ISI Admission Test 2016 Syllabus and Sample Paper M.Stat PSA and PSB

FOR ISI EXAM PREPARATION

ISI 2016
Syllabus and Sample
Paper · M.Stat PSA and
PSB
EXAM YEAR TYPE SUBJECT

ISI 2016 Syllabus and Sample Paper M.Stat PSA and PSB

Notes · Sample Papers · Previous Year Papers · Mock Tests

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TEST CODE: PSA (Objective type) and PSB (Short answer type) 2016

SYLLABUS
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Mathematics
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s emgeometry:
Arithmetic, geometric and harmonic progressions. Trigonometry. Two dimensional
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lElementary
coordinate
g
Straight lines, circles, parabolas, ellipses and hyperbolas.
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amutations and combinations, Binomial and multinomial theorem.
set theory. Functions and relations. Elementary combinatorics: Per-

Theory of equations.
Complex numbers and De Moivre’s theorem.
Vectors and vector spaces. Algebra of matrices. Determinant, rank, trace and
inverse of a matrix. Solutions of linear equations. Eigenvalues and eigenvectors of
matrices.
Limits and continuity of functions of one variable. Differentiation. Leibnitz for-

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mula. Applications of differential calculus, maxima and minima. Taylor’s theorem.
Indefinite integral. Fundamental theorem of calculus. Riemann integration and prop-
erties. Improper integrals.
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Statistics and Probability
Notions of sample space and probability. Combinatorial probability. Conditional
probability and independence. Bayes Theorem. Random variables and expectations.
Moments and moment generating functions. Standard univariate discrete and con-
tinuous distributions. Distribution of functions of a random variable. Distribution of
order statistics. Joint probability distributions. Marginal and conditional probability
distributions. Multinomial distribution. Bivariate normal and multivariate normal

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distributions.

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Sampling distributions of statistics. Statement and applications of Weak law of

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large numbers and Central limit theorem.

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Descriptive statistical measures. Contingency tables and measures of association.

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Product moment and other types of correlation. Partial and multiple correlation.
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Simple and multiple linear regression.
a Elementary theory of estimation (unbiasedness, minimum variance, sufficiency).
Methods of estimation (maximum likelihood method, method of moments). Tests
of hypotheses (basic concepts and simple applications of Neyman-Pearson Lemma).
Confidence intervals. Inference related to regression. ANOVA. Elements of nonpara-
metric inference.

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Basic experimental designs such as CRD, RBD, LSD and their analyses. Ele-
ments of factorial designs. Conventional sampling techniques (SRSWR/SRSWOR)
including stratification. Ratio and regression methods of estimation.

SAMPLE QUESTIONS: PSA
Note: For each question there are four suggested answers of which only one is correct.

1. The number of functions f : {1, 2, . . . , 10} → {1, 2, . . . , 10} such that f (x) 6= x
for all x is

(A) 10! (B) 910 (C) 109 (D) 1010 − 1.

2. The set of all ordered pairs of real numbers (x, y) satisfying satisfying y 2 − 2y −
x2 + 4x = 3 is a

(A) circle (B) point (C) hyperbola (D) pair of straight lines.

3. Let
log(2 + x) − x2n sin x
f (x) = lim for x > 0.
n→∞ 1 + x2n
Then
(A) f is continuous at x = 1
(B) lim f (x) 6= lim f (x)
x→1+ x→1−
(C) lim f (x) = sin 1
x→1+
(D) lim f (x) does not exist.
x→1−

4. Suppose a real matrix A satisfies A3 = A, A 6= I, A 6= 0. If Rank(A) = r and
Trace(A) = t, then
(A) r ≥ t and r + t is odd
(B) r ≥ t and r + t is even
(C) r < t and r + t is odd
(D) r < t and r + t is even.

5. Let
1 1
f (x) = x2 + 2
+x+ , x>0
x x
and let m = min{f (x)}. Then

(A) m = 1 (B) m = 4 (C) m = 27/4 (D) m does not exist.

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6. Let f be a convex function, i.e.,

f (tx + (1 − t)y) ≤ tf (x) + (1 − t)f (y)

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for all 0 ≤ t ≤ 1 and x, y ∈ R. Then which of the following is necessarily true?

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(A) 2f (0) + f (4) ≥ 2f (1) + f (2)
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(B) f g is a convex function whenever g is convex
e f is nondecreasing l as
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l (D) none of these. ag
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a 7. Suppose A is a 100 × 100 real symmetric matrix whose diagonal entries are all
positive. Then which of the following is necessarily true?
(A) All eigenvalues of A are greater than 0
(B) no eigenvalue of A is greater than 0
(C) at least one eigenvalue of A is greater than 0
(D) none of these.

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8. The integral
c
sin.x
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Z 1

s e x dx α

a
0

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(A) is finite only for α =a0
(B) is finite only for |α| < 1
(C) is finite for all α < 2
(D) is infinite for any value of α.

