ISI Admission Test 2020 Question Paper M.Stat PSA

FOR ISI EXAM PREPARATION

ISI 2020
Question Paper · M.Stat
PSA
EXAM YEAR TYPE SUBJECT

ISI 2020 Question Paper M.Stat PSA

Notes · Sample Papers · Previous Year Papers · Mock Tests

.
.co s e m

s em l a
a ag

1. How many distinct straight lines can one form that are given by an
equation ax + by = 0, where a and b are numbers from the set
m
{0, 1, 2, 3, 4, 5, 6, 7}?
m . co
c. o
(A) 63.
s e m
s em57.
(B)
g l a
g la (C) 37. a
a (D) 49.

m
.co
2. Which of the following sequences is not monotone?
m
s e
(A) an = n2 − 3n,
l a n ≥ 1.

ag 1.
2
(B) a = 3n − n, n ≥
n

(C) an = log( 43 )n , n ≥ 1.
(D) an = 7n − n2 , n ≥ 1.

m
m .co
m .co s e m
e la
3. Let A = (1, −1), B = (−2, 0), C = (1, 2) and D be the vertices of a

las ag
parallelogram in the X-Y plane listed clockwise. Then the point D is:

ag (A) (4, 1).
(B) (−2, −3).
(C) (3, 0).
(D) (−2, 1).

1
m . c
c. o s e m
s em
For more Question Papers, Sample Papers, Notes & Syllabus visit
gl a
Page 1 of 22

ROUGH WORK

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 2 of 22

.
.co s e m

s em l a
a ag

4. Let f be the function on the set of real numbers defined by

m
co
ex sin x cos x

om .
f (x) = e2x sin 2x cos 2x .

. c em
s
e3x sin 3x cos 3x
m
e l a
l as
Then f 0 (0) is
ag
g
a (A) 1. (B) 0. (C) −1. (D) 2.

m
m .co
s e
5. The period of the function f given by f (x) = sin( x3 − π2 ) , where x is
real, is:
g l a
(A) 6π.
a
(B) 9 .π
(C) 2π. (D) 13 π2 .
2

m
m .co
m .co s e m
e la
6. Let ! !
s
√

la X=
1 3 1
√ and A =
2 −1
a
.g
ag 2 −1 3 3 −2
Let X t denote the transpose of the matrix X. If B = XAX t , then
X t B 105 X equals

(A) I. (B) A. (C) X t AX. (D) B.

2
m . c
c. o s e m
s em
For more Question Papers, Sample Papers, Notes & Syllabus visit
gl a
Page 3 of 22

ROUGH WORK

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 4 of 22

.
.co s e m

s em l a
a ag

7. Suppose

m
n
X

om
aj (n) = kj , j = 0, 1, 2, 3, n = 1, 2, . . . .
. co
c m
k=1

m . s e
s e
Then limn→∞ aa10 (n)a 2 (n)
(n)a3 (n) equals
g l a
g a
l (A) 0. (D) 1. a
a
(B) 13 . (C) 32 .

m
8. Let S be the set of all 3 × 3 real matrices A = ((aij )) such that the matrix
((a3ij )) has rank one. Let R be the set
m .co
s e
l a R = {rank(A) : A ∈ S}.

Then R is equal to ag

(A) {1}. (B) {1, 2}. (C) {1, 3}. (D) {1, 2, 3}.

m
m .co
m .co 9. Consider the circles
s e m
s e S : x + y + 4x + 2y − 4 = 0, g
la
la a
2 2

g
1

a S2 : x2 + y 2 − 4x − 4y + 4 = 0.

Then the number of common tangents of S1 and S2 is

(A) 1. (B) 2. (C) 3. (D) 4.

3
m . c
c. o s e m
s em
For more Question Papers, Sample Papers, Notes & Syllabus visit
gl a
Page 5 of 22

ROUGH WORK

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 6 of 22

.
.co s e m

s em l a
a ag

10. Let A be the 3 × 3 matrix
 
m
co
1 2 4

om .
 
A = 0 0 3 

. c em
s
0 0 −1

e m l a
l as
Then the determinant of the matrix A17 + A10 − I is
ag
g
a (A) 1. (B) 2. (C) −1. (D) 0.

m
11. Let m and n be nonzero integers. Define
m .co

s e
a
Am,n = x ∈ R : n2 x3 + 2020x2 + mx = 0 .
l
ag (m, n) for which A
Then the number of pairs m,n has exactly two points
is

(A) 0. (B) 10. (C) 16. (D) ∞.

m
m .co
m .co s e m
s e a
12. The set of all real solutions of the inequality 2|x| > |x −l 1| is
g
g la a
a

(A) x : x < − 31 ∪ {x : x > 1}.

(B) x : x > 31 .
(C) {x : −1 < x < 1}.

(D) {x : x < −1} ∪ x : x > 31 .

