ISI Admission Test 2026 Sample Paper M.Stat PSB

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

ISI 2026 Sample Paper M.Stat PSB

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m
m .co

m .co s e m
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a
PSB

1. Let f : R ! R be a di↵erentiable function with derivative f 0 .
Suppose f (0) = 0 and f 0 (x) > f (x) for all x 2 R.
m
o m
(a) Prove that e x f (x) is an increasing function of x on (0, 1).
c m .co
m . that lim f (x) = 1. s e
a
(b) Show
s e l
ag
x!1

g l a
a

2. Find the number of functions f : {1, 2, . . . , 2n} ! {1, 2, . . . , 2n}
satisfying
m
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f (i) 6= i and f (f (i)) = i for all i = 1, 2, . . . , 2n .
m
s e
l a
ag

3. Suppose A is an n ⇥ n real matrix which has n linearly
independent eigenvectors v1 , v2 , . . . , vn corresponding to real
positive eigenvalues 1, 2, . . . , n.
o m
m . c
m .co (a) Let V be the n ⇥ n matrix whose i-th column is v for all
s e m i

s e i = 1, 2, . . . , n, and D be the diagonal matrix whose
l a i-th

g la diagonal entry is ag
for i = 1, 2, . . . , n. Show that
i

a V 1
AV = D.

(b) Show that there exist 2n distinct matrices B1 , B2 , . . . , B2n
such that Bj2 = A for all j = 1, 2, . . . , 2n .

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4. Let ⇢ 2 ( 1, 1), and X0 , X1 , X2 , . . . be a sequence of random
variables satisfying

Xi = ⇢Xi 1 + "i for all i 1.

Suppose "1 , "2 , . . . have a common mean 0 and a common
2 2 ⌧2
variance ⌧ , and X0 has mean 0 and variance = .
1 ⇢2
Assume further that X0 , "1 , "2 , . . . are independent.

(a) Find Var(Xi ) for i 1.
(b) Find Cov(Xi , Xj ) for j > i 0, and show that it is a
function of j i.

5. Suppose X1 , X2 , . . . , X2n are independent Exponential random
variables with mean 1/ , where > 0. Let X(1)  · · ·  X(2n)
denote the order statistics of X1 , . . . , X2n .

(a) Show that for all x > 0,
" 2n # !
X
P X(n)  x = P 1{Xi  x} n .
i=1

Here 1{A} denotes the indicator function of an event A,
which takes the value 1 if A occurs, and 0 otherwise.
(b) Hence or otherwise, prove that for all " > 0,
✓ ◆
1
lim P X(n)  loge 2 " = 0 .
n!1

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6. Suppose ⇡ is a permutation of {1, . . . , n} where n 2. A pair
(i, j) with i < j is called an inversion with respect to ⇡ if ⇡i > ⇡j .
m
m
Let X denote the number of inversions with respect to a
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permutation chosen at random from all possible permutations
m s e m
s e
of {1, . . . , n}. Find E(X).
l a
g l a ag
a
7. A population consists of units {1, 2, . . . , n + 1} for some n 1.
Consider the following sampling procedure.

• Start with S0 = {1, 2, . . . , n}, consisting of the first n units.
• Generate U from Uniform (0, 1).
n
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– If U > , define S1 = S0 .
n+1

em
– Otherwise, remove one of the units uniformly at

s
la S .
random from S0 (independently of U ), and add unit

a g
(n + 1) to it to create 1

Determine, with justification, whether S1 is a simple random
sample of size n from the population.

8. Consider the following bivariate data.

m
m i 1 2 3 4 5 6 7 8 9 10 11 12
.co
m .co xi 21 21 21 21 21
y 33 93 34 48 19 58 14 32 12 44 45 a62
35 35 35
s e
35m 35 35 35

s e g l
la
i

g a
a Determine, with justification, the values of a and b that minimise
the sum of absolute errors
12
X
yi a bxi .
i=1

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9. Let X1 , X2 , . . . , Xn be independent random variables having
common probability density function f✓ , ✓ 2 {0, 1}, where
8
<1 if 0 < x < 1,
f0 (x) =
:0 otherwise,

and 8
1
< p
> if 0 < x < 1,
f1 (x) = 2 x
:0
>
otherwise.

Suppose we wish to test H0 : ✓ = 0 vs. H1 : ✓ = 1 at significance
level ↵, where 0 < ↵ < 1. Find the rejection region of the most
powerful test based on X1 , X2 , . . . , Xn in terms of quantiles of
standard distributions.

10. Let ✓1 , ✓2 , and ✓3 denote the three angles of a triangle,
measured in degrees (i.e., ✓i > 0 for i = 1, 2, 3, and
✓1 + ✓2 + ✓3 = 180). Suppose that each angle is measured by an
2
instrument that is subject to independent additive N (0, )
error, and the measured values of ✓1 , ✓2 , and ✓3 are found to be
y1 = 83, y2 = 47, and y3 = 56, respectively. Find the maximum
likelihood estimates of ✓1 , ✓2 , and ✓3 .

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