ISI Admission Test 2021 Question Paper M.Stat PSA

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ISI 2021
Question Paper ·
M.Stat PSA
EXAM YEAR TYPE SUBJECT

ISI 2021 Question Paper M.Stat PSA

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1. Let f be a di↵erentiable function on R such that the number of ele-
ments in the set {x : f (x) = 0} is 50. Then the minimum number of
elements in the set {x : f 0 (x) = 0} is
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(A) 50
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s e m
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2. Let P3 be the vector space of all polynomials with real coefficients and
of degree less than or equal to 3 over the field of real numbers. The
dimension for the subspace of polynomials p(x) such that
p(1) = p(2) = 0 is
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(A) 0 (B) 1
s em (C) 3 (D) 2

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a

3. Let k denote the number of 2 ⇥ 2 real matrices X with
" #
10 9
X2 = .
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4 10
m
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s em
s e (C) k = 3 g
la (D) k = 1
g la (A) k = 2 (B) k = 0
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a

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sin x4
4. lim is
x!0 (sin x)3

(A) 1 (B) 1 (C) 0 (D) 34

5. Let p and q be two non-zero real numbers and pq 6= 1. Suppose the
roots of the equation
✓ ◆ ✓ ◆ ✓ ◆
1 2 1 1 1
p+ x p+ +q+ x+ q+ =0
q p q p

are distinct and rational. Then

(A) p = q
(B) p and q MUST be rational numbers
(C) p + q MUST be a rational number
p
(D) MUST be a rational number
q

6. Let
[x3 ] [x]3
↵ = lim and = lim ,
x!1 x3 x!1 x3

where [x] denotes the greatest integer less than or equal to x. Then

(A) ↵ = 1, =0
(B) ↵ = 1, =1
(C) ↵ = 1, does not exist
(D) Neither ↵ nor exists

2

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7. For every natural number n, let
Z n
In = ({x}n + { x}n ) dx
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0

o m m
c
where {x} = x [x] where [x] denotes the greatest integer less than

m . to x. Then I equals s e
a
or equal
e 2n l
n

a s g
l (A) n + 1 (D) n a
2nn+1
ag (B) 0 (C)
n+1

8. The lengths of the sides of a right-angled triangle are numerically
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equal to the arithmetic mean, the geometric mean and the harmonic
a

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mean of two distinct positive real numbers a, b, a > b. Then is
b
p
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(A) 5 + 2
p
(B) 2 + 3
p
(C) 2 + 5
p
(D) 3 + 2

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9. The number of non-real roots of the polynomial equation
s e g
la
x12 + 2x6 = 5 is
g a
a (A) 10 (B) 0 (C) 2 (D) 6

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3
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10. A matrix is chosen at random from the set of all 2 ⇥ 2 matrices with
elements 0 or 1 only. The probability that the determinant of the
chosen matrix is non-zero is

3
(A) 16 (B) 38 (C) 81 (D) 14

11. If ( n C0 + n C1 )( n C1 + n C2 ) . . . ( n Cn 1 + n Cn ) = k n C0 n C1 . . . n Cn 1 ,
then k is equal to
n
(A) nn!
(B) (n+1)
n

n!

(C) (n+1)
n

nn!

(D) (n+1)
n+1

n!

12. Let ⇡ = (a1 , a2 , · · · , a2021 ) be a permutation of (1, 2, · · · , 2021). For
every such permutation ⇡,
2021
Y
P (⇡) = (aj j).
j=1

Then

(A) P (⇡) = 0 for all ⇡
(B) P (⇡) is odd for all ⇡
(C) P (⇡) is even for all ⇡
(D) There exists ⇡1 and ⇡2 for which P (⇡1 ) is even and P (⇡2 ) is odd.

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" # " # " #
1 3 2 0 4 0
13. Let A = ,B= and C = . Then the sum
4 2 0 3 0 1

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BAC C 2 AB 2 B 2n 1 AC 2n 1 C 2n AB 2n
trA + tr + tr + · · · + tr + tr +···
e m
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9 92 92n 1 92n
l as
s
(A) is equal to 14
la(B) is equal to 9 ag
g
a (C) is 1
(D) is equal to 12

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14. Let F be the set of all bijections f. from the set S = {1, 2, 3, 4, 5, 6}
m
onto itself such that f (f (i)) 6=siefor any i 2 S. The number of elements
g la
in F are
a
(A) 160 (B) 265 (C) 120 (D) 200

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15. Let f : [0, 1] ! R be a monotonic function such that the number of its
R1
e m
e m las
discontinuity points is finite. For t 0 let mt = 0 xt f (x)dx. Then

las which of the following statements is true?
ag
ag (A) There exists a t 0 for which m is not defined. t

(B) mt < 1 for all t 0.
(C) mt is defined and is equal to 1 for all t 0.
(D) mt is defined for all t 0 and there exists t1 , t2 0 such that
mt1 is equal to 1 and mt2 < 1.

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16. Let F (x) and G(x) be cumulative distribution functions of random
variables X and Y , respectively. Then which of the following would
NOT be a cumulative distribution function?

(A) H(x) = (F (x))2

(B) H(x) = (1 F ( x) + F (x)) /2

(C) H(x) = (F (x) + G(x))/2
p
(D) H(x) = G(x)

17. Suppose that X and Y are independent and identically distributed
exponential random variables with mean 1/ where > 0. Then
P (X > 4Y |X < 5Y ) equals

(A) 1/25

(B) 4/5

(C) e 5 /e 4

(D) e 4 e 5 / 1 e 4

18. Suppose X1 and X2 are independent and identically distributed
exponential random variables with mean 2. The expectation of
max{X1 , X2 } is

(A) 4 (B) 3.5 (C) 3 (D) 2.5

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19. Let X be a random variable. Let F (t) denote its cumulative dis-
tribution function and M (t) denote its moment generating function.
Consider the following two statements.
m
m F ( t) for t 2 R .co
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(I) F (t) = 1
s e m
s emM (t) = M ( t) for t 2 R
(II)
g l a
l a a
ag Which of the above statements are individually equivalent to random
variable X being symmetric about zero?

