ISI Admission Test 2023 Question Paper M.Stat PSA

FOR ISI EXAM PREPARATION

ISI 2023
Question Paper ·
M.Stat PSA
EXAM YEAR TYPE SUBJECT

ISI 2023 Question Paper M.Stat PSA

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1. A sequence of real numbers {an }n≥1 has a peak at n if an ≥ ak
for all k ≥ n. Consider the following statements.
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(I) No sequence of real numbers can have only nitely many
c. opeaks. m .co
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s em(II) No sequence of real numbers can have in nitely many l a
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peaks.
(III) Any sequence of real numbers having nitely many peaks
must have the property that an ≥ 0 for all n greater than
some k.

Then

(A) only (I) is true
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(B) none of (I), (II) and (III) are true
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(C) both (II) and (III) are true
(D) only (III) is truegl
a
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2. Let C denote the set of complex numbers and let Im(z) denote
the imaginary part of z ∈ C. Consider the set
o m
m c
S = {s ∈ R : there exists z ∈ C such that Im(z) ̸= 0 and s .= z +2z−1}.
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2

e m la s
las Then,
ag
ag (A) S ̸= R, but contains in nitely many elements
(B) S is a non-empty nite set
(C) S = R
(D) S is the empty set

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3. For a set S, let S c denote the complement of S. Also, for two
sets P and Q, let P \ Q = P ∩ Qc . Let A, B1 , B2 and B3 be four
sets. Which of the following statements is NOT true?

(A) (A ∪ B1 ∪ B2 ∪ B3 )c = Ac ∩ B1c ∩ B2c ∩ B3c
(B) (A \ B1 ) \ (B2 ∪ B3 ) = A \ (B1 ∪ B2 ∪ B3 )
(C) A \ (B1 ∪ B2 ∪ B3 ) = (A \ B1 ) ∪ (A \ B2 ) ∪ (A \ B3 )
(D) A ∩ (B1 ∪ B2 ∪ B3 ) = (A ∩ B1 ) ∪ (A ∩ B2 ) ∪ (A ∩ B3 )

4. For a complex number z, let z̄ be its complex conjugate. Then
the equation
z z̄ 2 + z 2 z̄ = 0

has

(A) exactly three roots
(B) exactly two roots
(C) in nitely many roots
(D) only real roots

5. Let f (x) = x2 + (2a + 1)x + (a2 + 2). The number of values of a
for which one of the roots of the equation f (x) = 0 is twice the
other root is

(A) more than 2 (B) 2 (C) 1 (D) 0

2
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6. Let A be a nite set of real numbers having m (≥ 2) elements.
De ne a function f : R → R, given by
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c o m f (x) = min{|a − x| : a ∈ A}.
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s e
Then,
l a
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l (A) f is continuous everywhere ag
a (B) f is continuous only at nitely many points
(C) f is discontinuous everywhere
(D) f has m discontinuities

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7. Let A be the set of functions f : R → R for which |f (x)−f (y)| ≤
2

g
2|x − y| for all x, y ∈ Rlaand f (0) = 0. Then, for any f ∈ A,
a
(A) the functions g(x) = P (f (x)) ∈ A for every polynomial P
(B) the function g(x) = x + f (x) ∈ A
(C) the function g(x) = xf (x) ∈ A
(D) the function g(x) = ef (x) ∈ A

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s e 8. The number of values of a for which the three linesla

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a 2x + y − 1 = 0, ax + 3y − 3 = 0, 3x + 2y − 2 = 0

are concurrent is

(A) more than 2 (B) 1 (C) 0 (D) 2

3
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9. For a non-constant geometric progression for which the second
term is 2 and the common ratio is an integer, the 10th, 20th and
30th terms are in arithmetic progression. Then, the fourth term
is

(A) −2 (B) −4 (C) 4 (D) 2

√ √
x + 8 − 8x + 1
10. lim √ √ equals
x→1 5 − x − 7x − 3

2 1 7
(A) does not exist (B) (C) (D)
3 2 12

11. The rank of the matrix
 
0 1 0 0 0
0 1 2 0 0
 
 
.
0 0 0 3 0

0 1 0 a 0
 

1 0 0 0 b

(A) depends on the values of both a and b
(B) is independent of the values of both a and b
(C) depends on the value of a but not on the value of b
(D) depends on the value of b but not on the value of a

