ISI Admission Test 2017 Question Paper M.Stat PSB

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ISI 2017
Question Paper ·
M.Stat PSB
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ISI 2017 Question Paper M.Stat PSB

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GROUP A

1. Let a and b be real numbers. Show that there exists a unique 2 × 2

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real symmetric matrix A with trace(A) = a and det(A) = b if and only

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if a2 = 4b.
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eformsome n ≥ 1,
2. Let f : R → R be an infinitely differentiable function and suppose that

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f (1) = f (0) = f (0) = f (0) = · · · = f (0) = 0, (n)

where f (k) denotes the k-th derivative of f for k ≥ 1. Prove that there
exists x ∈ (0, 1) such that f (n+1) (x) = 0.

3. Consider an urn containing 5 red, 5 black, and 10 white balls. If balls
are drawn without replacement from the urn, calculate the probability
that in the first 7 draws, at least one ball of each colour is drawn.
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GROUP B
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4. Let X1 , X2 , . . . , Xn be independent random variables, with Xi having

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probability mass function
a  i  1 k
P (Xi = k) = , for k = 0, 1, 2, . . .
i+1 i+1
and for all i = 1, . . . , n. Let M = min{Xi : 1 ≤ i ≤ n}. Derive the
probability mass function of M .

5. The lifetime in hours of each bulb manufactured by a particular com-

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pany follows an independent exponential distribution with mean λ.

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To test the null hypothesis H0 : λ = 1000 against the alternative
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H : λ = 500, a statistician sets up an experiment with 50 bulbs,
5 bulbs in each of 10 different locations, to examine theirslifetimes.
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las e To get quick preliminary results, the statistician a gla to stop the
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decides
a experiment as soon as one bulb fails at each location. Let Yi denote
the lifetime of the first bulb to fail at location i. Obtain the most
powerful test of H0 against H1 based on Y1 , Y2 , . . . , Y10 , and compute
its power.

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6. Suppose you have a 4-digit combination lock, but you have forgotten
the correct combination. Consider the following three strategies to find
the correct one:

(i) Try the combinations consecutively from 0000 to 9999.
(ii) Try combinations using simple random sampling with replace-
ment from the set of all possible combinations.
(iii) Try combinations using simple random sampling without replace-
ment from the set of all possible combinations.

Assume that the true combination was chosen uniformly at random
from all possible combinations. Determine the expected number of
attempts needed to find the correct combination in all three cases.

7. Consider independent observations {(yi , x1i , x2i ) : 1 ≤ i ≤ n} from the
regression model

yi = β1 x1i + β2 x2i + i , i = 1, . . . , n ,

where x1i and x2i are scalar covariates, β1 and β2 are unknown scalar
coefficients, and i are uncorrelated errors with mean 0 and variance
σ 2 > 0. Instead of using the correct model, we obtain an estimate β̂1
of β1 by minimizing
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X
(yi − β1 x1i )2 .
i=1

Find the bias and mean squared error of β̂1 .

8. Let θ > 0 be an unknown parameter, and X1 , X2 , . . . , Xn be a random
sample from the distribution with density

2x/θ2 , 0 ≤ x ≤ θ ,
f (x) =
0 , otherwise.

Find the maximum likelihood estimator of θ and its mean squared
error.

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