4
m . c
c. o s e m
s em
For more Question Papers, Sample Papers, Notes & Syllabus visit
gl a
Page 7 of 22

ROUGH WORK

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 8 of 22

.
.co s e m

s em l a
a ag

13. Let S and T be two non-empty sets and f : S → T be a function such
that for all subsets A and B of S, we have f (A ∩ B) = f (A) ∩ f (B).
m
Then
m . co
c. o exists an A ⊆ S such that f (f (A)) 6= A.
(A) There −1

s e m
s emThere exist disjoint subsets A and B of S such that
(B)
g l a
g la f (A) ∩ f (B) 6= φ. a
a (C) f is injective.
(D) None of the above statements is true.

14. Let S be the set of all 3 × 3 matrices A such that among the 9 entries

m
of A, there are exactly three 0’s, exactly three 1’s and exactly three 2’s.

.co
The number of matrices in S that have trace divisible by 3 is
m
s e
(A) 580.
l a
(B) 600. (C) 150. (D) 120.

ag
15. Let x1 , . . . , xn ∈ R be distinct reals. Define the set
n
o
A = f1 (t), . . . , fn (t) : t ∈ R ,

where for 1 ≤ k ≤ n
m
.co

m 1

.co
if xk ≤ t,
fk (t) =
0
e m
e m
otherwise.

las
las Then A contains
ag
ag (A) exactly n distinct elements.
(B) exactly (n + 1) distinct elements.
(C) exactly 2n distinct elements.
(D) infinitely many distinct elements.

5
m . c
c. o s e m
s em
For more Question Papers, Sample Papers, Notes & Syllabus visit
gl a
Page 9 of 22

ROUGH WORK

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 10 of 22

.
.co s e m

s em l a
a ag

16. Let X, Y be independent and identically distributed random variables
with
m
o
P (X = k) = (0.75)(0.25)k ,
m> Y ) is equal to
k = 0, 1, 2, . . . .
. co
Then Pc(X
. e m
m
e 0.5 l as
l as(A) (B) 0.4 (C) 0.2 (D) 0.6
ag
ag

17. Let X1 , X2 , . . . be independent and identically distributed Bin(n, p) ran-
dom variables. Define

Sk = X12 + X22 + · · · + Xk2 ,
o m k = 1, 2, . . . ,
m .c
and let ε > 0. Then as k → ∞,
s e
 
l a
ag
(A) P Skk − n2 p2 > ε → 0 for all p ∈ (0, 1).
 
(B) P Skk − np2 > ε → 0 for all p ∈ (0, 1).
 
(C) P Skk − n4 > ε → 0 when p = 1/2.
 
(D) P Skk − n(n+1)
4 > ε → 0 when p = 1/2.

m
m .co
m .co 18. Let X, Y have joint probability density function
s e m
s e 
g la
g la  1 ye−xy if x > 0, 2 ≤ y ≤ 4,
f (x, y) = 2 a
a 0 otherwise.

Then E(XY ) is given by

(A) 3. (B) 1. (C) 2. (D) 1.5

6
m . c
c. o s e m
s em
For more Question Papers, Sample Papers, Notes & Syllabus visit
gl a
Page 11 of 22

ROUGH WORK

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 12 of 22

.
.co s e m

s em l a
a ag

19. Let X and Y be random variables with V (X) = 9 and V (Y ) = 4. Then
which of the following is true for non negative reals a and b?
m
om
(A) V (aX + bY ) = 0 if and only if a = b = 0.
. co
. c em
m
(B) V (aX + bY ) lies between (3a − 2b)2 and (3a + 2b)2 .
e V (aX + bY ) lies between 3a and 2b. l as
as(C)
l (D) V (aX + bY ) lies between (3a) and (2b) . ag
ag 2 2

m
.co
20. The moment generating function of a continuous random variable X

m
is given by M (t) = et(t+1) , −∞ < t < ∞. Let Φ denote the cumula-

s e
tive distribution function of a standard normal random variable. Then
P (X ≥ 0) is
g l a
(A) Φ(0).
a
(B) Φ(− ). √1 (C) Φ( √12 ). (D) Φ( 12 ).
2

o m
m . c
m .co e m
21. A multiple choice test uses the following scoring procedure to discourage
students from guessing. For each correct response the scoresis 5, for each
s e incorrect response the score is 0 and for no responsegthelascore is 2. If
g la a
a there are 4 choices for each question, what is the minimum number of in-
correct choices that the student must eliminate before it is advantageous
to guess among the rest than leave the question unanswered?

(A) 0. (B) 1. (C) 2. (D) 3.