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!
a 1
12. If the matrix A = has 1 as an eigenvalue, then the
2 3
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determinant of A is
m
(A) 5
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13. Let a < 500 be a positive integer. Consider a box containing
balls numbered a, a+1, . . . , 500. Suppose that the ball numbered
x is picked with probability
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2xa
for x = a, a + 1, . . . , 500.
(500 + a)(500 − a + 1)
e m
a s
gl
Then the value of a is

(A) 251
a(B) 1 (C) 499 (D) 2

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14. Let f (x) = ax + b for some a, b ∈ R. De ne fn (x) inductively

m.co by setting
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f1 (x) = f (x)

las ag
ag and
fn+1 (x) = f (fn (x)) for n > 1.

If f7 (x) = 128x + 381, then ab equals

1 1
(A) (B) 32 (C) (D) 8
8 32

5
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15. Let n = aaaaaaaaabcd be a 12-digited number divisible by 45
where the digits a, b, c, d are not necessarily distinct and a ̸= 0.
How many such numbers are there?

(A) 216 (B) 207 (C) 189 (D) 198

16. Let a, b and c be the sides of a triangle such that c2 = a2 +b2 −ab.
Then which of the following is always true?

(A) a ≤ c and b ≤ c
(B) a ≤ c ≤ b or b ≤ c ≤ a
(C) c ≤ a and c ≤ b
(D) None of the above

17. Let X be a discrete random variable and Y be a continuous
random variable which is independent of X. Let U = X + Y
and V = XY . Choose the correct statement from the options
given below.

(A) Both U and V are continuous random variables
(B) U is a continuous random variable but V need not be
(C) U is a discrete random variable but V need not be a con-
tinuous random variable
(D) U is a discrete random variable and V is a continuous ran-
dom variable

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18. Suppose that the sample mean and sample standard deviation
for a set of n observations x1 , x2 , . . . , xn are m and s (> 0),
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respectively. These values are updated to m1 and s1 after one

o m m
c
more observation xn+1 is added to the data set.
. on the above information, choose the correct statement s e
m
efrom the options given below.
Based
l a
l as ag
g
a (A) If m = m then s < s
1 1

(B) If m1 = m then s1 = s
(C) If m1 < m then s1 = s
(D) If m1 < m then s1 < s

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19. Suppose that X1 , X2 , . . . , Xn are independent and identically
distributed random variables with probability density function

λe−λx if x ≥ 0,
fλ (x) =
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0 otherwise,

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la
where λ > 0. Let Y = X1 + X2 + · · · + Xn . Then the conditional
s e g
la
distribution of Xn given Y = 1 is
g a
a (A) uniform on (0, 1)
(B) exponential with mean 1
(C) beta with parameters n − 1 and 1
(D) beta with parameters 1 and n − 1

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20. Let X1 , X2 , . . . be a sequence of random variables such that
1
E(Xi ) = 1, Var(Xi ) = 1 for all i and Cov(Xi , Xj ) = for
n
2
1X
all i ̸= j. Let Zn = Xi . Then, lim Var(Zn ) equals
n i=1 n→∞

1 1
(A) (B) (C) 0 (D) 1
2 4

21. Suppose that P(A|B) = 0.4 and P(Ac |B c ) = 0.6. Then, the two
equations are su cient to nd

(A) neither P(A) nor P(B)
(B) both P(A) and P(B)
(C) P(B) but not P(A)
(D) P(A) but not P(B)

22. Let X1 , . . . , Xn be independent and identically distributed nor-
mal random
√ variables with mean 0 and variance σ 2 > 0. De ne
n Xn
where X n = n1 ni=1 Xi and Sn2 = n1 ni=1 Xi2 .
P P
Tn =
Sn
Then, the distribution of Tn is

(A) Student’s t with n degrees of freedom
(B) Student’s t with (n − 1) degrees of freedom
(C) normal with mean 0 and variance 1
(D) None of the above

