ISI Admission Test 2018 Question Paper M.Stat PSA

FOR ISI EXAM PREPARATION

ISI 2018
Question Paper · M.Stat
PSA
EXAM YEAR TYPE SUBJECT

ISI 2018 Question Paper M.Stat PSA

Notes · Sample Papers · Previous Year Papers · Mock Tests

.
.co s e m

s em l a
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1. Let A be a 2 × 2 nonzero real matrix. Which of the following is true?

m
co
(A) A has a nonzero eigenvalue.

. c om
(B) A2 has at least one positive entry.
e m .
emtrace (A ) is positive.
2
l as
s
(C)

la (D) All entries of A cannot be negative. ag
g
2

a

o m
c
eigenvalue of A, the determinant of.A equals
2. Let A be a 3 × 3 real matrix with zero diagonal entries. If 1 + i is an

e m
l as
(B) g
(A) 4.
a −4. (C) 2. (D) −2.

o m
m c
3. Let A be an n × n matrix and let b be an n × 1 vector such that Ax. = b

m .co s emwhich
la
0
has a unique solution. Let A denote the transpose of A. Then
s e g
la a
of the following statements is false?

ag (A) A0 x = 0 has a unique solution.
(B) A0 x = c has a unique solution for any non-zero c.
(C) Ax = c has a solution for any c.
(D) A2 x = c is inconsistent for some vector c.

om
1
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m as
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4. Let A and B be n × n matrices. Assuming all the inverses exist,

m
co
(A−1 − B −1 )−1

equals co
m m .
m . s e
s e g l a
g a
l (B) A(B − A) B.
−1 −1
(A) (I − AB ) B.
a
a −1

(C) B(B − A)−1 A.
(D) B(A − B)−1 A.

m
.co
5. Let f be a function defined on (−π, π) as

e m
l as
f (x) = (| sin x| + | cos x|) · sin x.

Then f is differentiable g
a at
(A) all points.
(B) all points except at x = −π/2, π/2.
(C) all points except at x = 0.
(D) all points except at x = 0, −π/2, π/2.

m
m .co
m .co s em
s e g laat x = 1 is
la
2 3
6. The equation of the tangent to the curve y = sin (πx /6)

g a
a
√
3π
(A) y = 41 + 4 (x − 1).
√ √
(B) y = 43π x + 1− 4 3π .
√ √
(C) y = 43π x − 1− 4 3π .
√
(D) y = 14 − 43π (x − 1).

om
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7. Let f be a function defined from (0, ∞) to R such that

m
co
lim f (x) = 1 and f (x + 1) = f (x) for all x.

Then fco
m x→∞

m .
m . is
s e
s e g l a
g a
l (B) continuous but not necessarily bounded.
(A) continuous and bounded.
a
a
(C) bounded but not necessarily continuous.
(D) neither necessarily continuous nor necessarily bounded.

m
m .co
s e

la
8. The value of lim log xg
a
1/x
x→∞

(A) is e. (B) is 0. (C) is 1. (D) does not exist.

m
m .co
m .co s e m
s e g la
la
9. The number of real solutions of the equation,

g a
a x7 + 5x5 + x3 − 3x2 + 3x − 7 = 0

is

(A) 5. (B) 7. (C) 3. (D) 1.

om
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10. Let x be a real number. Then
 
m
co
lim lim cos2n (m!πx)
m m→∞ n→∞
.
c. o not exist for any x.
(A) does
s e m
s emexists for all x. g l a
la (C) exists if and only if x is irrational.
(B)

g a
a (D) exists if and only if x is rational.

11. Let {an }n≥1 be a sequence such that a1 ≤ a2 ≤ · · · ≤ an ≤ · · · .
Suppose the subsequence {a2n }n≥1 is bounded. Then

(A) {a2n }n≥1 is always convergent but {a2n+1 }n≥1 need not be con-

o m
vergent.
. c
}m are always convergent and have
(B) both {a } and {a
2n n≥1
s e 2n+1 n≥1

the same limit.
g a
l convergent.
(C) {a } a
is not necessarily
3n n≥1

(D) both {a2n }n≥1 and {a2n+1 }n≥1 are always convergent but may
have different limits.

