ISI Admission Test 2026 Sample Paper M.Stat PSA

FOR ISI EXAM PREPARATION

ISI 2026
Sample Paper · M.Stat
PSA
EXAM YEAR TYPE SUBJECT

ISI 2026 Sample Paper M.Stat PSA

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PSA

Notations and Abbreviations

The following are used throughout the question paper.
m
c o mof real numbers m .co
det(A) m. Determinant of the matrix A e
R Set

e Natural logarithm of x l as
log xs
g l a ag
aE
exp(x) e x

Expectation
Var Variance
Cov Covariance
c.d.f. cumulative distribution function
c.d.f. of the standard Normal distribution
i.i.d. independent and identically distributed
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s e
l a
ag

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s e g la
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1. The function f : R ! R given by

f (x) = ex e2x 6e3x

(A) has no real root.
(B) has exactly three real roots.
(C) has exactly one real root.
(D) has exactly two real roots.

2. Let S be the set of all ordered pairs (x, y) such that x, y are
integers satisfying the relation

x2 y 2 = 154.

What is the number of elements in S?

(A) 0 (B) 4 (C) 8 (D) 2

3. Suppose A and B are two 5 ⇥ 5 matrices with real entries.
Consider the following two statements:

I. det(A + B) det(A) + det(B).
II. trace(AB)  trace(A) trace(B).

Then

(A) both I and II are false.
(B) both I and II are true.
(C) II is true but I is false.
(D) I is true but II is false.

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4. Suppose (an )n 1 is a convergent sequence of real numbers. Let
1 2n
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.co
bn = an + , cn = an + for all n 1.
n n+1
m
Define
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s e l a
g l a (xn )n 1 = (a1 , b1 , a2 , b2 , . . . , an , bn , . . . ),
ag
a (yn )n 1 = (a1 , c1 , a2 , c2 , . . . , an , cn , . . . ).

Then

(A) neither (xn )n 1 nor (yn )n 1 is convergent.
(B) both (xn )n 1 and (yn )n 1 are convergent.
(C) (xn )n 1 is convergent but (yn )n 1 is not.
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(D) (yn )n 1 is convergent but (xn )n 1 is not.
.co
s em
g la
a that contains all the elements of
5. Identify the smallest interval
⇢
m 4n
A= + : m, n are positive integers .
n m

(A) [5, 1) (B) (4, 1) (C) [4, 1) (D) (0, 1)

o m
m . c
co 6. Let f (x) = x log x for x 2 (0, 1). Which of the following
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p
. s e
s em statements is not true?
g l a
g la a
a (A) f has a local maxima.
(B) f is di↵erentiable everywhere on (0, 1).
(C) f (x) has a limit as x ! 0+.
(D) f has a local minima.

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7. How many solutions does the following system of equations have?

x1 x2 + 2x3 = 1
2x1 + 2x3 = 1
x1 3x2 + 4x3 = 2

(A) 2 (B) 1 (C) 0 (D) 3 or more

8. Let ↵ > 0. Find the value of
Z 1
(2↵/n) ↵ ↵
lim 2 x n +1 (1 x) n 1 dx ,
n!1 (↵/n) 0

where (x) denotes the Gamma function evaluated at x.

(A) ↵/2 (B) 1/2 (C) 1 (D) 0

9. How many permutations of the twenty-six letters A, B, . . . , Z of
the alphabet are there in which Y is adjacent to at least one of
the five vowels A, E, I, O, U ?

(A) 5550 · (23)! (B) 230 · (24)!

(C) 5 · (25)! (D) 10 · (25)!

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10. If A is a 3 ⇥ 3 real symmetric matrix with trace(A) = 0, then
what is the set of possible values of rank(A)?
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(A) {0, 1, 3}
c
m (B) {0, 2, 3}
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e
(C) {0,
s
1, 2} (D) {0, 1, 2, 3}
l a
g l a ag
a
11. Let A and B be two events such that
1 4 3
P (A) = , P (B) = , and P (A | B) = .
2 9 4
Let X be the random variable which takes the value 1 if A occurs
and 0 otherwise. Similarly, Y is the random variable which takes
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the value 1 if B occurs and 0 otherwise. Consider the following
statements:
e m
1
l as
ag
I. E(XY ) = .
3
II. P XY = X 2 Y 2 = 1.

