ISI Admission Test 2017 Question Paper M.Stat PSA

FOR ISI EXAM PREPARATION

ISI 2017
Question Paper · M.Stat
PSA
EXAM YEAR TYPE SUBJECT

ISI 2017 Question Paper M.Stat PSA

Notes · Sample Papers · Previous Year Papers · Mock Tests

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1. Let f : R → R be a nonconstant function satisfying f (x + 2) = f (x)
for every real x. Then lim f (x)
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x→∞

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(A) does not exist.
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(B) exists and equals +∞ or −∞.
e exists and is finite. l as
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l (D) may or may not exist depending on f .
(C)
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ag

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2. Let A be a square matrix with real entries such that A2 = A. Then
l a
10
ag
(A) (A + I) = I + 1024A.
(B) (A + I)10 = I + 511A.
(C) (A + I)10 = I + 1023A.
(D) (A + I)10 = I + 512A.

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m .co
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s e g la
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a 3. Let A be a 2 × 2 matrix such that trace(A) = det(A) = 3. What is
trace(A−1 )?

(A) 1/2 (B) 3 (C) 1 (D) 1/3

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4. Let A be a nonzero 2 × 2 matrix such that A3 = 0. Then which of the
following statements is always false?
m
om
(A) trace(A − A2 ) = 0 . co
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m
(B) trace(A + A2 ) = 0
e det(I + A) = 0 l as
as
l (D) A = 0
(C)
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ag 2

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a
5. The vertices A, B, C, D of a square are to be coloured with one of three
l
ag
colours red, blue, or green such that adjacent vertices get different
colours. What is the number of such colourings?

(A) 18 (B) 12 (C) 20 (D) 24

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a 6. Let S ⊂ R. Consider the statement: “If f is a continuous function
from S to S, then f (x) = x for some x.” This statement is true if S
equals

(A) [0, 1]. (B) (0, 1]. (C) R. (D) [−3, −2] ∪ [2, 3].

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7. The coordinates of the point at which the tangent of the curve
y = x2 − 1 makes an angle of 45◦ with the positive X-axis are
m
(A) ( om
√ √
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c m
5−1 1− 5
(B) ( 23 , − 59 ). (C) ( 43 , − 16
7
(D) ( 21 , − 34 ).
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2

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8. Let f : R → R be defined by

x2 sin 1


x if x 6= 0 ,
f (x) =
m
c. o
0 if x = 0 .

Then f is
e m
l as
ag and its derivative is continuous.
(A) continuous everywhere but not differentiable at 0.
(B) differentiable everywhere
(C) discontinuous at 0.
(D) differentiable everywhere and its derivative is discontinuous at 0.

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s e 3
g lawhat is the
la
9. If α, β, and γ are the roots of x − px + q = 0, then

g determinant of a
a
 
α β γ
 β γ α ?
 

γ α β

(A) p2 + 6q (B) 1 (C) p (D) 0

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10. Suppose the rank of  
1 1 2 2
m
co
 1 1 1 3 
 

. c om a b b 1
e m .
em
is 2 for some real numbers a and b. What is the value of b ?
l as
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l (A) ag
ag 1
3 (B) 3 (C) 1 (D) 12

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11. What is the infinite sum x + 2x2 + 3x3 + · · · for |x| < 1 ?
l a
x
(A) 1−x 2 (B)ag x
(1+x)2
1
(C) (1−x) 2
x
(D) (1−x) 2

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s e 12. A function f : R → R, such that f (x) = f (−x) for all x, has left
g la
g la derivative 5 at x = 0. Then, the right derivative of f at x = 0
a
a (A) exists and equals −5.
(B) may or may not exist.
(C) does not exist.
(D) exists and equals 5.

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13. Let f : R → R be defined by
b|x|c
m
co
X 1

om .
f (x) = ,
2k
. cbyc denotes the greatest integer less than or equal to y. Then,
k=0
e m
em
where
l as
s
la (A) f is a left continuous function which is not right continuous. ag
g
a (B) f is a continuous function.
(C) f is neither left continuous nor right continuous.
(D) f is a right continuous function which is not left continuous.

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14. A function f : R → R is differentiable at 0. Suppose that
a
f (x) < f (0) < f (y) for all x < 0 < y.

Then, what is the set of all possible values that f 0 (0) can take?

(A) {0} (B) [0, ∞) (C) (0, ∞) (D) R

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a 15. Suppose that the events A, B, and C are pairwise independent such
that each of them occurs with probability p. Assume that all three of
them cannot occur simultaneously. What is P (A ∪ B ∪ C) ?

(A) 1 − (1 − p)3 (B) p3 (C) 3p(1 − p) (D) 3p

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16. Let X1 , X2 , . . . be a sequence of independent random variables dis-
tributed exponentially with mean 1. Suppose that N is a random
m
variable, independent of the Xi -s, that has a Poisson distribution with
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mean λ > 0. What is the expected value of X1 + X2 + · · · + XN 2 ?
e m
em l as
l as(A) N 2 (B) λ + λ2 (C) λ2 (D) 1/λ2
ag
ag

17. For which value of k does the inequality
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Var(X − Y ) ≥ |σX − σY |k
m
s e
a
hold for all choices of random variables X and Y with finite variances
l
ag
2 and σ 2 , respectively?
σX Y

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

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18. Consider a diagnostic test for a disease that gives the correct result with
e m
s e probability 0.9, independently of whether the subject being diagnosed
g la
g la has the disease or not. Suppose that 20% of the population has the
a
a disease. For a particular subject, given that the test indicates presence
of the disease, what is the conditional probability that the subject has
the disease?