12. Let {an }n≥1 be a sequence of positive numbers such that an+1 ≤ an
for all n, and lim an = a. Let pn (x) be the polynomial
n→∞

m
m pn (x) = x2 + an x + 1,
.co
m .co s e m
la
and suppose pn (x) has no real roots for every n. Let α and β be the
s e g
la
2

a
roots of the polynomial p(x) = x + ax + 1. Then

ag (A) α = β, α and β are not real.
(B) α = β, α and β are real.
(C) α 6= β, α and β are real.
(D) α 6= β, α and β are not real.

om
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13. Consider the set of all functions from {1, 2, . . . , m} to {1, 2, . . . , n},

m
where n > m. If a function is chosen from this set at random, what is

om
the probability that it will be strictly increasing?
. co
. c
/m . (B)
/n . (C) e m
e
(A)m n n n m m+n−1


/mn . (D) m+n−1

m
m−1 /n .
l as
s g
m m m

g l a a
a

o m
14. A flag is to be designed with 5 vertical stripes using some or all of the
c
. yellow. In how many ways can
m
four colours: green, maroon, red and
e stripes have the same colour?
as
this be done so that no two adjacent
l
(A) 576. (B)
g
a 120. (C) 324. (D) 432.

m
m .co
m .co s e m
s e 15. Suppose x , . . . , x are real numbers which satisfy
1 6
g la
g la Y a
a x = i x , for all i = 1, . . . , 6.
j6=i
j

How many choices of (x1 , . . . , x6 ) are possible?

(A) Infinitely many. (B) 2. (C) 3. (D) 1.

om
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16. Suppose X is a random variable with P (X > x) = 1/x2 , for all x > 1.
The variance of Y = 1/X 2 is
m
(A) 1/4.om . co
. c (B) 1/12. (C) 1. (D) 1/2.
em
em l as
las ag
ag

17. Let X ∼ N (0, σ 2 ), where σ > 0, and


−1 if X ≤ −1,
m



.co
Y = X if X ∈ (−1, 1),

mif X ≥ 1.


e

s is correct?
1

l a
ag
Which of the following statements

(A) Var(Y ) = Var(X).
(B) Var(Y ) < Var(X).
(C) Var(Y ) > Var(X).
(D) Var(Y ) ≥ Var(X) if σ ≥ 1, and Var(Y ) < Var(X) if σ < 1.

m
m .co
m .co s e m
s e g la
g la a
a 18. If a fair coin is tossed 5 times, what is the probability of obtaining at
least 3 consecutive heads?

(A) 1/8. (B) 5/16. (C) 1/4. (D) 3/16.

om
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19. Let X and Y be random variables with mean λ. Define

min(X, Y ) with probability 1 ,
m
co
2

om .
Z=
max(X, Y ) with probability 1 .
. c 2
e m
em is E(Z)? l as
sWhat

la (A) λ. ag
ag (B) 4λ/3. (C) λ2 . (D)
√
3λ/2.

20. A finite population has N (≥ 10) units marked {U1 , . . . , UN }. The

m
.co
following sampling scheme was used to obtain a sample s. One unit

m
is selected at random: if this is the i-th unit, then the sample is s =

s e
{Ui−1 , Ui , Ui+1 }, provided i 6∈ {1, N }. If i = 1 then s = {U1 , U2 } and
Ua }. The probability of selecting U in s is
,l
ag
if i = N then s = {U N −1 N 2

(A) N2 . (B) N3 . (C) (N 1−2) + N2 . (D) (N 3−2) .

o m
c
21. Suppose X1 , . . . , Xn are i.i.d. observations from a distribution assum-
m .
.co m
ing values −1, 1 and 0 with probabilities p, p and 1 − 2p, respectively,
Q
s e = 1),
em
1 n

b = P (Z = −1), c = P (Z = 0). Then as n → ∞, la
where 0 < p < . Define Z =
2 X and a = P (Z
n i=1 i n n

las n n n n
ag
ag (A) an → 41 , bn → 21 , cn → 14 .
(B) an → 31 , bn → 31 , cn → 13 .
(C) an → 0, bn → 0, cn → 1.
(D) an → p, bn → p, cn → 1 − 2p.

