This is the Indian Statistical Institute JRF in Statistics Exam Previous Years Question Paper:

Syllabus for RSI and RSII
Mathematics
Functions and relations. Matrices - determinants, eigenvalues and eigenvectors,
solution of linear equations, and quadratic forms.
Calculus and Analysis - sequences, series and their convergence and divergence;
limits, continuity of functions of one or more variables, di_erentiation, applications,
maxima and minima. Integration, de_nite integrals, areas using integrals, ordinary
linear di_erential equations.

Statistics
(a)Probability: Basic concepts, elementary set theory and sample space, conditional
probability and Bayes theorem. Standard univariate and multivariate distribu-
tions. Transformations of variables. Moment generating functions, characteristic
functions, weak and strong laws of large numbers, convergence in distribution and
central limit theorem. Markov chains.

(b) Inference: Su_ciency, minimum variance unbiased estimation, Bayes estimates,
maximum likelihood and other common methods of estimation. Optimum tests for
simple and composite hypotheses. Elements of sequential and non-parametric tests.
Analysis of discrete data - contingency chi-square.

(c) Multivariate Analysis: Standard sampling distributions. Order statistics with
applications. Regression, partial and multiple correlations. Basic properties of
multivariate normal distribution, Wishart distribution, Hotelling's T-square and
related tests.

(d) Design of Experiments: Inference in linear models. Standard orthogonal and
non-orthogonal designs. Inter and intra-block analysis of general block designs.
Factorial experiments. Response surface designs. Variance components estimation
in one and two-way ANOVA.

(e) Sample Surveys: Simple random sampling, Systematic sampling, PPS sampling,
Strati_ed sampling. Ratio and regression methods of estimation. Non-sampling
errors, Non-response.