9. Given θ in the range 0 ≤ θ < π, the equation

2x2 + 2y 2 + 4x cos θ + 8y sin θ + 5 = 0
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represents a circle for all θ in the interval

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(A) 0 < θ < π/3
a
(B) π/4 < θ < 3π/4
(D) 0g≤l θ < π.
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(C) 0 < θ < π/2

ag 10. How many 5 × 5 matrices are there such that each entry is 0 or 1 and each row
sum and each column sum is 4?

(A) 64 (B) 32 (C) 120 (D) 96.

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π k
11. For n ≥ 1, let Gn be the geometric mean of {sin( · ) : 1 ≤ k ≤ n}. Then
2 n
lim Gn is
n→∞

(A) 1/4 (B) log 2 (C) 12 log 2 (D) 1/2.

12. Suppose a, b, x, y are real numbers such that a2 + b2 = 81, x2 + y 2 = 121 and
ax + by = 99. Then the set of all possible values of ay − bx is

     
9 9 9
(A) {0} (B) 0, (C) 0, (D) ,∞ .
11 11 11

13. In a triangle with sides of length a, b, c, suppose b + c = x and bc = y. If also
(x + a)(x − a) = y, then the triangle is necessarily

(A) equilateral (B) right angled
(C) acute angled (D) obtuse.

14. Three distinct squares are selected at random from a 8 × 8 chess board. Then
the probability that they form an L-shaped pattern (looked at from one fixed
side only) as drawn below is

196 49 36
(A) 64

(B) 64

(C) 64

(D) greater than 1/2.
3 3 3

15. Suppose X is distributed as Poisson with mean λ. Then E(1/(X + 1)) is

eλ − 1 1 1 − e−λ 1 − e−λ
(A) (B) (C) (D) .
λ λ+1 λ λ+1

16. A permutation of 1, 2, . . . , 100 is chosen at random. Then the probability that
the numbers 1 and 100 appear next to each other equals

(A) 1/100 (B) 1/50 (C) 1/99 (D) 1/98.

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17. There are 10 boxes each containing 6 white and 7 red balls. Two different boxes
are chosen at random, one ball is drawn simultaneously at random from each
and transferred to the other box. Now a box is again chosen from the 10 boxes
and a ball is chosen from it. Then the probability that this ball is white is
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(A) 6/13 om . co
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emnumber of cars (X) arriving at a service station per day is a random variable glas
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lawith
18. The
a
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mean 4. The service station can provide service to a maximum of 4 cars
a per day.
equals
Then the expected number of cars per day that do not get serviced

∞
X ∞
X
(A) 4 (B) 0 (C) iP (X = i + 4) (D) iP (X = i − 4).
i=0 i=4

19. Suppose X1 and X2 are independent random variables distributed as Ber(p1 )
and Ber(p2 ) respectively. Then Y = max(X1 , X2 ) is distributed as a Bernoulli
random variable with success probability
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(A) 1 − p p (B) p + p − p p (C) .max{p , p } (D) min{1 − p , 1 − p }.
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1 2 1 2 1 2 1 2 1 2

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20. Assume that (X, Y ) is bivariateaNormal with E(X) = E(Y ) = 0, Var(X) = σ , 2

Var(Y ) = σ and Cor(X, a
1
2
2 Y ) = ρ for some ρ ∈ (−1, 1). The probability that X
is larger than Y is

   
σ1 −σ2 σ12 +σ22
(A) 1/2 (B) Φ √ (C) Φ √ (D) none of the above.
1−ρ2 1−ρ2

21. Suppose that the bivariate data (x1 , y1 ), . . . , (xn , yn ) lie on the straight line
y = a + bx for some a, b ∈ R. Assume further that neither all the xi ’s are same,
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nor are all the yi ’s. Which of the following values is not a possibility for the

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correlation coefficient calculated from the above data?

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(C) 0
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(D) None of the above.

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a 22. In the randomised block design for ANOVA where k is the number of treatments
and b is the number of blocks, the degrees of freedom for error is given by

(A) bk − 1 (B) kb + 1 (C) (b − 1)(k − 1) (D) k + b − 1.

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23. Suppose X1 , X2 , . . . , Xn is a random sample from a population with mean µ
and finite variance. Consider the following two estimators of µ2 :

1 XX
E1 = Xi Xj
n(n − 1)
1≤i6=j≤n
1 X 2
E2 = X̄ 2 − (Xi − X̄) .
n(n − 1)
1≤i≤n

Then,

(A) E1 is unbiased but E2 is biased
(B) E1 is biased but E2 is unbiased
(C) both E1 and E2 are unbiased
(D) both E1 and E2 are biased.

24. It is known that the proportion of smokers (p) in a population lies in the interval
[1/3, 2/3]. In a random sample of N individuals selected from the population, it
was found that M were smokers. The maximum likelihood estimate of p based
on the above data is

(A) max{1/3, M/N }
(B) min{M/N, 2/3}
(C) M/N
(D) none of the above.

