ISI Admission Test 2022 Question Paper M.Stat PSB

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ISI 2022
Question Paper ·
M.Stat PSB
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ISI 2022 Question Paper M.Stat PSB

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GROUP A

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. (0, 1). Further, assume that r converges to an irrational
1. Consider a sequence {rk }k≥1 of rational numbers lying in the
m s e
s enumber as k → ∞. Suppose r = , for all k ≥ 1, where m
interval
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k

g a
l and n are positive integers with no common divisors. Show
k
mk
nk k

a k

that the set of integers {nk : k ≥ 1} is not bounded.

2. Consider a 4 × 4 real matrix A which has positive trace and
negative determinant.

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(a) Show that A must have at least two real eigenvalues.

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(b) Show that A can have non-real eigenvalues.

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3. Let P be a regulara polygon with 24 sides. Consider all the
triangles whose vertices are also vertices of P . Find the number
of such triangles that are neither isosceles nor equilateral.

GROUP B
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4. Suppose U and V are independent and identically distributed

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random variables following a binomial distribution

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parameters n and . Let the random variable T denote the
2
number of distinct real roots of the quadratic equation

x2 + 2U x + V 2 = 0.

Find E(T ) and Var(T ).

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5. Suppose r ≥ 1 distinct books are distributed at random among
n ≥ 3 children.

(a) For each j ∈ {0, 1, 2, . . . , r}, compute the probability that
the first child gets exactly j books.
(b) Let X be the number of children who do not get any book,
and Y be the number of children who get exactly one book.
Show that
r(n − 1)(n − 2)r−1 r(n − 1)2r−1
Cov(X, Y ) = − .
nr−1 n2r−2

6. Consider two random variables (X, Y ) distributed as bivariate
normal with parameters (µ1 , µ2 , σ12 , σ22 , ρ). Based on a random
sample from this bivariate distribution, the fitted least squares
regression line of Y on X and that of X on Y were as follows:

Y = 22 − 3X

X = 5.84 − 0.12Y

(a) Compute the maximum likelihood estimates of the
following parametric functions:
σ1
(i) min{µ1 , µ2 } (ii) (iii) ρ
σ2

You are not required to derive the expressions for the
maximum likelihood estimators of µ1 , µ2 , σ12 , σ22 and ρ.
(b) If a new observation (5, 5) is included in the above set
of observations, determine whether each of the maximum
likelihood estimates obtained in (a) will increase, decrease
or remain unchanged.

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7. Suppose N students arriving at a college are all equally likely
to have a particular disease with an unknown probability p.
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The disease status (affected / not affected) of all students are

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independent. Blood samples are collected from all N students.
. to estimate p, two strategies are proposed. s e
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Strategy 1 Test all samples separately to obtain the status for all N
students.

Strategy 2 Randomly partition the students into m disjoint groups,
each comprising K = N/m students (with K ≥ 2 being

m
an integer). For each group, pool (mix) the blood samples

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from all K students within the group and test the pooled
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sample. If the pooled sample tests positive, then at least
one student withinathat group is affected. If the pooled
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sample tests negative,
unaffected.

(a) Based on the data obtained from Strategy 2, find a real-
valued sufficient statistic for p and the maximum likelihood
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estimator of p.
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(b) If a group tests positive, then all students within that group

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are further tested individually. Suppose that each

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individual sample or pooled sample) has equal
a which of the two strategies would you prefer to identify
all students affected with the disease when the underlying
p = 0.5, N = 200, and m = 20?

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8. Based on historical data, a positive random variable X arising
from an unknown distribution is believed to have the chi-square
distribution with 1 degree of freedom. A new theory suggests
√
that it may be better to model X as exponentially distributed
with mean λ, where λ is such that E(X) for this model is the
same as that for the earlier model.

(a) Compute λ.
(b) Suppose X1 , X2 , . . . , Xn are independent observations from
this unknown distribution. For α ∈ (0, 1), consider the
most powerful level α test for the null hypothesis that the
earlier model is correct against the alternative that the new
model is correct. Show that the rejection region of this test
is
� �
� √ �√
(x1 , x2 , . . . , xn ) : xi − 2 2 xi > c
i i

where c is a constant that depends on α.

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9. The following ANOVA table for three factors A, B, and C was
obtained (under a suitable model) from some data, but several
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values were illegible and marked ‘∗’. It is known that A had two
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levels and there were 24 observations in all.

m Source df Sum of squares Mean Square F ratio s e
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a B ∗ ∗ ∗ 0.62
C ∗ ∗ 25.4 ∗
AB 2 ∗ 5.4 0.54
BC 2 ∗ 4.2 ∗
Error ∗ ∗ ∗
Total ∗ 257.4

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(a) Fill in the missing values, providing suitable justification.

m .access only to tables for t distri-
ethe hypothesis of equality of effects
(b) Suppose that a student has
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butions. Can she testa
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of levels of A and
both at 5% level of significance? Justify your answer.

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