ISI Admission Test 2024 Question Paper M.Stat PSB

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ISI 2024
Question Paper ·
M.Stat PSB
EXAM YEAR TYPE SUBJECT

ISI 2024 Question Paper M.Stat PSB

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Notations and Abbreviations
The following are used throughout the question paper.
m
R
m Set of real numbers
c. o n - dimensional Euclidean space m .co
Rn
m s e
sQe Set of rational numbers
l a
g la E Expectation ag
a V Variance
P Probability
Cov Covariance
T
A Transpose of the matrix A
i.i.d. independent and identically distributed
p.d.f. probability density function

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1. Let m ̸= n and let A = ((a )) be an m × n real matrix such that
ij
T
A A = I. Let b1 , . . . , bm ∈ R. Prove that the system of linear
equations in the unknowns x1 , . . . , xn

a11 x1 + a12 x2 + ··· + a1n xn = b1 ,
a21 x1 + a22 x2 + ··· + a2n xn = b2 ,
.. .. .. ..
. . . .
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am1 x1 + am2 x2 + · · · + amn xn = bm
e m
e m has at most one solution.
las
las ag
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2. Use the identity n2 + (2n + 1) = (n + 1)2 to prove that the set
S = {(x, y) ∈ Q × Q : x2 + y 2 = 1} has infinitely many elements.

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3. A box contains 7 indistinguishable green balls, 5 indistinguish-
able white balls and 6 indistinguishable black balls. Suppose
balls are drawn using simple random sampling with replacement
until balls of all colours are obtained. Let N denote the mini-
mum number of draws needed to achieve this.

(a) For each nonnegative integer n, calculate P(N > n).
(b) Using (a) or otherwise, compute E(N ).

4. Suppose (X, Y ) is uniformly distributed over the region

(x, y) ∈ R2 : 0 < y < 1, |x| < 1 − y .
{ }

Calculate the p.d.f. of |X| + Y .

5. Let X1 , X2 , X3 be i.i.d. N (0, 1) random variables. Define, for
1 ≤ i ̸= j ≤ 3, 
1 if X > X ,
i j
Wij =
0 otherwise.

(a) Calculate E(Wij ) and V(Wij ) for all 1 ≤ i ̸= j ≤ 3.
(b) Calculate Cov(W12 , Wij ) for all 1 ≤ i ̸= j ≤ 3.

6. Assume that the length, in minutes, of a phone-call of an in-
dividual follows an exponential distribution with an unknown
parameter λ > 0 with density function f (x) = λ e−λx , x > 0.
However, when the phone company calculates the length of a
phone-call, it always considers the nearest integer greater than
or equal to the actual length. For example, a 22.09 minutes long
phone-call will have a call-length of 23 minutes in the phone
company records. Suppose you have the data on the lengths
of n independent phone-calls T1 , T2 , . . . , Tn of that individual as
reported by the phone company. Based on this data, compute
the maximum likelihood estimator of λ.

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7. Suppose that X1 , X2 are i.i.d. U (0, θ) random variables. We
want to test H0 : θ = 1 against H1 : θ = θ1 where θ1 > 1 is fixed.
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For this, we adopt two testing strategies.
m
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s e m
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Test 2: Reject H if max{X , X } > κ.
l
0 1 2

ag (a) Find κ such that both the tests have the same size.
(b) Find the powers of the two tests. For Test 2, use the value
of κ obtained in part (a).
(c) Which of the two tests would you prefer and why?

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. parametric model may be
8. Suppose we have paired observations (X1 , Y1 ), . . . , (Xn , Yn ). A

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statistician thinks that the following
a s
gl α, β and unknown ρ ∈ (−1, 1),
appropriate for this data:
For unknown odd aintegers
the pairs (Xiα , Yiβ ) are i.i.d. bivariate normal with parameters
(0, 0, 1, 1, ρ).
Write down the likelihood function for this model.

9. Let Y1 , . . . , Y2n be i.i.d. Bernoulli(p) random variables, where
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p ∈ (0, 1) is unknown. Consider the estimators
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2n−1
s e 1∑
n m
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T1,n = Yi Yi+1 and T2,n = Y2i−1 Y2i .

las 2n − 1 i=1
a g n i=1

ag (a) Show that both estimators are consistent for p . 2

(b) Find lim (V(T1,n )/V(T2,n )).
n→∞
(c) For large n, which estimator would you prefer and why?

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10. A city has N households numbered {1, . . . , N }. Let yi denote the
size of the i-th household. A simple random sample s1 of size
m is drawn without replacement from households {1, . . . , M }.
Another independent simple random sample s2 , also of size m, is
drawn without replacement from households {N −M +1, . . . , N }.
Assume m < M and M > N/2.

(a) Compute πi = P(i ∈ s1 ∪ s2 ), the probability that the
i-th household is included in either of the two samples, for
i = 1, . . . , N .
(b) Show that T is an unbiased estimator of the average house-
hold size in the city, where

1 ∑
( )
yi
T = .
N i∈s ∪s πi
1 2

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