25. Suppose that the least squares linear regression equation of y on x is y = a + bx
and that of x on y is x = c + dy. If it is known that b 6= 0 and d 6= 0, then the
ratio of the standard deviation of x to the standard deviation of y

(A) is √1bd
p
(B) is d/b
p
(C) is b/d
(D) cannot be determined from the above information.

26. Let bxc denote the largest integer not larger than x. If X is distributed as
N (0, 1), then E(bXc)

(A) is 0 (B) is 0.5 (C) is −0.5 (D) does not exist.

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27. X and Y are i.i.d. random variables with finite variances. Then

(A) V ar(XY ) = V ar(X)V ar(Y )
(B) V ar(XY ) ≥ V ar(X)V ar(Y )
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(C) V ar(XY ) ≤ V ar(X)V ar(Y )
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(D) none coof the above is necessarily true. s e m
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ladensity
28. Suppose
a
1 2 n

ag f (x) =
α
x e
p
,x > 0p−1 −αx
Γ(p)
where p is a known positive constant and α > 0 is an unknown parameter. Let
α̂ = p/X̄ be a proposed estimator of α. Then,

(A) E(α̂) = α
(B) E(α̂) = 1−α 1
np
α
(C) E(α̂) = 1−np
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(D) none of the above statements is true.

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s identical means.
29. The sign test is a nonparametric procedure for testing

a
(A) whether two populationslhave
ag have identical medians.
(B) whether two populations
(C) whether two populations have identical probability distributions.
(D) whether two populations are independent.

30. Let X1 , . . . , Xn be i.i.d. from N (0, σ 2 ). What is the form of the most powerful
test for testing the null hypothesis

H0 : σ = 1 ,
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(A) Reject H0 if i=1 Xi > c

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Pn2
i=1 Xi < c a
a (C)
Pn
Reject H0 if i=1 (Xi − X̄)2 > c
Pn 2
(D) Reject H0 if i=1 (Xi − X̄) < c.

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SAMPLE QUESTIONS: PSB

1. In the diagram below, L(x) is a straight line that intersects the graph of a
polynomial P (x) of degree 2 at the points A = (−1, 0) and B = (5, 12). The
area of the shaded region is 36 square units.
Obtain the expression for P (x).

L(x)

B

A
P (x)

2. Let f : [−1, 1] → R be a continuous function. Suppose that f 0 (x) exists and
f 0 (x) ≤ 1 for all x ∈ (−1, 1). If f (1) = 1 and f (−1) = −1, prove that

f (x) = x for all x ∈ [−1, 1].

3. Suppose A is an n × n real symmetric matrix such that

T r(A2 ) = T r(A) = n.

Show that all the eigenvalues of A are equal to 1.

4. For each c ∈ R, define a function Tc : R4 → R4 by




Tc x1 , x2 , x3 , x4 := (1 + c)x1 , x2 + cx3 , x3 + cx2 , (1 + c)x4 .

For every c ∈ R, find the dimension of the null space of Tc .

5. A box contains 50 red balls, 30 green balls and 20 blue balls. Suppose balls are
drawn successively at random with replacement from the box. Let N denote
the minimum number of draws required to obtain balls of all three colours.
Compute P (N > n) for all positive integers n.

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6. Let X1 , . . . , Xn be i.i.d. random variables from a continuous distribution whose
density is symmetric around 0. Suppose E(|X1 |) = 2. Define
n n

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X X
Y = Xi and Z= 1(Xi > 0) .

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i=1 i=1

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Calculate o
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7. Suppose

laon (x , . . . , x ) based on the first n observations is the same as that based on a
variate
i 1i
observations.
2i
It is found
ki
that the least squares
1
linear
2
regression fit of y g
g
a the remaining n observations, and is given by
1 k 1

2

k
X
y = β̂0 + β̂j xj .
j=1

If the regression is now performed using all (n1 + n2 ) observations, will the
regression equation remain the same? Justify your answer.

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8. Suppose that (X1 , Y1 ), (X2 , Y2 ), . . . , (Xn , Yn ) are the coordinates of n points

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chosen independently and uniformly at random within a circle with centre (0, 0)
and unknown radius r.
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(a) Obtain the MLE r̂n of r.
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(b) Examine whether r̂n is sufficient for r.
(c) For any ε > 0, show that lim P (|r̂n − r| > ε) = 0.
n→∞

9. Suppose X1 , X2 , . . . , Xn is a random sample from an exponential distribution
with mean λ. Assume that the observed data is available on [X1 ], . . . , [Xn ],
instead of X1 , . . . , Xn , where [x] denotes the largest integer less than or equal
to x. Consider a test for H0 : λ = 1 vs H1 : λ > 1 which rejects H0 when
Xn
[Xi ] > cn . Given α ∈ (0, 1), obtain values of cn such that the size of the test
i=1
converges to α as n → ∞.
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that the errors in obtaining X and Y are independent and normally distributed

las with mean zero and the same (unknown) variance. Devise a test for the hy-
ag
ag pothesis that the true weights of the two pieces are equal.

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