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X −Y
23. Consider a matrix M = where X and Y are indepen-
Y X
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dent standard normal random variables. Then the probability
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that M is a non-singular matrix is
e m
e(A)m 0 1 1
l as
las (B) 1 (C) √
2
(D)
2
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24. Let (U, V ) be a point chosen uniformly at random from the unit
circle {(u, v) ∈ R : u +av = 1}. Then Var(U ) is
gl
2 2 2

1 a 1 1
(A) (B) (C) 1 (D)
3 2 4

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g la 25. Suppose that we choose 2 cards simultaneouslyaat random from
a a deck of 20 cards numbered 1, 2, . . . , 20. What is the probability
that the smaller of the two numbers divides the larger?

36 46 56 66
(A) (B) (C) (D)
190 190 190 190

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26. Let X be a random variable with probability density function
displayed in the following graph.

0.20

0.15
Density f(x)

0.10

0.05

0.00

−10 −5 0 5 10

x

Match the following random variables with their respective prob-
ability density functions
X
(i) X + 3 (ii) X − 3 (iii) 2X (iv) .
2

(a) (b)
0.4

0.3

0.2

0.1
Density f(x)

0.0

(c) (d)
0.4

0.3

0.2

0.1

0.0

−10 −5 0 5 10 −10 −5 0 5 10

x

(A) (i)–(d), (ii)–(c), (iii)–(b), (iv)–(a)
(B) (i)–(d), (ii)–(c), (iii)–(a), (iv)–(b)
(C) (i)–(c), (ii)–(d), (iii)–(a), (iv)–(b)
(D) (i)–(c), (ii)–(d), (iii)–(b), (iv)–(a)

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27. Suppose that we want to t the regression model

y = β1 x + β2 x2 + ϵ
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.values, 0 and 1. Which of the following can be estimated
to 10 pairs of observations (x1 , y1 ), . . . , (x10 , y10 ) where xi ’s take
m s e
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two
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a (A) Both β and β 1 2

(B) β1 but not β2
(C) β2 but not β1
(D) β1 + β2

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28. Consider a bivariate sample (X , Y ), . . . , (X , Y ) where X = i
1 1 9 9 i

for i = 1, 2, . . . , 9. The least squares regression line for this
dataset is obtained as y = 3 + 2x. Later it turns out that Y5 was
recorded wrongly. When the revised regression line is obtained
which of the following are possible?

(A) Intercept can change but slope cannot
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(B) Slope can change but intercept cannot

m s e
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g la
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a

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29. Assume X1 , . . . , Xn are independent and identically distributed
N (µ, 1) random variables with µ ∈ R. We want to test H0 :
µ = 0 versus H1 : µ ̸= 0. Consider the following two one-sided
testing problems

H0,A : µ = 0 versus H1,A : µ > 0
and H0,B : µ = 0 versus H1,B : µ < 0.

Let ϕA,η (x) and ϕB,η (x) denote the most powerful tests of size
η ∈ (0, 1) for H0,A and H0,B , respectively. Then, for testing H0
versus H1 ,

(A) ϕ(x) = ϕA,η (x) + ϕB,η (x) is a test of size η
(B) ϕ(x) = ϕA,η (x) ϕB,η (x) is a test of size 2η
1
(C) ϕ(x) = max{ϕA,η (x), ϕB,η (x)} is a test of size η2
2
(D) ϕ(x) = ϕA,η (x) + ϕB,η (x) is a test of size 2η

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30. Let ϕ denote the probability density function of the standard
normal distribution. Let fθ , for θ ∈ {0, 1}, be de ned as
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

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ϕ(x) if θ = 0,
fθ (x) =
e m
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 1 ϕ x−1
s


a
if θ = 1.
s eAssume 2 2
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1 n ag
a from the density f (x). Which of the following is a
θ

su cient statistic for θ?
n
X
(A) Xi
i=1
n n
!
X X
(B) Xi , Xi2

m
i=1 i=1

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n
X
(C) Xi2
i=1
s em
n n
l a !

ag | ≥ 2)
X X
(D) Xi , 1(|X i
i=1 i=1

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