ISI Admission Test 2021 Question Paper M.Stat PSB

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ISI 2021
Question Paper ·
M.Stat PSB
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ISI 2021 Question Paper M.Stat PSB

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m
m .co

m .co s e m
se g l a
a

Group A

m
1. Define f : R → R as

m f (x) = .co


.co m
cos(2x) if x is rational,
s e
em a
sin2 (x)
s
if x is irrational.
l
g laFind all the real numbers where ag
a
(a) f is continuous,
(b) f is differentiable.

2. Consider the matrix  
m .
.co
5 3
 
P= 3 2 

s em8 5
la
3×2

Find a matrix G such ag
that AGA = A where A = PP . T

3. Consider the set of all five digit integers formed by permuting the
digits 1, 2, 4, 6 and 7. Let X denote a randomly chosen integer from
this set and let Y denote the position, from the right, of the digit 4
in the randomly chosen integer. For example, if the integer chosen is
m
.co
12647, then Y is 2 and for 41276, Y is 5.
m
m .co s e m
la
(a) Find E(X) and E(Y ).
s e (b) Find E[X|Y = y] for all possible values of y. g
g la a
a (c) Show that X and Y are uncorrelated but not independent.

m .
c. o
1
e m
em l as
las ag
ag For more Question Papers, Sample Papers, Notes & Syllabus visit Page 1 of 4

Group B

4. Let U1 , U2 , U3 be i.i.d. random variables which are uniformly dis-
tributed on (0, 1). Let X = min(U1 , U2 ) and Y = max(U2 , U3 ).

(a) Find P (X ≤ x, Y ≤ y) for all x, y ∈ R.
(b) Find P (X = Y ).
(c) Find E[XI{X=Y } ] where IA is the indicator function of A.

5. Consider a game with six states 1, 2, 3, 4, 5, 6. Initially a player starts
either in state 1 or in state 6. At each step the player jumps from one
state to another as per the following rules.
A perfectly balanced die is tossed at each step.

(i) When the player is in state 1 or 6: If the roll of the die results
in k then the player moves to state k, for k = 1, . . . , 6.
(ii) When the player is in state 2 or 3: If the roll of the die results in
1, 2 or 3 then the player moves to state 4. Otherwise the player
moves to state 5.
(iii) When the player is in state 4 or 5: If the roll of the die results in
4, 5 or 6 then the player moves to state 2. Otherwise the player
moves to state 3.

The player wins when s/he visits 2 more states, besides the starting
one.

(a) Calculate the probability that the player will eventually move
out of states 1 and 6.
(b) Calculate the expected time the player will remain within states
1 and 6.
(c) Calculate the expected time for a player to win, i.e., to visit 2
more states, besides the starting one.

2

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m
m .co

m .co s e m
se g l a
a

6. Let X1 , . . . , Xn be i.i.d. random variables which are uniformly dis-
tributed on (θ, 2θ), θ > 0.

m
.co
X(n)
(a) Show that is the maximum likelihood estimator (MLE) of θ
m
.co
2
where X(n) = max(X1 , . . . , Xn ).
e m
em l as
s(b) Find an unbiased estimator for θ based on the MLE.
la(c) Given any  > 0, show that ag
ag  
X (n)
lim P −θ > = 0.
n→∞ 2

7. There are two urns each contains N balls numbered from 1 to N .
From each urn a sample of size n is selected without replacement.

m
Denote the set of numbers appearing in the first and second samples

.co
by s1 = {i1 , . . . , in } and s2 = {j1 , . . . , jn } respectively. Let

s em
la
X = |s1 ∩ s2 | = number of common elements in s1 and s2 .
g
a distribution of X.
(a) Find the probability
(b) Suppose n = 6 and the observed value of X is 4. Obtain a
method of moments estimate of N .

8. Let X1 , X2 , X3 be i.i.d. random variables from N (µ, σ 2 ). Let

m
.co
3 3 3
1X X 1X
m
.co
X̄ = Xi , T1 = Xi 2 , T2 = (Xi − X̄)2 .
3 i=1 3 i=1
e m
s
i=1

e m la
las ag
ag (a) Compute E[T1 |X̄] and E[T1 |T2 ]
(b) Obtain the exact critical region of a level α (0 < α < 1) test for
X̄ 2
H0 : µ = 0 vs H1 : µ 6= 0 that rejects H0 if and only if is
T1
sufficiently large.

m .
c. o
3
e m
em l as
las ag
ag For more Question Papers, Sample Papers, Notes & Syllabus visit Page 3 of 4

9. Consider a linear regression model:

yi = α + βxi + ei , i = 1, 2, . . . , n

where xi ’s are fixed and ei ’s are i.i.d. random errors with mean 0 and
variance σ 2 .
Define two estimators of β as follows
Pn Pn
y xi yi
βb1 = Pn
i
i=1
and βb2 = Pi=1
n 2
.
i=1 xi i=1 xi

(a) Obtain an unbiased estimator of β as a linear combination of βb1
and βb2 .
(b) Find mean squared errors of βb1 and βb2 . Which, between βb1 and
βb2 , has lower mean squared error?

4

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