ISI Admission Test 2023 Question Paper M.Stat PSB

FOR ISI EXAM PREPARATION

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

ISI 2023 Question Paper M.Stat PSB

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GROUP A

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1. Let An = ((aij )) be the n × n matrix de ned by
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

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

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

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

a


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aij = 1 if |i − j| = 1,

ag


2 if i = j.


Find the determinant of An for n ≥ 1.

2. Consider a random permutation of the eight numbers 1, 2, . . . , 8.
Compute the probability that no two adjacent numbers in this
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permutation have a product which is odd. Give justi cation for

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your computations.

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3. Identify, with justi a
g
cation, all cumulative distribution functions
F that satisfy F (x) = F (x2023 ) for every x ∈ R.

GROUP B

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4. A six-faced fair die is rolled repeatedly till 1 appears. Let X
s e g laof times 6
la
be the total number of rolls and Y be the number
g a
a
appeared in these X rolls.

(a) Find E[Y |X = x].
(b) Find E[Y ].

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5. Suppose X1 , . . . , Xn (n ≥ 2) are independent and identically
distributed observations from a distribution having probability
density function

e−(x−θ) if x ≥ θ,
fθ (x) =
0 if x < θ,

where θ ∈ R. Let
Z ∞
ψ(θ) = fθ (x)dx.
1

De ne θbn = min{X1 , . . . , Xn }. Consider ψ(θbn ) as an estimator
of ψ(θ) and let Bn (θ) denote the associated bias.

(a) Show that Bn (θ) > 0 for every θ < 1 and Bn (θ) = 0 for
every θ ≥ 1.
(b) Show that lim Bn (θ) = 0 for every θ < 1.
n→∞

6. Suppose that two observations X1 and X2 are drawn at ran-
dom from a distribution with the following probability density
function
1

 if 0 ≤ x ≤ θ or 2θ ≤ x ≤ 3θ,
 2θ


fθ (x) =



0 otherwise,

where θ > 0. Determine the maximum likelihood estimator of θ
for each of the following observed values of X1 and X2 .

(a) X1 = 7 and X2 = 9.
(b) X1 = 4 and X2 = 9.
(c) X1 = 5 and X2 = 9.

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7. Mr. X wants to compare ve di erent varieties of wheat. He has
access to a piece of land which has been divided into 25 smaller
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plots as shown in the gure below.
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Mr. X has asked forayour help in his venture.

(a) Give an allocation of the wheat varieties to the plots so
that you are able to compare them. Justify your allocation
rule. Clearly state all the assumptions you make.
(b) Once the crop is harvested, how would you analyze the data
and compare the di erent varieties of wheat?
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a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 3 of 5

8. Let X1 , . . . , Xn be independent and identically distributed
Bernoulli(θ) random variables where θ ∈ (0, 1). The maximum
likelihood estimator of θ is the sample mean X n . However, a
statistician feels that the sample size n is too small, and decides
to increase the sample size. In order to do so, he records the
observed values of the data points, {x1 , . . . , xn }, and then se-
lects a random sample of size m = kn with replacement from
{x1 , . . . , xn }, where k is a positive integer. The values in the
observed sample are recorded as {Y1 , . . . , Ym }. The statistician
proposes the new estimator T = 12 (Y m + X n ), where Y m is the
sample mean of Y1 , . . . , Ym .

(a) Show that T is unbiased for θ.
(b) Is T a better estimator of θ than X n ? Justify your answer.

4

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9. To test whether the heights of siblings are correlated, a re-
searcher devised the following plan: She identi ed a random
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sample of n families with at least two adult male children.

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c
For the ith family, suppose that Xi and Yi are the heights
. rst and second male child, respectively. Assume that s e
m
of the
e(X , Y ), . . . , (X , Y ) are independent bivariate normal random l a
as
l vectors with parameters (µ, µ, σ , σ , ρ), where µ and σ are
1 1 n n
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a known from previous studies. She is interested in testing the
2 2 2

null hypothesis H0 : ρ = 0 against the alternative H1 : ρ = 0.5.
Unfortunately, due to a mistake in the questionnaire, she was
only able to observe (Ui , Vi ) for each i, where Ui = max(Xi , Yi )
and Vi = min(Xi , Yi ).

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(a) Based on the observed sample, obtain the test statistic cor-

m . test of H against H .
responding to the most powerful
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(b) Find a critical valueasso that the size of the test converges

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to 0.05 as n → ∞.

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