Then

(A) both I and II are wrong. (B) both I and II are correct.

(C) I is correct and II is wrong. (D) I is wrong and II is correct.

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s e la
12. Suppose (X1 , X2 ) follows a Bivariate Normal distribution with
g
g la E(X ) = E(X ) = 0, Var(X ) = Var(X ) = 1, Cov(Xa , X ) = ⇢
a 1 2 1 2 1 2

for some 0  ⇢  1. Then E(X12 X22 ) equals

(A) 2⇢. (B) 1 + 2⇢2 . (C) ⇢2 . (D) ⇢ + 2⇢2 .

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13. A random variable U with c.d.f. F stochastically dominates a
random variable V with c.d.f. G if F (t)  G(t) for every t 2 R.
Suppose

X ⇠ N (1, 1), Y ⇠ N (2, 1), Z ⇠ N (2, 2).

Consider the following statements:

I. Y stochastically dominates X.
II. Z stochastically dominates Y .

Then

(A) both I and II are false. (B) I is true but II is false.

(C) both I and II are true. (D) II is true but I is false.

14. Let X1 , . . . , X7 be i.i.d. with c.d.f. F and density f . Then the
density of the sample median is

(A) 140F (x)3 (1 F (x))3 . (B) 7F (x)3 (1 F (x))3 f (x).

(C) 140F (x)3 (1 F (x))3 f (x). (D) 840F (x)6 f (x).

15. Suppose two players A and B take turns in conducting a
sequence of trials. Each trial consists of rolling a pair of fair
dice, and recording the sum of the two numbers obtained. The
first player who obtains exactly 7 in a trial is the winner. If A
rolls first, what is the probability that B will win?

1 6 5 7
(A) (B) (C) (D)
2 11 11 36

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16. Suppose a random sample of four observations is drawn from the
Poisson distribution
⇣ with1 mean > 0. Let X denote the sample
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⌘
mean. What is P X < ?
o m 2
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(A) (1 +. 4 + 8 )e s e
em a
2 4
(B) (4 + 1)e 4
s l
g l(C)a (2 + 1)e 2
(D) e 4 ag
a

17. Graduating students at a certain college bring zero, one, or
two parents to attend the graduation ceremony with equal
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probability, independently of other students. In a class of 600
graduating students, let N be the number of parents who attend.
em
What is the variance of N ?
l as
(A) 200 a
(B) 800
g (C) 600 (D) 400

18. If A1 , A2 , A3 , . . . are independent events with
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P (An ) = pn for all n 1,

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e la
1
!

las g
[
a
then what does P An equal?

ag n=1

1
Y 1
Y
(A) 1 pn (B) (1 pn )
n=1 n=1
1
X 1
Y
(C) pn (D) 1 (1 pn )
n=1 n=1

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! ! !!
X1 0 4 2
19. Suppose ⇠ Normal , . Find the value of
X2 0 2 4
p
P (X1 + 3X2 > 2 13).

p p
(A) 1 (2 13) (B) 1 ( 13/7)

(C) 1 (1) (D) 1 ( 1)

20. Suppose X1 and X2 are independent Bernoulli(1/2) random
variables. If M = max{X1 , X2 }, find the value of Cov(X2 , M ).

1 1 1
(A) (B) (C) (D) 0
8 2 4

21. Suppose X1 , X2 , . . . , Xn are independent Uniform(0, 1) random
variables. Find the smallest value of n such that

P max{X1 , X2 , . . . , Xn } 0.99 0.95.

