ISI Admission Test 2020 Question Paper M.Stat PSB

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ISI 2020
Question Paper · M.Stat
PSB
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ISI 2020 Question Paper M.Stat PSB

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GROUP A

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1. Let f (x)o= x − 2x + 2. Let L and L be the tangents to its graph at
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. cand x = 2 respectively. Find the area of the region enclosed by
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s
1 2

sethemgraph of f and the two lines L and L .
x= 0
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g la 1 2
a
a
2. Find the number of 3 × 3 matrices A such that the entries of A belong
to the set Z of all integers, and such that the trace of At A is 6.
(At denotes the transpose of the matrix A).

3. Consider n independent and identically distributed positive random vari-

m
ables X1 , X2 , . . . , Xn . Suppose S is a fixed subset of {1, 2, . . . , n} con-
sisting of k distinct elements where 1 ≤ k < n.
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(a) Compute

g l a X i

a
i∈S
E P . n
X i=1 i

(b) Assume that Xi ’s have mean µ and variance σ 2 , 0 < σ 2 < ∞. If
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j 6∈ S, show that the correlation between ( i∈S Xi )Xj and i∈S Xi
1 1
lies between − √k+1 and √k+1 .

GROUP B
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4. Let X , X , . . . , X be independent and identically distributed random

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1 2 n

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variables. Let S = X + · · · + X . For each of the following
n 1 n

a determine whether they are true or false. Give reasons in each case.

(a) If Sn ∼ Exp with mean n, then each Xi ∼ Exp with mean 1.
(b) If Sn ∼ Bin(nk, p), then each Xi ∼ Bin(k, p).

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5. Let U1 , U2 , . . . , Un be independent and identically distributed random
variables each having a uniform distribution on (0, 1). Let

X = min{U1 , U2 , . . . , Un }, Y = max{U1 , U2 , . . . , Un }.

Evaluate E [X|Y = y] and E [Y |X = x].

6. Suppose individuals are classified into three categories C1 , C2 and C3 .
Let p2 , (1 − p)2 and 2p(1 − p) be the respective population proportions,
where p ∈ (0, 1). A random sample of N individuals is selected from
the population and the category of each selected individual recorded.
For i = 1, 2, 3, let Xi denote the number of individuals in the sample
belonging to category Ci . Define U = X1 + X23 .

(a) Is U sufficient for p? Justify your answer.
U
(b) Show that the mean squared error of N is p(1−p)
2N .

7. Consider the following model:

yi = βxi + εi xi , i = 1, 2, . . . , n,

where yi , i = 1, 2, . . . , n are observed; xi , i = 1, 2, . . . , n are known posi-
tive constants and β is an unknown parameter. The errors ε1 , ε2 , . . . , εn
are independent and identically distributed random variables having the
probability density function
 
1 |u|
f (u) = exp − , −∞ < u < ∞,
2λ λ

and λ is an unknown parameter.

(a) Find the least squares estimator of β.
(b) Find the maximum likelihood estimator of β.

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8. Assume that X1 , . . . , Xn is a random sample from N (µ, 1), with µ ∈ R.

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We want to test H0 : µ = 0 against H1 : µ = 1. For a fixed integer

om
m ∈ {1, . . . , n}, the following statistics are defined:
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em T1 = (X1 + . . . + Xm )/m,
l as
l as T2 = (X2 + . . . + Xm+1 )/m,
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.=
..
.
Tn−m+1 = (Xn−m+1 + . . . + Xn )/m.

Fix α ∈ (0, 1). Consider the test

reject H0 if max {Ti : 1 ≤ i ≤ n − m + 1} > cm,α .

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Find a choice of cm,α ∈ R in terms of the standard normal distribution

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function Φ that ensures that the size of the test is at most α.

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9. A finite population hasgN
a
with x being the value associated
i
th
with the i unit, i = 1, 2, . . . , N . Let x̄ be the population mean.
N
A statistician carries out the following experiment.

• Step 1: Draw a SRSWOR of size n (< N ) from the population.
Call this sample S1 and denote the sample mean by X̄n .
• Step 2: Draw a SRSWR of size m from S1 . The x-values of the
sampled units are denoted by {Y1 , . . . , Ym }.
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An estimator of the population mean is defined as,

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e la
m
1 X

las Tbm =
m
Yi .
ag
ag i=1

(a) Show that Tbm is an unbiased estimator of the population mean.
(b) Which of the following has lower variance: Tbm or X̄n ?

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