9 9
(A) 10 (B) 13 (C) 21 (D) 51

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19. Let X1 , X2 , X3 , X4 be independent standard normal random variables.
Then what is the distribution of
m
om
(X1 + X2 + X3 + X4 )2
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m
?
. c (X1 − X2 + X3 − X4 )2
s e
m
eF l a
l as(A) 4,4 (B) χ24 (C) F1,1 (D) F2,2
ag
ag

20. Suppose that X ∼ Uniform (0, 1) and Y ∼ Bernoulli (1/4), indepen-

m
dently of each other. Let Z = X + Y . Then,

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(A) the distribution of Z is symmetric
e function.
as
(B) Z has a probability density
l
(C) E(Z) = 5/4.
a g
(D) P (Z ≤ 1) = 1/4.

m
m 21. Assume that the events A and B are such that A ⊂ B and P (B) > 0.
.co
m .co For every event E, define 1E to be the random variable which takes
s e m
s e value 1 or 0 depending on whether E occurs or not, respectively. Then,
g la
g la Cov(1A , 1B ) equals
a
a (A) P (A|B)P (B c ).
(B) P (A)P (B c ).
(C) P (A)P (B).
(D) P (B)P (Ac ).

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22. Suppose that X is chosen uniformly from {1, 2, . . . , 100}, and given
X = x, Y is chosen uniformly from {1, 2, . . . , x}. What is P (Y = 30) ?
m
1
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m
(A) 100
. c s e
m
70 1

a
(B) 100 × 30
s e g l
a a
1
1 + 12 + 31 + · · · + 30 1

l (D)


(C) 100 ×

ag 1 1 1
100 × 30 + · · · + 100



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23. Three numbers are chosen at random from {1, 2, ..., 10} without re-
l a
ag
placement. What is the probability that the minimum of the chosen
numbers is 3 or their maximum is 7 ?

(A) 13
40
19
(B) 60 3
(C) 10 (D) 11
40

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a 24. Let X be a random variable taking values 1, 2, and 3 with respective
probabilities 73 , 27 , and 27 . Find the set of values of α for which E|X −α|
is minimized.

(A) [1, 2] (B) { 13
7 } (C) {2} (D) {1}

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25. Suppose that X1 , . . . , Xn are independent and identically distributed
normal random variables with mean θ and variance θ, where θ > 0.
m
Then,
o m . co
c
(A) . X is sufficient for θ.
n
e m
em as
X

l
i

s
la (B) X X is sufficient for θ.
i=1

ag
g
n

a
2
i
i=1
n
X
Xi + Xi2 is sufficient for θ.


(C)
i=1
n
X n
X
(D) neither Xi nor Xi2 is sufficient for θ.
i=1 i=1

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l a
g with mean µ and variance σ > 0.
a
26. Let X be normally distributed 2

What is the variance of e ? X

2 2
(A) eµ+σ (eσ − 1)
2 2
(B) e2µ+2σ (e2σ − 1)
2 2
(C) e2µ+2σ (eσ − 1)
2 2
(D) e2µ+σ (eσ − 1)
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a 27. Consider data from a single replication of a 2 factorial experiment.
Assume that all interactions of orders higher than two are negligible.
4

What is the error degrees of freedom in the ANOVA table?

(A) 16 (B) 2 (C) 12 (D) 5

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28. For n ≥ 2, let X1 , . . . , Xn be independent and identically distributed
normal random variables with mean 0 and variance σ 2 . Consider the
m
most powerful test for the null hypothesis H0 : σ = 2 against the
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alternative H1 : σ = 1, at significance level α for some 0 < α < 1.
e m
m
What is the power of this test?
e l as
as
l χ random variable with m degrees of freedom, and χ satisfies the
In what follows, F denotes the cumulative distribution function of a
m
ag
g
a equation F (χ ) = 1 − β, for every m ≥ 1 and β ∈ (0, 1).
2

2
2
m,β
m m,β

(A) 1 − Fn−1 ( 14 χ2n−1,α )
(B) Fn (4χ2n,1−α )
(C) Fn−1 (4χ2n−1,1−α )
(D) 1 − Fn ( 41 χ2n,α )

o m
c
. and identically distributed
m
e which depends on a parameter θ.
29. Suppose that X , . . . , X are independent
1 n

as
random variables from a distribution
l distributions is the maximum likelihood
a
For which of the followingg
estimator of θ not unbiased?

(A) Normal with mean θ and variance 1 where θ ∈ R
(B) Bernoulli with parameter θ ∈ [0, 1]
(C) Poisson with mean θ ≥ 0
(D) Normal with mean 0 and variance θ2 where θ > 0

o m
m . c
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30. Suppose that X , . . . , X are independent observations from a uniform
e
1 n
distribution on [0, θ], where θ ∈ N. Then, the maximumslikelihood
e m la
las estimator of θ
ag
ag (A) may not exist in some cases.
(B) is bX(n) c, where bac = greatest integer ≤ a.
(C) is dX(n) e, where dae = smallest integer ≥ a.


(D) is bX(n) c, if X(n) − bX(n) c < 1/2, otherwise it is dX(n) e.

om
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