Here dae denotes the smallest integer larger than or equal to a.
⇠ ⇡ ⇠ ⇡
log(0.05) exp(0.99)
(A) (B)
log(0.99) exp(0.05)
⇠ ⇡ ⇠ ⇡
0.99 0.99 0.95
(C) (D) +
0.05 1 0.05 1 0.01

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22. Each worker in a factory takes lunch in a canteen
with probability 0.1 and elsewhere with probability 0.9,
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independently of other workers. If n workers work in the factory,
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the factory management wishes to find ↵n such that
s e
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lim P (At most ↵n workers take lunch in the canteen) = 0.95.

g a
lWhich
n!1

a of the following values of ↵ ensures the above?
n

(A) ↵n = n ⇥ 0.1
p 1
(B) ↵n = n ⇥ 0.1 + n ⇥ 0.3 ⇥ (0.95)
p 1
(C) ↵n = n ⇥ 0.1 + n ⇥ 0.3 ⇥ (0.975)
p 1
(D) ↵n = n ⇥ 0.1 + n ⇥ 0.09 ⇥ (0.95)
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✓ ◆
1 X
m 23. Suppose X ⇠ Uniform(0, ✓), ✓ > 0, and Y = log .

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✓ ✓

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Let X1 , X2 , . . . , Xn be a random sample from the distribution

s e l
of X. Let M = max{X , X , . . . , X }. What is the maximum
g a
la
1 2 n

g a of Y ?
likelihood estimator of the median of the distribution
a (A)
log 2
(B) M log 2
M
✓ ◆
2 n log 2
(C) log (D) Pn
M
Xi
i=1

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24. Suppose X is a random variable with density
8
<✓x✓ 1 if 0 < x < 1,
f✓ (x) =
:0 otherwise,

where ✓ > 0. The rejection region of the most powerful test for
H0 : ✓ = 4 vs. H1 : ✓ = 3 at significance level 0.05 based on X is

(A) {X < (0.05)1/4 }. (B) {X > (0.95)1/3 }.

(C) {X < (0.05)1/3 }. (D) {X > (0.95)1/4 }.

25. A researcher collects data on the average daily air quality index
(AQI) and average daily temperature (Temp) at a monitoring
station for 365 consecutive days. She obtains the following least
squares regression line with response AQI and predictor Temp.

AQI = 289 0.0049 ⇥ Temp

The coefficient of determination R2 of the model is 0.3844. What
is the value of the sample correlation coefficient between AQI and
Temp?

(A) 0.62 (B) 0.62 (C) 0.07 (D) 0.07

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26. Suppose X1 and X2 are jointly distributed as Bivariate Normal
2 2
with means µ1 , µ2 , variances 1,
2 , and correlation ⇢. Which of

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the following is equivalent to H0 : 12 = 22 ?
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(A) X1 and X2 are independent.
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(B) X1 and X2X are independent.
l
1

ag (C) X + X and X X are independent.
1 2 1 2

(D) None of the above.

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27. Given positive valued data X1 , X2 , . . . , Xn , what is the value of
✓ that minimises
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s
n
e1
a
X
l
(✓) = ?

ag i=1
Xi ✓

(A) Arithmetic mean of X1 , X2 , . . . , Xn
(B) Median of X1 , X2 , . . . , Xn
1 1 1
(C) Median of , ,...,
X1 X2 Xn
(D) Harmonic mean of X1 , X2 , . . . , Xn

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28. Suppose X , X , . . . , X are independent Poisson(✓)glrandom
g la 1 2 n
a
a variables with ✓ > 0. Let m
estimator of ✓m exists
1 be an integer. An unbiased

(A) only for odd m. (B) only for m  n.

(C) only for m 2 {1, 2}. (D) for all integers m.

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2 2
29. Suppose X1 , . . . , Xn are i.i.d. N (1, ) for some unknown > 0.
2
What is the minimum variance unbiased estimator of ?
n
1 X
(A) (Xi 1)2
n 1 i=1
n n
1 X 1X
(B) (Xi X)2 , where X = Xi
n 1 i=1 n i=1
n
1X 2
(C) (X 1)
n i=1 i
n
1X
(D) (Xi 1)2
n i=1

30. Suppose we have bivariate data (xi , yi ), i = 1, 2, . . . , n with
n > 2 such that xi 6= xj and yi 6= yj whenever i 6= j. Let
r1 be the sample Pearson’s correlation coefficient, and r2 be the
sample Spearman’s rank correlation coefficient between x and y.
Consider the following statements.

I. If |r1 | = 1, then r2 = r1 .
II. If |r2 | = 1, then r1 = r2 .
III. If r1 < 0, then r2  0.

Then

(A) only II and III are correct.
(B) only I and III are correct.
(C) only I and II are correct.
(D) only I is correct.

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