(A) Neither (I) nor (II)
(B) (II) but not (I)
(C) (I) and (II)
(D) (I) but not (II)
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20. Suppose that X1 , X2 , . . . are independent and identically distributed
exponential random variables with mean > 0. For n 1, define
Yn = X12 + X22 + · · · + Xn2 . Then

V (Yn )
lim =
n
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n!1

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(A) 2 2 (2 2 1)
e m
e m (B) 2 2
las
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21. The mean and the variance of a set of observations are 10 and 25
respectively. When two new observations are included in the set, both
the mean and the variance remain the same. Which of the following
is a correct statement?

(A) the two new observations are both 10
(B) the two new observations are 20 and 0
(C) the two new observations cannot be uniquely determined unless
the number of observations are known
(D) the two new observations are 15 and 5

22. Consider data on three variables X, Y and Z. Suppose least squares
regressions of Y on X, Y on Z and X on Z are performed. If the slope
coefficient in each of the three regressions is 2, what can be said about
the slope coefficient of the least squares regression of Y on (X + Z)?

(A) It is greater than 2
(B) It is less than 2
(C) It is equal to 2
(D) It cannot be determined whether it is more than 2 or less than 2
but it cannot be equal to 2

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23. Suppose that the probability of a diagnostic test detecting the presence
of an infection is 0.9 for a diabetic individual and 0.8 for a non-diabetic
individual. An individual is randomly selected from the population.
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What should be the ratio of the proportions of diabetic to non-diabetic
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individuals in the population so that the probability of the selected

s
individual
laa negative result is the same as that of the individual being non- a
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a diabetic given that the test produces a positive result?

(A) 4 : 3 (B) 2 : 1 (C) 3 : 2 (D) 5 : 3

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24. Consider independentarandom samples X , . . . , X and Y , . . . , Y
1 m 1 n
2 2
(m, n 2) from N (µ1 , ) and N (µ2 , ), respectively. Suppose we
want to test H0 : µ1 = µ2 vs H1 : µ1 6= µ2 at level of significance
↵ (0 < ↵ < 1). Then

(A) the two sample t-test statistic yields more power than the
ANOVA statistic for any non-zero value of µ1 µ2 and for any
m
m m, n.
.co
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lafor any m, n.
(B) the two sample t-test statistic and the ANOVA statistic yield the
s e g
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same power for any non-zero value of µ1 µ and
2

ag (C) the two sample t-test statistic yields less power than the ANOVA
statistic for any non-zero value of µ1 µ2 and for any m, n.

(D) the two sample t-test statistic and the ANOVA statistic will yield
the same power for any non-zero value of µ1 µ2 if m = n but
will not necessarily yield the same power if m 6= n.

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25. In order to compare between t( 2) treatments, one carries out t
replications of an experiment using a randomized block design with
t blocks. The error degrees of freedom for testing the equality of
treatment e↵ects in such a design is

(A) t + 1
(B) 3(t 1)
(C) (t 1)(t2 + t 1)
(D) t(t 1)(t + 1)

26. Let Xi be independently distributed Poisson(i ) random variables,
n
X
i = 1, . . . , n and > 0, is an unknown parameter. Assume Xi > 0.
i=1
Then the MLE of is
Pn
i=1 iXi
(A)
(n + 1)
Pn
i=1 Xi
(B)
n(n + 1)
2 ni=1 Xi
P
(C)
n(n + 1)
Pn
i=1 iXi
(D)
n

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27. Suppose X1 , X2 , X3 are independent normally distributed random
2
variables with mean µ and variance . However, instead of
X1 , X2 , X3 , we only observe Y1 = X2 X1 and Y2 = X3 X2 . Which
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2
of the following statistics is sufficient for ?
e m
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l as
s
(A) YY
ag
1 2

g la(B) Y + Y + 2Y Y
2 2

a (C) Y + Y
1 2 1 2

2 2
1 2

(D) Y12 + Y22 + Y1 Y2

o m
c
. (BMI) and fasting blood sugar
28. Consider data on body mass index
e m
l as
levels (fgl) for two groups of individuals. Suppose that the correlation
coefficient between BMIgand fgl is equal to 0.5 in each of the two
a
groups. If the two groups are combined, the correlation coefficient
between BMI and fgl:

(A) can be negative
(B) will be equal to 0.5

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(C) will be positive but not necessarily be equal to 0.5

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(D) can be zero but cannot be negative
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29. A random variable Z is obtained as follows. Let X ⇠ U (0, 1), and
Y given X = x be Bernoulli with probability of success x. If Y = 1,
Z is defined to be X. Otherwise, the experiment is repeated until a
pair (X, Y ) with Y = 1 is obtained. Then, the probability density
function of Z on (0, 1) is

(A) 2(1 z)
(B) 2z
(C) 12z 2 (1 z)
(D) 6z(1 z)

30. Suppose X1 , X2 , . . . , X9 are independent variables such that Xi is
distributed as Poisson with mean i, i = 1, 2 . . . , 9.
5
X 9
X
The conditional variance of Xi given Xi = 60 is
i=1 i=1

(A) 20 (B) 100/3 (C) 400/27 (D) 40/3

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