om
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22. Suppose X1 , X2 and X3 are i.i.d. positive valued random variables.
Xi

m
Define Yi = X1 +X 2 +X3
, i = 1, 2, 3. The correlation between Y1 and Y3
is
om . co
. c e m
e m
(A) 0. (B) −1/6. (C) −1/3. (D) −1/2.
l as
l as ag
ag

23. Assume (yi , xi ) satisfies the linear regression model,

yi = βxi + i , for i = 1, . . . , n,
m
.co
where, β ∈ R is unknown, {xi : 1 ≤ i ≤ n} are fixed constants and
m
e
{i : 1 ≤ i ≤ n} are i.i.d. errors with mean zero and variance σ 2 ∈
s
l a
(0, ∞). Let βb be the least squares estimate of β and ybi = βx
g each n ≥ 1, define
b i be the
predicted value of y . a
Fori

n
1 X
an = Cov(yi , ybi ).
σ2
i=1

Then,

(A) an = 1. (B) an ∈ (0, 1). (C) an = n. (D) an = 0.
m
m .co
m .co s e m
s e g la
g la a
a 24. Let X and Y be two random variables with E(X|Y = y) = y 2 , where
Y follows N (θ, θ2 ), with θ ∈ R. Then E(X) equals

(A) θ. (B) θ2 . (C) 2θ2 . (D) θ + θ2 .

om
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25. Suppose X is a random variable with finite variance. Define X1 = X,

m
X2 = αX1 , X3 = αX2 , . . . , Xn = αXn−1 , for 0 < α < 1. Then

om
Corr(X1 , Xn ) is
. co
. c e m
em
(A) αn . (B) 1. (C) 0. (D) αn−1 .
l as
l as ag
ag

26. Let X be a random variable with P (X = 2) = P (X = −2) = 1/6 and
m
.co
P (X = 1) = P (X = −1) = 1/3. Define Y = 6X 2 + 3. Then

e m
(A) Var(X − Y ) < Var(X).
l as
a g Y ).
(B) Var(X − Y ) < Var(X +
(C) Var(X + Y ) < Var(X).
(D) Var(X − Y ) = Var(X + Y ).

m
m .co
m .co s em
s e 27. Suppose X is a random variable on {0, 1, 2, . . .} with g la p.m.f.
la a unknown

ag p(x). To test the hypothesis H : X ∼ Poisson(1/2) against H :
0 1

p(x) = 2−(x+1) for all x ∈ {0, 1, 2, . . .}, we reject H 0 if x > 2. The
probability of type-II error for this test is

(A) 14 . (B) 1 − 13
8 e
−1/2 . (C) 1 − 23 e−1/2 . (D) 78 .

om
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28. Let X be a random variable with

Pθ (X = −1) =
(1 − θ) 1
, Pθ (X = 0) = , and Pθ (X = 1) =
θ
o m
m 2 2 2
. c
for 0 <cθo< 1. In a random sample of size 20, the observed frequencies m
ofem
. s e
s −1, 0 and 1 are 6, 4 and 10, respectively. The maximum likelihood
g l a
g la estimate of θ is a
a
(A) 1/5. (B) 4/5. (C) 5/8. (D) 1/4.

o m
c
. Suppose there are no ties and
29. Two judges evaluate n individuals, with (Ri , Si ) the ranks assigned to

e m
the i-th individual by the two judges.
S = R + 1, for i = 1, . . a . ,s
i i
g l between the two evaluations is 0, what is
(n − 1), and S = 1 if R = n. If the
i i

a
Spearman’s rank correlation
the value of n?

(A) 7. (B) 11. (C) 4. (D) 5.

m
m .co
m .co s e m
e la
30. Let X1 , X2 , . . . be a sequence of i.i.d. random variables with variance 2.

las Then for all x, ! g
a
ag lim P √
n→∞
1 X
n
(−1) X ≤ x
n
i
i
i=1

equals

√ √

(A) Φ(x 2). (B) Φ x/ 2 . (C) Φ(x). (D) Φ(2x).

om
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