ISI Admission Test 2018 Question Paper M.Stat PSB

FOR ISI EXAM PREPARATION

ISI 2018
Question Paper · M.Stat
PSB
EXAM YEAR TYPE SUBJECT

ISI 2018 Question Paper M.Stat PSB

Notes · Sample Papers · Previous Year Papers · Mock Tests

.
.co s e m

s em l a
a ag

GROUP A

m
co
1. Find all real solutions (x1 , x2 , x3 , λ) for the system of equations

. c om e m .
s
x2 − 3x3 − x1 λ = 0,
em l a
l as x1 − 3x3 − x2 λ = 0,
ag
ag x1 + x2 + x3 λ = 0.

2. Let {xn }n≥1 be a sequence defined by x1 = 1 and
 1 1/3
xn+1 = x3n + , n ≥ 1.
n(n + 1)(n + 2)

Show that {xn }n≥1 converges and find its limit.

m
3. Consider all permutations of the integers 1, 2, . . . , 100. In how many

.co
of these permutations will the 25th number be the minimum of the
m
s e
first 25 numbers and the 50th number be the minimum of the first 50

l a
agGROUP B
numbers?

4. An urn contains r > 0 red balls and b > 0 black balls. A ball is drawn
at random from the urn, its colour noted, and returned to the urn.
Further, c > 0 additional balls of the same colour are added to the
urn. This process of drawing a ball and adding c balls of the same

m
.co
colour is continued. Define Xi = 1 if at the i-th draw the colour of the

m
P
ball drawn is red, and 0 otherwise. Compute E( ni=1 Xi ).

m .co s em
e la X ) = 7/3
5. Suppose X1 and X2 are identically distributed random variables, not

las a g
necessarily independent, taking values in {1, 2}. If E(X 1 2

ag and E(X ) = 3/2, obtain the joint distribution of (X , X ).
1 1 2

6. A fair 6-sided die is rolled repeatedly until a 6 is obtained. Find the
expected number of rolls conditioned on the event that none of the
rolls yielded an odd number.

om
1
. c
. c e m
m as
se
For more Question Papers, Sample Papers, Notes & Syllabus visit
glPage 1 of 2

7. Suppose {(X1 , Y1 ), . . . , (Xn , Yn )} is a random sample from a bivariate
normal distribution with E(Xi ) = E(Yi ) = 0, Var(Xi ) = Var(Yi ) = 1
and unknown Corr(Xi , Yi ) = ρ ∈ (−1, 1), for all i = 1, . . . , n. Define
P
Wn = n1 ni=1 Xi Yi .

(a) Is Wn an unbiased estimator of ρ? Justify your answer.
(b) For large n, obtain an approximate level (1 − α) two-sided confi-
dence interval for ρ, where 0 < α < 1.

8. Let {X1 , . . . , Xn } be an i.i.d. sample from f (x : θ), θ ∈ {0, 1}, with
 
1 if 0 < x < 1,  √1 if 0 < x < 1,
f (x : 0) = and f (x : 1) = 2 x
0 otherwise, 0 otherwise.

Based on the above sample, obtain the most powerful test for testing
H0 : θ = 0 against H1 : θ = 1, at level α, with 0 < α < 1. Find the
critical region in terms of the quantiles of a standard distribution.

9. Suppose (yi , xi ) satisfies the regression model,

yi = α + βxi + i , for i = 1, . . . , n,

where {xi : 1 ≤ i ≤ n} are fixed constants and {i : 1 ≤ i ≤ n} are i.i.d.
N (0, σ 2 ) errors, where α, β and σ 2 (> 0) are unknown parameters.

(a) Let α
e denote the least squares estimate of α obtained assuming
β = 5. Find the mean squared error (MSE) of α
e in terms of the
model parameters.
(b) Obtain the maximum likelihood estimator of this MSE.

2

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 2 of 2