ISI Admission Test 2022 Question Paper M.Stat PSA

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ISI 2022
Question Paper ·
M.Stat PSA
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ISI 2022 Question Paper M.Stat PSA

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1. Let {xj }j≥1 be a sequence of positive numbers in geometric
progression. Let Pn denote the product of the first n terms
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of {xj }. Which of the following statements is true for all n ≥ 1?
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(A) P3n = P2n
m P = P /P
/Pn3
s e m
s e(B) l a
la (C) P = P P ag
2
3n 2n n

ag 3n
1/2
2n
2
n

(D) P3n = Pn P2n

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2. Suppose that f : R → R is a differentiable non-decreasing
function whose derivative f � is continuous. Fix a < b, and let
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S = {x ∈ (a, b) : f � (x) = 0} .
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Then, f (a) < f (b) if and only if

la ag
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(B) S is a countably infinite set.
(C) S is the empty set.
(D) S is a proper subset of (a, b).

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3. Let x > 1. Then lim x−n n(−1) x
n→∞

(A) equals ex .
(B) does not exist.
(C) equals +∞.
(D) equals 0.

4. Let f and g be monotonic functions on a closed interval [ a, b ].
Consider the following statements.

(I) f + g is monotonic on [ a, b ].
(II) f g is monotonic on [ a, b ].
(III) The maxima of f + g is attained at either a or b.
(IV) The maxima of f g is attained at either a or b.

Which of the above statements are correct?

(A) Only (I) and (III).
(B) Only (III) and (IV).
(C) Only (II) and (IV).
(D) None of (I), (II), (III), (IV).

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5. What is the range of the function
� �
1 −x2
2
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f (x) = x + e , x ∈ (−∞, ∞) ?
m�
2

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(A) . , e s e
�1 � � � � � �
em a
−1/2
(B) 0, 21 (C) 0, e−1/2 (D) 0, e−1/2
s
2
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6. Let (a, b) be a pair of real numbers. Consider the function


 −ax − b, if x ≤ −1,


f (x) = a2 x2 + b, if − 1 < x ≤ 1,

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


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5x2 + 1, if x > 1.

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For how many distinct pairs (a, b) is the function f continuous

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everywhere?

(A) Exactly 1 (B) ∞ (C) Exactly 2 (D) 0

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7. Consider a continuous function f : [−1, 1] → R which is

m differentiable everywhere in the interval (−1, 1).
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. 1). Further,

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1 �
suppose that f (−1) = − and f (x) ≤ 1 for all x ∈ (−1,

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2

la s
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Which of the following statements is false?

la a g
ag (A) f (1) can be greater than 1 but not greater than 2.
(B) f (1) can be greater than 2.
(C) f (1) can be less than −2.
(D) f (1) can be less than −1 but not less than −2.

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8. For θ ∈ R with θ �= 0, what is the value of
� ∞
x 2 2
√ eθx−θ x /2 dx ?
−∞ 2π
√ √
1 e e 1
(A) (B) (C) (D)
θ|θ| θ2 θ|θ| θ2

9. Let S� be the� set of all real numbers a such that the matrix
1 −2
M= has no real eigenvalue. Then,
8 a

(A) S = (−9, 7).
(B) S = (−7, 9).
(C) The interval (−9, 7) is a proper subset of S.
(D) S is a proper subset of the interval (−7, 9).

10. Let M be the set of all n × n real matrices. Which of the
following statements is false?

(A) rank(AB) = rank(BA) for all A, B ∈ M.
(B) AB = I if and only if BA = I for all A, B ∈ M.
(C) det(AB) = det(BA) for all A, B ∈ M.
(D) trace(AB) = trace(BA) for all A, B ∈ M.

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11. Let M (x) denote the matrix
 
x 0 1
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 
m 0 1 0  ,
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m x is a real number. Consider the function s e m
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a �
F (x) = trace M (x) M (x)
� T

where AT denotes the transpose of the matrix A. Which of the
following statements is true?