Sample Questions : RSI
1. Let A and B be two given subsets of . For each subset C of de_ne
Ic(!) = _ 1 if ! 2 C
0 if ! =2 C:
Does there exists a subset C of such that j IA(!)_IB(!) j= Ic(!) for each
! 2 ? Justify your answer.
2. (a) Evaluate the determinant
_________
1 + x1y1 1 + x1y2 : : : 1 + x1yn
1 + x2y1 1 + x2y2 : : : 1 + x2yn
...
...
...
1 + xny1 1 + xny2 : : : 1 + xnyn
_________
:
(b) Find the inverse of the matrix A + __0, where A is a diagonal matrix
diag(a1; a2 _ _ _ ; ak) and _0 = ( 1
_1
; _ _ _ ; 1
_k
), _i > 0 for i = 1; _ _ _ ; k.
3. Let x1; x2; x3; y1; y2; and y3 be real numbers. De_ne two 3 _ 3 matrices A1
and A2 by
A1 = 0@
x2
1 x1x2 x1x3
x2x1 x2
2 x2x3
x3x1 x3x2 x2
3
1A
;
A2 = 0@
x2
1 + y2
1 x1x2 + y1y2 x1x3 + y1y3
x2x1 + y2y1 x2
2 + y2
2 x2x3 + y2y3
x3x1 + y3y1 x3x2 + y3y2 x2
3 + y2
3
1A
:
Find det(A1). Hence, or otherwise, _nd det(A2).
4. Let A be a 3 _ 3 real orthogonal matrix. Prove that there exists a vector w
in R3 such that Aw = w or Aw = _w.
5. Let An_m and Bm_n be two real matrices. Show that the nonzero eigenvalues
of AB and BA are the same.
6. Let f, g and h be de_ned on [0; 1] as follows:
f(x) = g(x) = h(x) = 0 whenever x is irrational;
f(x) = 1 and g(x) = x whenever x is rational;
h(x) =
1
n
if x is the rational number m=n (in lowest terms);
h(0) = 1:
Prove that
2
(a) f is not continuous anywhere in [0; 1],
(b) g is continuous only at x = 0, and
(c) h is continuous only at the irrational points in [0; 1].
7. Consider a function f which satis_es the relations :
2f0(x) = f(
1
x
) if x > 0; f(1) = 2:
(a) Let y = f(x). Find suitable values for constants a and b such that y
satis_es a di_erential equation of the form x2y00 + axy0 + by = 0:
(b) Determine a solution of the above di_erential equation of the form f(x) =
cxn.
8. Give an example of a function f : [a; b] ! R such that
j f(x) _ f(y) j_j x _ y j for all x; y 2 [a; b]:
Prove that any function f satisfying the above condition also satis_es
j Z b
a
f(x)dx _ (b _ a)f(a) j _
1
2
(b _ a)2
provided f is integrable on [a; b].
9. Find the maximum of Z Z 4
dxdy as a function of m for 0 < m < 1 where
4 = f(x; y) : x2
m
+ y2
1 _ m _ 1g
10. (a) Find R4
_2[x]dx, where [x] is the largest integer less than or equal to x.
(b) If f(x); x _ 0 is a di_erentiable function such that f0(x) ! 1 as x ! 1,
then show that f(x) ! 1 as x ! 1.
11. (a) Find the limit
lim
n!1
n Xk=0
n
n2 + k2
(b) Let f(t) be a continuous function on (_1;1) and let
F(x) = Z x
0
(x _ t)f(t)dt: Find F0(x):
12. (a) Find max(xyz) subject to x2 + 2y2 + 9z2 = 6:
3
(b) Let f(x) = Pn
k=0 akxk, where ak's satisfy Pn
k=0
ak
k+1 = 0:
Show that there exists a root of f(x) = 0 in the interval (0; 1).
13. One of the sequences of letters AAAA;BBBB;CCCC is transmitted over a
communication channel with respective probabilities p1; p2; p3, where (p1 +
p2 + p3 = 1). The probability that each transmitted letter will be correctly
understood is _ and the probabilities that the letter will be confused with two
other letters are 1
2 (1 _ _) and 1
2 (1 _ _). It is assumed that the letters are
distorted independently. Find the probability that AAAA was transmitted if
ABCA was received.
14. In a sequence of independent tosses of a fair coin, a person is to receive Rs.2
for a head and Re.1 for a tail.
(a) For a positive integer n, _nd the probability that the person receives
exactly Rs. n at some stage.
(b) What is the limit of this probability as n ! 1 ?
15. Let X1; : : : ;Xn be i.i.d. random variables with P(Xi = _1) = P(Xi = 1) = 1
2 .
Let Sn = X1 + _ _ _ + Xn for n _ 1. Let N = minfn _ 1; such that Sn = 0g.
(a) Show that P(N = 2k + 1) = 0 for every integer k _ 0.
(b) Find P(N = 2k; Sk = k) for every integer k _ 1.
16. Let X1; _ _ _ ;Xn be random variables such that E(X2
i ) _ 1 for all i = 1; _ _ _ ; n.
Let Si = Xi + _ _ _ + Xi for i = 1; 2; _ _ _ ; n. Show that
E( max
1_i_n
S2
i ) _ n2:
17. For n _ 1, let Xn have probability density function fn given by
fn(x) = _ 1 + sin(2_nx) if 0 < x < 1;
0 otherwise:
Does the sequence fXn : n _ 1g converge in distribution? Give reasons.
18. Suppose fXng is a sequence of independent random variables such that for
n = 2; 3; : : : ;
P(Xn = 0) = 1 _
1
n log n
; P(Xn = _n) =
1
2n log n
:
Let Sn = X1 + _ _ _ + Xn. Does Sn=n _! 0 in probablity?
19. Let X1; _ _ _ ;Xn; _ _ _ ; be a sequence of independent random variables with
E(Xr) = 0, Var(Xr) = pr for r = 1; _ _ _ ; n; _ _ _. Prove that 1
n(X1 + _ _ _ + Xn)
converges almost surely and _nd the limit.

