ISI Admission Test 2019 Question Paper M.Stat PSB

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ISI 2019
Question Paper ·
M.Stat PSB
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ISI 2019 Question Paper M.Stat PSB

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2019

GROUP A

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1. Let f (x) = x3 − 3x + k, where k is a real number. For what values of

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k will f (x) have three distinct real roots?
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a 2. Let A and B be 4 × 4 matrices. Suppose that A has eigenvalues
x1 , x2 , x3 , x4 and B has eigenvalues 1/x1 , 1/x2 , 1/x3 , 1/x4 , where each
xi > 1.

(a) Prove that A + B has at least one eigenvalue greater than 2.
(b) Prove that A − B has at least one eigenvalue greater than 0.

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(c) Give an example of A and B so that 1 is not an eigenvalue of AB.

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g la on any seven days in April and May. In
how many ways canathe seven days be chosen such that elections are
3. Elections are to be scheduled

not scheduled on two consecutive days?

GROUP B

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s e ables with mean µ > 0 and taking values in {0, 1, 2, .g. l
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4. Let X and Y be independent and identically distributed random vari-

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a all m ≥ 0,
1
P(X = k | X + Y = m) = , k = 0, 1, . . . , m.
m+1
Find the distribution of X in terms of µ.

1

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5. Suppose X1 , X2 , . . . , Xn are independent random variables such that

P(Xi = 1) = pi = 1 − P(Xi = 0),

where p1 , p2 , . . . , pn ∈ (0, 1) are all distinct and unknown. Consider
�n
X = Xi and another random variable Y which is distributed as
i=1
�
n
Binomial(n, p̄), where p̄ = n1 pi . Between X and Y , which is a
i=1
�
n
better estimator of pi in terms of their respective mean squared
i=1
errors?

6. Suppose X1 , X2 , . . . , Xn is a random sample from Uniform(0, θ) for
some unknown θ > 0. Let Yn be the minimum of X1 , X2 , . . . , Xn .

(a) Suppose Fn is the cumulative distribution function (c.d.f.) of nYn .
Show that for any real x, Fn (x) converges to F (x), where F is
the c.d.f. of an exponential distribution with mean θ.
(b) Find lim P(n[Yn ] = k) for k = 0, 1, 2, . . . , where [x] denotes the
n→∞
largest integer less than or equal to x.

7. Suppose an SRSWOR of size n has been drawn from a population
labelled 1, 2, . . . , N , where the population size N is unknown.

� of N .
(a) Find the maximum likelihood estimator N
�.
(b) Find the probability mass function of N
(c) Show that n+1 �
n N − 1 is an unbiased estimator of N .

2

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8. Suppose {(xi , yi , zi ) : i = 1, 2, . . . , n} is a set of trivariate observations
on three variables: X, Y , and Z, where zi = 0 for i = 1, 2, . . . , n − 1
and zn = 1. Suppose the least squares linear regression equation of Y
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on X based on the first n − 1 observations is
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y=α
�0 + α
�1 x
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y = β�0 + β�1 x + β�2 z.

�1 = β�1 .
Show that α

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9. Let Z be a random variable
1
f (z) = e−|z−µ| , z ∈ R
2
with parameter µ ∈ R. Suppose we observe X = max(0, Z).

(a) Find the constant c such that the test that “rejects when X > c”
has size 0.05 for the null hypothesis H0 : µ = 0.
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(b) Find the power of this test against the alternative hypothesis
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H1 : µ = 2.

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a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 3 of 3