(A) The range of F contains 1.
(B) The range of F has a finite lower bound.
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(C) The range of F contains 0.
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(D) The range of F has a finite upper bound.
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12. Saurabh went to an amusement park with 9 tokens. There are

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4 different rides available, with 3 of them costing 2 tokens each

s e and one costing 5 tokens. Saurabh can take each ride
g la as many
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times as he likes as long as he has tokens available.
a different ways can he choose the rides so that he has no tokens
left, if the order in which the rides were taken is ignored?

(A) 15 (B) 12 (C) 7 (D) 6

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13. A six digit number is chosen at random. What is the probability
that at least two consecutive digits in the chosen number are
equal?

(A) 1 − (0.9)6 (B) (0.9)5 (C) (0.9)6 (D) 1 − (0.9)5

14. A multiple-choice test consists of 20 questions. Each question
has four choices, exactly one of which is correct. For each
question, a student is able to correctly identify one of the choices
as wrong, and chooses one answer at random from the remaining
three choices. The student will get a scholarship if she answers
at least 18 questions correctly. What is the probability that she
gets the scholarship?

(A) 1771/320 (B) 801/320 (C) 801/420 (D) 1771/420

15. The probability that a lie detector correctly determines whether
a person is lying or telling the truth is 0.8, independently of
whether the person is actually lying. Assume that people tell
the truth 95% of the times. Given that for a particular person
the lie detector says that the person is telling the truth, what is
the conditional probability that he is actually telling the truth?

(A) 36
37
(B) 76
77
(C) 0.95 (D) 0.8

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16. Suppose that X has a uniform distribution on {1, 2, . . . , n}.
Further suppose that conditioned on the event {X = x}, Y
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has a uniform distribution on {1, 2, ..., x}. Let
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s e m
s eThen l a
l a for each y = 1, 2, . . . , n,
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ag (A) p (x) > p (x + 1) for x = y, y + 1, . . . , n − 1.
y y

(B) py (x) < py (x + 1) for x = y, y + 1, . . . , n − 1.
(C) py (x) = 1/n for x = 1, 2, . . . , n.
(D) py (x) = 1/(n − y + 1) for x = y, y + 1, . . . , n.

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17. For n ≥ 3, let (X , Yg),l(X , Y ), . . . , (X , Y ) be independently
a 1 1 2 2 n

and identically distributed bivariate normal random variables
n

with parameters (0, 0, 1, 1, ρ). Suppose (i1 , . . . , in ) is a
random permutation of 1, 2, . . . , n such that all n! permutations
�
n
are equally likely. What is the expected value of T = X k Yi k ?
k=1

ρ √
(A) ρ (B) 1−ρ (C) 0 (D) nρ
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function f satisfying f (3) = f (4). What is E[X(X − 1)(X − 2)]?

(A) 27 (B) 3 (C) 4 (D) 64

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19. Let X be a random variable with the Student’s t-distribution
with 1 degree of freedom. Let M (t) be its moment generating
function. Which of the following statements is correct?

(A) M (t) is defined only on (−a, a) for some a ∈ (0, ∞).
(B) E(X α ) is defined for all α ∈ (0, 1) but not for α = 1.
(C) M (t) is defined for all real numbers t.
(D) E(X α ) is defined for all α ∈ (0, 2) but not for α = 2.

20. Which of the following distributions has a unique mode at 21 ?

(A) Beta distribution on (0, 1) with parameters 21 and 21 .
(B) Gamma distribution with scale parameter 1 and shape
parameter 12 .
(C) Gamma distribution with scale parameter 1 and shape
parameter 23 .
(D) Beta distribution on (0, 1) with parameters 1 and 1.

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21. Suppose X1 , X2 , . . . , X100 are independent and identically
100
�
distributed as Bernoulli(p), where p ∈ (0, 1). If Y = Xi and
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i=1

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Z = min{X1 , X2 , . . . , X100 }, then
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(C) Y and Z neither have the same mean nor the same variance.
(D) Y and Z have the same distribution.