20. Suppose die A has 4 red faces and 2 green faces while die B has 2 red faces
and 4 green faces. Assume that both the dice are unbiased. An experiment is
started with the toss of an unbiased coin. If the toss results in a Head, then
die A is rolled repeatedly while if the toss of the coin results in a Tail, then
die B is rolled repeatedly. For k = 1; 2; 3; _ _ _ ; de_ne
Xk = _ 1 if the kth roll of the die results in a red face
0 otherwise.
(a) Find the probability mass function of Xk.
(b) Calculate _(X1;X7).
21. A model that is often used for the waiting time X to failure of an item is
given by the probability mass function
pX(kj_) = _k_1(1 _ _); k = 1; 2; : : : ; 0 < _ < 1:
Suppose that we only record the time to failure if the failure occurs on or before
r, and otherwise just note that the item has lived at least (r+1) periods. Let
Y denote this censored waiting time.
(a) Write down the probability mass function of Y .
(b) If Y1; Y2; : : : ; Yn is a random sample from the censored waiting time, write
down the likelihood function and _nd the MLE of _.
22. Let X be exponentially distributed with mean 1 and let U be a U[0; 1] random
variable, independent of X. De_ne
I = 1 if U _ e_x and I = 0 otherwise .
Show that the conditional distribution of X given fI = 1g is exponential with
mean 0.5.
23. Let X be an exponential r:v: with mean 1
2 . Let Y be the largest integer less
than or equal to X. Find the probability distribution of Y .
24. Consider two coins with probabilties of head _1 and _2, respectively. The two
coins are tossed together independently. Let X be the number of heads in the
joint tossings. Let pi = P(X = i); i = 0; 1; 2:
(a) Is it possible to choose _1 and _2 such that p1 = p2 = p0? Justify your
answer.
(b) Is it possible to choose _1; _2 such that X may be used to simulate the
outcomes of a fair coin? Justify your answer.
25. Two points are chosen at random from a line segment of length a. Find the
probability that the three parts thus obtained can form a triangle.
5
26. Let X1;X2;X3 be independent standard normal variables. Find the distribu-
tion of
U = X1 + X2 + X3
X2 _ 2X3 + X1
:
27. Let Y have a chi-squared distribution with k(_ 3) degrees of freedom. Express
E(1=Y ) in terms of k.
28. Let X follow an Np(_; _) distribution, where _ is a known positive de_nite
matrix and _ is either _1 or _2; _i's being known. Show that (_2__1)0P_1 X
is a su_cient statistic.
29. Let X1;X3; _ _ _ ;Xn be i.i.d. observations from the N(_; 1) distribution, where
_ is unknown. For a _xed real number u, consider the problem of estimating
g(_) =
1
p2_
exp[_
1
2
(u _ _)2];
on the basis of X1;X2; _ _ _ ;Xn:
(a) Show that X = 1
n_Xi is a su_cient statistic for _.
(b) Write down the probability density function of X.
(c) Work out the conditional density of X1 given X at u.
(c) Obtain the UMVUE of g(_).
30. Show that X1 + 2X2 is not su_cient for _ where X1 and X2 are two random
observations from N(_; 1) distribution.
31. The number of accidents X per year in a manufacturing plant may be mod-
elled as a Poisson random variable with mean _. Assume that accidents in
successive years are independent random variables and suppose that you have
n observations.
(a) How will you _nd the minimum variance unbiased estimator for the prob-
ability that in a year the plant has at most one accident ?
(b) Suppose that you wish to estimate _. Suggest two unbiased estimators
of _ and hence _nd the UMVUE of _.
32. Let the life time (in hours) of a bulb be random and follow exponential dis-
tribution with mean 1=_ hours. Let Xn = number of bulbs having lasted for
more that 5 hours in a random sample of n bulbs. Construct a consistent
estimate of _ on the basis of Xn.
33. Let X1;X2; _ _ _ ;Xn be a random sample of size n from a negative exponential
distribution with mean _. The experiment is terminated after the _rst r(r _ n) smallest observations have been noted. Write down the likelihood for _
based on these censored observations. Find the mle of _. Obtain the UMP
test for testing H0 : _ = _0 vs. H1 : _ > _0 at level _.
6
34. Consider the following _xed e_ects linear model:
Y1 = _1 + _2 + _4 + _1
Y2 = _1 + _3 + 2_4 + _2
Y3 = _1 + _2 + _4 + _3
Y4 = __2 + _3 + _4 + _4
_ = (_1; _2; _3; _4) _ N(0; _2I4)
(a) Find the full set of error functions.
(b) Hence or otherwise, _nd an unbiased estimator of _2 with maximum
possible degrees of freedom.
35. An experimenter wanted to use a Latin Square Design (LSD) but instead,
used the following row-column design
A B C
B A C
C A B
Are all treatment contrasts estimable ? Give reasons for your answer.
36. An unbiased coin is to be used to select a Probability Proportional to Size
With Replacement (PPSWR) sample in 2 draws from the following population
of 3 units, where X is the size measure.
i 1 2 3
Xi 2 4 1
Consider the following procedure.
a) Toss the coin thrice independently.
b) If the outcome is fHHHg or fHHTg, select the _rst unit.
c) If the outcome is fHTHg, fHTTg, fTHHg or fTHTg, then select the
second unit.
d) If the outcome is fTTHg, then select the third unit.
e) If the outcome is fTTTg, do not select any of the units, go back to (a)
above.
f) Continue till a unit is selected.
Show that the above procedure is a PPS method. If W = number of tosses
required to select a unit, _nd E(W).
37. Show that under PPSWOR of size 2 or 3, inclusion probability of ith unit
exceeds that of jth unit, whenever ith unit has a larger size measure than
unit jth unit.

38. