ISI Admission Test 2024 Question Paper M.Stat PSA

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ISI 2024
Question Paper ·
M.Stat PSA
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ISI 2024 Question Paper M.Stat PSA

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m
m .co

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Notations and Abbreviations

The following are used throughout the question paper.
m
m of real numbers
c. o Set m .co
e
R
E
e m Expectation
l as
l asV Variance
ag
ag Cov⌊x⌋
Covariance
Largest integer less than or equal to x
iid independent and identically distributed
pmf probability mass function
cdf cumulative distribution function
MLE Maximum likelihood estimator

m
.co
s em
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a integer. For any permutation σ of
1. Let n be an odd positive
{1, 2, 3, . . . , n}, consider the product

∏
n
P (σ) = (j − σ(j)) = (1 − σ(1))(2 − σ(2)) · · · (n − σ(n)).
j=1

Then

m
m (A) P (σ) = 0 for all σ.
.co
m .co (B) P (σ) is odd for all σ.
s e m
s e g la
la
(C) P (σ) is even for all σ.
g a
a (D) None of the above.

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2. Let a, b, c be three distinct non-zero real numbers. The expres-
sion
(x − b)(x − c) (x − c)(x − a) (x − a)(x − b)
a +b +c −x
(a − b)(a − c) (b − c)(b − a) (c − a)(c − b)

is equal to zero for
(A) exactly 2 values of x. (B) exactly 3 values of x.

(C) infinitely many values of x. (D) no value of x.

3. Equation of the straight line passing through the centres of the
two circles x2 + y 2 − 4x = 0 and 2x2 + 2y 2 + 4x − 5y = 0 is

(A) (x − 2)y + (x − 1)(y + 5/4) = 0.
(B) 4(x + 1)y − (x − 2)(4y − 5) = 0.
(C) (x + 1)y − (x − 2)(y − 5) = 0.
(D) 2x + y = 3.

4. Let p be a positive real and T be the set of all triangles that
have perimeter p. A triangle in T has maximum area if its
angles are

(A) 60◦ , 60◦ and 60◦ (B) 90◦ , 45◦ and 45◦

(C) 120◦ , 30◦ and 30◦ (D) 50◦ , 60◦ and 70◦

5. For x ≥ 0, let f (x) denote the non-negative square root of x.
The expression
( ) ( )
f sin4 x + 4 cos2 x − f cos4 x + 4 sin2 x

equals

(A) sin 2x (B) cos x (C) sin x (D) cos 2x

2

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6. Let S be the set of all 3 × 3 matrices A such that A2 = 0 and
the entries of A belong to {0, 1}. Then the number of elements
m
.co
in the set S is
m
m
(A) 9 .co (B) 12 (C) 15 (D) 6
s e m
s e l a
g l a ag
a
7. Let A be an m × n matrix and let P be an n × n idempotent
matrix. If P ̸= I, then which of the following statements is true?

(A) rank (AP ) < m
(B) rank (AP ) < min{m, n}
(C) rank (AP ) = min{m, n}
m
.co
em
(D) rank (AP ) < n

la s
a g
8. Let a , a , a , a be four distinct positive numbers. Consider the
1 2 3 4

4 × 4 matrix B = ((bij )) where bij = ai + aj for i, j = 1, 2, 3, 4.
Then rank (B) is

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

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m .co e m
9. Let S be the set of vectors x = (x1 , x2 , x3 ) in R3 such that
s
s e l a
x ∈ {1, 2, 3, 4, 5} for all i and not all x ’s are equal. For how
la ag dependent
i i

ag many ordered pairs (x, y) ∈ S×S are x and y linearly
vectors in R3 ?

(A) 137 (B) 125 (C) 126 (D) 132

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10. Let f : [0, 1] → [0, 1] be a function such that

|f (x) − f (y)| < |x − y|

for all x ̸= y ∈ [0, 1]. Consider the following two statements:

(i) There exists at least one point a ∈ [0, 1] such that f (a) = a.
(ii) There exists at most one point a ∈ [0, 1] such that f (a) = a.

Choose the correct option.

(A) Both (i) and (ii) are true.
(B) (ii) is true but (i) is false.
(C) (i) is true but (ii) is false.
(D) Both (i) and (ii) are false.

11. The number of subsets {a, b, c} of {1, 2, . . . , 24} such that a, b
and c are in arithmetic progression is

(A) 66 (B) 132 (C) 276 (D) 138

12. Let f and g be two real-valued differentiable functions on R. Let
h : R → R be defined as h(x) = max{f (x), g(x)}. Consider the
following two statements:

(i) If h is differentiable then f ≥ g or f ≤ g.
(ii) If f ≥ g or f ≤ g then h is differentiable.

Choose the correct option.

(A) (ii) is true but (i) is false.
(B) h is always differentiable.
(C) Both (i) and (ii) are true.
(D) (i) is true but (ii) is false.

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13. Let X(1) ≤ · · · ≤ X(n) denote the order statistics from a random
sample of size n, drawn from an exponential distribution with
m
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mean equal to 1. For z > y > 0, the conditional density of X(n)
m
.co m
given X(1) = y is

m n exp{−(z − y)} (1 − exp{−(z − y)}) s e
s e(A) n−1
l a
g la (B) (n − 1) exp{−(z − y)} (exp{−(z − y)}) n−2 ag
a (C) (n − 1) exp{−(z − y)} (1 − exp{−(z − y)}) n−2

(D) exp{−(z − y)}

m
.co
14. Let F and G be two cdfs and let α, β > 0. Which of the following
is not a cdf?
s em
g la/(α + β) (B) F (αx)G(βx)
(A) (αF (αx) + βG(βx))
a
(C) (F (αx) + G(βx)) /(α + β) (D) (F (αx)) (G(βx)) α β

15. Suppose X1 , X2 and X3 are three random variables such that

m
.co
X1 , (X1 + X2 )/2 and (X1 + X2 + X3 )/3 are iid standard normal
m
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random variables. Based on the above information choose the
e m
e m
correct statement from the options given below.
las
las (A) This is not possible.
ag
ag (B) X1 and X3 are independent normal random variables.
(C) X1 , X2 and X3 are identically distributed normal random
variables.
(D) X1 , X2 and X3 are independent normal random variables.

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16. Suppose E1 , . . . , Em are disjoint events satisfying 0 < P(Ej ) < 1,
∪
m
for each j = 1, . . . , m. Let E = Ej . Assume 0 < P(E) < 1.
j=1
Let A be any other event and define p = P(A | E). Which of
the following statements is true?
∑
m
(A) p = P(A | Ej ) P(E)
j=1
∑
m
(B) p = P(A | Ej ) P (Ej | E)
j=1
∑
m
(C) p = P(A | Ej )
j=1
∑
m
(D) p = P(A | Ej ) P(Ej )
j=1

17. Suppose A is the set of all 6-digit numbers formed using each
of the digits 1, 2, 3, . . . , 6 exactly once. If a number X is chosen
uniformly at random from A, then what is the probability that
X is divisible by 6 but not by 9 or 11?

(A) 1/2 (B) 1/6 (C) 1/11 (D) 2/33

18. Suppose X is a random variable taking values an with probability
(1 − p) pn for n ≥ 0, where a ̸= 0 is a real number and 0 < p < 1.
Consider the following statements.

(i) E[X] is finite for all values of a and p.
√
(ii) V(X) is finite whenever |a| < 1/ p.

Which of the following statements is correct?

(A) Only (ii) is true.
(B) Both (i) and (ii) are true.
(C) Only (i) is true.
(D) Neither (i) nor (ii) is true.

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19. A coin with success probability 1/4 is independently tossed
N times and the outcomes are recorded. Given that the total
m
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number of successes is m, what is the probability that the first
m
.co m
k tosses are successes, where k ≤ m?

m s e
s e(A) l a
a ag
k
l
k N −k
(1/4) (3/4) (B)

ag (C) m(m − 1) · · · (m − k + 1)
m )
(
m
(D) (1/4)k (3/4)m−k
N (N − 1) · · · (N − k + 1) k

20. Suppose X and Y are two independent random variables both
following Poisson distribution with parameter 10. What is the
value of E(X − Y )2 ?

m
(A) 30 (B) 10
c. o (C) 40 (D) 20

s em
g la bounded by the parallelogram with
a
21. Let M denote the region
vertices at (0, 1), (2, 1), (0, −1) and (−2, −1). Let (X, Y ) denote
the x and y co-ordinates of a randomly chosen point from M .
Consider the folowing statements.

(i) The marginal distribution of X is uniform on (−2, 2).
(ii) E(XY ) = 0.
m
m (iii) X and Y are independent.
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la
Which of the following statements is correct?
s e g
g la (A) Only (ii) and (iii) are true. a
a (B) Only (ii) is true.
(C) Only (i) is true.
(D) None of the statements is true.

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22. Suppose {Xi : i ≥ 1} are iid Poisson random variables with
standard deviation 2. Then as n → ∞

1∑
n
Xi (Xi − 2)
n i=1

converges in probability to

(A) 16 (B) 8 (C) 12 (D) 2

23. For x1 , x2 , . . . , x2n ∈ [−1, 1], where the xi ’s are not necessarily
distinct, define
2n 2n
1 ∑ 2 1 ∑
m= xi and s = (xi − m)2 .
2n i=1 2n i=1

Let s2max denote the maximum possible value that s2 can take
over all such x1 , x2 , . . . , x2n .
Based on the above information, choose the FALSE statement
from the options given below.

(A) s2 ∈ [0, 1] for all such x1 , x2 , . . . , x2n .
(B) m ∈ [−1, 1] for all such x1 , x2 , . . . , x2n .
(C) If m = 0 then the corresponding s2 = s2max .
(D) If s2 = s2max then the corresponding m = 0.