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22. Suppose that X1 , X2 , . . . are independent and identically

m .co distributed bounded random variables with common meanm
s e 3 and

s e common variance 5. For n ≥ 1, define
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g la Y = (X − X ) + (X − X ) + . . . + (X a − X ) .
a
2 2 2
n 1 2 3 4 2n−1 2n

1
Then 2n Yn converges in probability to

(A) 5. (B) 0. (C) 14. (D) 20.

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23. Let X be a random variable. Define the function
P(X < x) + P(X ≤ x)
g(x) = , for all x ∈ R.
2
Which of the following statements is always true?

(A) g is right continuous at all x ∈ R.
(B) g is not a monotone function.
(C) g is left continuous at all x ∈ R.
(D) lim g(x) = 1.
x→∞

24. Consider positive numbers x1 , x2 , . . . , xn . Let θ̂1 be the value of
�n
θ that minimizes [log(xi /θ)]2 , and θ̂2 be the value of θ that
i=1
�
n
minimizes [(xi − θ)2 /xi ]. If g and h are the geometric mean
i=1
and the harmonic mean of x1 , x2 , . . . , xn , respectively, then,

(A) θ̂1 = g and θ̂2 = h.
(B) θ̂1 �= g but θ̂2 = h.
(C) θ̂1 �= g and θ̂2 �= h.
(D) θ̂1 = g but θ̂2 �= h.

25. Let xi = i and yi = i2 for i = 1, 2, 3, 4. Which value of β̂ satisfies
4
� 4
�
|yi − β̂xi | ≤ |yi − βxi | for all β ∈ R?
i=1 i=1

(A) 2 (B) 10/3 (C) 5/2 (D) 3

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26. Suppose X1 , X2 , . . . , Xn is a random sample from the uniform
distribution on (0, θ), and let Yi = log Xi for i = 1, 2, . . . , n.
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Which of the following statistics is an unbiased estimator of
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log θ?

m Y −1 s e
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a 1 2 n

(C) Y + 1
(D) log(2X)

27. Suppose X1 , X2 , . . . , X5 is a random sample from a Poisson dis-
tribution with mean λ. Consider a test for H0 : λ = 0.2 vs

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H1 : λ = 0.4 that rejects H0 if and only if X > 0.4. What is
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the probability of type 2 error of. this test?
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(B) 5 g
a (C) 1.4 e
a
−2 −2 −0.4
(A) 3 e (D) 1.44 e−0.4

28. Suppose a single observation X is obtained from a distribution
with a probability density function f , which is either f0 or f1 ,
where 
2x if x ∈ (0, 1),
f0 (x) =
0 if x ∈
/ (0, 1),
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and 
s
5x4
e m f1 (x) =
if x ∈ (0, 1),
/ (0, 1). gl
a
las 0 if x ∈
a
ag What is the maximum possible power of a level α test, based on
this observation, for H0 : f = f0 against H1 : f = f1 ?

� � √ √
(A) 1 − (1 − α)5 (B) (1 − α)5 (C) 1 − α5 (D) α5

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29. Suppose Y1 , Y2 , . . . , Y5 is a random sample from the exponential
distribution with mean µ. Which of the following intervals
contains µ with 100(1−α)% confidence? Here xk,p is the number
�
5
such that P (χ2k > xk,p ) = p, and T = Yi .
i=1
� �
T
(A) 0,
x5,1−α
� �
2x10,α
(B) 0,
T
� �
5T
(C) 0,
− log(1 − α)
� �
2T
(D) 0,
x10,1−α

30. Let X be normally distributed with mean θ and variance 1,
where θ ∈ R. Assume that φ(X) is a most powerful (MP) test
of size α (with 0 < α < 1) for testing H0 : θ = 0 against
H1 : θ = 2. Which of the following statements is correct?

� 0 : θ = −1 against H1 : θ = 2, φ is a test of
(A) For testing H
size α.
� 0 : θ = −1 against H1 : θ = 2, φ is a MP test
(B) For testing H
of size α� , where α� < α.
� 1 : θ = −2, φ is a MP test
(C) For testing H0 : θ = 0 against H
of size α.
� 0 : θ = 1/2 against H1 : θ = 2, φ is a test of
(D) For testing H
size α� , where α� < α.

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