7
m . c
c. o s e m
s em
For more Question Papers, Sample Papers, Notes & Syllabus visit
gl a
Page 13 of 22

ROUGH WORK

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 14 of 22

.
.co s e m

s em l a
a ag

22. Each of 100 laboratory rats has available plain water and a mixture of
water and caffeine in their cages. After 24 hours, two measurements
m
were recorded for each rat: the amount of caffeine consumed (X) and
om . co
. c
the blood pressure (Y ). Based on the data, the correlation coefficient
e m
m
between X and Y was found to be 0.428. Which of the following con-
e l as
as clusions is justified on the basis of the study?
l (A) The correlation between caffeine consumed and blood pressure in ag
ag
the population of rats is 0.428.
(B) If rats stop drinking the mixture of water and caffeine, their blood
presure will go down.
(C) Rats with low blood pressure do not consume the mixture of water
and caffeine as much as rats with high blood pressure.

m
(D) About 18% of variation in blood pressure can be explained by a

.co
linear relationship between blood pressure and caffeine consumed.

m
s e
l a
ag
23. Suppose X is distributed as Bin(3, 32 ) and the conditional distribution
of Y given X is N (X, X 2 + 1). Then the variance of Y is

(A) 19
3 . (B) 17
3 . (C) 13
3 . (D) 11
3 .

m
m .co
m .co s e m
e la
24. Suppose X and Y are independent observations from N (0, 1). For t > 0,

las define g(t) = P (|X − Y | > t | (X + Y ) > t). Then
ag
ag (A) g is a strictly increasing function.
(B) g is a strictly decreasing function.
(C) g is a constant function.
(D) g is not a monotonic function.

8
m . c
c. o s e m
s em
For more Question Papers, Sample Papers, Notes & Syllabus visit
gl a
Page 15 of 22

ROUGH WORK

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 16 of 22

.
.co s e m

s em l a
a ag

25. Let X1 , X2 , . . . , Xn be independent and identically distributed N (1, σ 2 )
random variables. Let σ c2 denote the maximum likelihood estimator of
m
σ 2 . Then
cco
mP (X − X̄) and it is biased. . co
. 1 n 2
e m
s
(A) σ =2

m
i

a
n i=1

s e
(B) c
σ = 2
P 1 n
(X − X̄) and it is unbiased. 2
g l
la (C) σc = a
n−1 i=1 i

g
P 1 n

a (D) σc = P (X − 1) and it is unbiased.
2 (X − 1) and it is biased. 2
n−1 i=1 i

2 1 n 2
n i=1 i

m
m .co
26. Let X1 , X2 , . . . , Xn be independent and identically distributed exponen-

s e
tial random variables with mean λ. Then maximum likelihood estimator

g l a
of the median of the distribution is

(A) Sample median.
a (B) (log 2)X̄. (C) log 2/X̄. (D) X̄/ log 2.

o m
m c
. with
.co e m
27. Suppose X is a discrete random variable taking values −1, 0, 1, each
s
em a
1

l
probability . Let Y = |X|. Which one of the following statements is
s
3

l a correct?
ag
ag (A) Correlation between X and Y is ±1.
(B) X and Y are positively correlated.
(C) X and Y are negatively correlated.
(D) X and Y are uncorrelated.

9
m . c
c. o s e m
s em
For more Question Papers, Sample Papers, Notes & Syllabus visit
gl a
Page 17 of 22

ROUGH WORK

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 18 of 22

.
.co s e m

s em l a
a ag

28. Let X1 , X2 , . . . , Xn be independent and identically distributed random
variables with probability density function
m
om
1 −| x−1 |
. co
m
f (x) = e λ , −∞ < x < ∞,
. c 2λ
s e
e m l a
s
where λ > 0 is an unknown parameter. Which one of the following is
a
l H : λ = 1?
a form of the most powerful test of its size for testing H : λ = 2 vs
ag
g
0

a 1
P n 2
(A) Reject H0 if i=1 (Xi − 1) > C.
Pn
(B) Reject H0 if i=1 |Xi − 1| > C.
Pn 2 < C.
(C) Reject H0 if i=1 (Xi − 1)
Pn
(D) Reject H0 if i=1 |Xi − 1| < C.

o m
c
. and identically distributed
e
29. Suppose X and Y are independent m
continuous random variables s
±1) = 1/2. Then whichg
a
ofl the following statements is always true?
and are independent of Z. Assume P (Z =

a
(A) X + Y has the same distribution as Z(X + Y ).
(B) X − Y has the same distribution as Z(X − Y ).
(C) X/Y has the same distribution as Z(X/Y ).
(D) XY has the same distribution as Z(XY ).

o m
m c
30. Suppose T and T are two different real valued statistics each of.which

m .co 1 2
is sufficient for an unknown parameter θ. Then, which of s emfollowing
e la
the

las is always true?
a g
ag (A) T1 + T2 is sufficient for θ.
(B) max{T1 , T2 } is sufficient for θ.
(C) (T1 , T2 ) is sufficient for θ.
(D) min{T1 , T2 } is sufficient for θ.

10
m . c
c. o s e m
s em
For more Question Papers, Sample Papers, Notes & Syllabus visit
gl a
Page 19 of 22

ROUGH WORK

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 20 of 22

.
.co s e m

s em l a
a ag

ROUGH WORK

m
om . co
. c e m
em l as
l as ag
ag

m
m .co
s e
l a
ag

m
m .co
m .co s e m
s e g la
g la a
a

m . c
c. o s e m
s em
For more Question Papers, Sample Papers, Notes & Syllabus visit
gl a
Page 21 of 22

ROUGH WORK

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 22 of 22