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24. Consider a random variable with pmf given by


 p if x = 1,
m
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

m



.co
3p
m
if x = 2,
P (X = x) =
s e
em

s



2p if x = 3,
l a
ag

la

1 − 6p if x = 4,
g
a where the unknown parameter p ∈ [1/20, 1/10]. A random
sample of size 4 from this distribution yielded the observations
x1 = 2, x2 = 3, x3 = 2, x4 = 1. What is the maximum likelihood
estimate of p based on these observations?
√
(A) 1/10 (B) 1/ 4 18 (C) 1/18 (D) 1/20

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25. Suppose that we havegl
a
a observations (x , y , z ), i = 1, . . . , n. Let
i

βb be the estimated coefficient obtained by least-squares linear
x
i i

regression of z on x, without intercept. Let γ
bx , γ
by be the esti-
mated coefficients obtained by similarly regressing z on x and
y jointly, also without intercept. Which of the following state-
ments is true?
∑
n ∑ n
(zi − βbx xi )2 > (zi − γ
bx xi )2 .
m
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(A)
m
i=1 i=1

.co m
∑n ∑n

m
(B) (zi − βbx xi )2 < (zi − γ by yi )2 .
bx xi − γ
s e
e la
i=1 i=1

las ∑n
(zi − βbx xi )2 ≤
∑n
by yi )2 .
(zi − γ ag
g
(C)
a i=1
∑n
i=1
∑n
(D) (zi − βbx xi )2 ≥ (zi − γ by yi )2 .
bx xi − γ
i=1 i=1

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26. Suppose that we have observations (x1 , y1 ), . . . , (xn , yn ). We fit
a simple linear regression of y on x, with intercept, using these
observations via the least-squares method. Let ei , i = 1, . . . , n,
be the residuals of the fitted regression model. Consider the
following statements.
∑
n
(i) n1 ei = 0.
i=1
∑n ∑
n ∑
n
(ii) n1 (ei − ē)(xi − x̄) = 0 where x̄ = n1 xi and ē = n1 ei .
i=1 i=1 i=1
∑n
(iii) n1 ei xi = 0.
i=1

Which of the following statements is correct?

(A) (i), (ii) and (iii) are all true.
(B) Only (ii) and (iii) are true.
(C) Only (i) and (ii) are true.
(D) Only (i) and (iii) are true.

27. Let X1 , . . . , Xn be iid N (µ, σ 2 ) random variables, where µ ∈ R
and σ > 0 are unknown parameters. Let Φ(·) and Φ−1 (·) denote
the cdf of N (0, 1) and its inverse respectively. Define
1∑ ∑
n n
1 2
X̄n = Xi and Sn2 = (Xi − X̄n ) .
n i=1 n i=1

For α ∈ (0, 1), let ξα (µ, σ) be a parametric function satisfying

P(X1 ≤ ξα (µ, σ)) = α.

What is the MLE of ξα (µ, σ)?
( )
(A) Φ X̄n + αSn .
(B) X̄n + αSn .
(C) X(⌊nα⌋) , the ⌊nα⌋-th order statistic of X1 , . . . , Xn .
(D) X̄n + Φ−1 (α) Sn .

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28. A random sample is obtained from a N (µ, 1) distribution, where
µ ∈ R is an unknown parameter. Using this sample, a most
m
.co
powerful (MP) test ϕ of size α = 1/2 is constructed for testing

o m m
c
H0 : µ = 0 against H1 : µ = 1. Let β denote the power of this
test.. Which of the following statements is correct? s e
s em l a
g la (A) ϕ is an MP test of size < 1/2 with power β for testing ag
a ′
H : µ = −1 against H : µ = 1.
0
′
1

(B) ϕ is an MP test of size 1/2 with power > β for testing
H0′ : µ = 0 against H1′ : µ = 1/2.
(C) ψ = (1 − ϕ) is not an MP test of size (1 − β) for testing
H0′ : µ = 1 against H1′ : µ = 0.

m
(D) ϕ is an MP test of size 1/2 for testing H0′ : µ = 0 against
H1′ : µ = a for some a ∈ (−∞, 0).
m .co
s e
l a
ag

m
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29. Let λ > 0. Let X1 , X2 , X3 be independent Poisson random vari-
e m
e m
ables with parameters λ, 2λ and λ/2 respectively. Which of the
las
las following statistics is sufficient for λ?
ag
ag (A) T = 2X1 + 4X2 + X3
(B) T = 2X1 + X2 + 4X3
(C) T = max{2X1 , 4X2 , X3 }
(D) T = X1 + X2 + X3

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30. Let N ≥ 3 be an integer. A box contains N balls numbered
{1, 2, . . . , N }. A simple random sample s of size n (1 < n < N )
is selected with replacement from the box. For i = 1, . . . , N ,
define 
1 if the i-th ball is selected in s,
Wi =
0 otherwise.

Which of the following statements is true?

(A) Cov(W1 , W2 ) < 0.
∑
N
(B) Wi = n.
i=1

(C) E(Wi ) = n/N for each i = 1, . . . , N .
(D) Cov(W1 , W2 ) = 0.

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