the error sum of square is
equal to
A) ( − 1) B) ( − 1) C) ( − 2)( − 1) D) ( − 2)

67. Which of the following statements is/are true for symmetric BIBD?
1. Number of treatment is equal to the number of blocks.
2. Number of replications of each treatment is equal to the block size.
3. The number of treatments common in any two blocks is equal to the number of
blocks in which each pair of treatments occurs together.

A) 1 only B) 1 and 2 only
C) 1 and 3 only D) 1, 2 and 3

68. Suppose a 2 design confounded in 8 blocks by selecting 3 effects to generate the blocks.
Then the number of effects that are confounded in addition to the selected effects is equal to
A) 3 B) 4 C) 5 D) 6

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69. The Gauss-Markov theorem will not hold if
A) the error term has the same variance given any values of the independent variables
B) the error term has an expected value of zero given any values of the independent
variables
C) the independent variables have exact linear relationships among them
D) the regression model relies on the method of random sampling for collection of data

70. If is distributed as ( , ), then ( − )′ ( − ) has the distribution
A) distribution with ( − 1) degrees of freedom
B) distribution with degrees of freedom
C) Wishart distribut ion wit h degrees of freedom
D) ( , )
4 1 0
71. Let = ( , , )′ be distributed as ( , ) with = 1 3 0 . Which of the
0 0 2
following statement is/are true?

1. and are independent. 2. ( , ) and are independent.

A) 1 only B) 2 only
C) Both 1 and 2 D) Neither 1 nor 2

72. Let , , … . , be random sample from ( , ) with = ∑ , and is known.
Then the test statistic to test : = against : ≠ is

A) ( − )′ ( − ) B) ( − 1)( − )′ ( − )

C) ( − )′ ( − ) D) ( − )′ ( − )

73. The multiple correlation coefficients
A) can vary within the range from −1 to+1
B) can vary within the range from 0 to+1
C) can be any nonnegative value
D) cannot be zero

74. The state space of a stochastic process is
A) always discrete B) always continuous
C) may be continuous or discrete D) neither discrete nor continuous

75. Consider a Markov chain with transition probability matrix
0 1 0
1 1
= 2 0 2 .
0 1 0
Then

A) The Markov chain is irreducible B) All states are periodic with period 2
C) All the states are persistent D) All the above

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76. If { ( )} is a Poisson process then { (2) = 1| (4) = 4} is

A) B) C) D) 0

77. In an M/M/1 queue with arrival rate , departure rate and infinite queue capacity, the
steady state probabilities
A) exist when <
B) exist when >
C) always exist
D) none of the above

78. Cyclical variation in time series has a period of oscillation of
A) Less than one year B) More than one year
C) Both A and B D) None of the above

79. Method of simple averages for a time series data is used to measure
A) Seasonal variation B) Trend
C) Cyclical variation D) Irregular variation

80. Which of the following index number is generally expected to have an upward bias?
A) Paasche’s index number B) Fisher’s index numbers
C) Laspeyre’s index number D) All the above

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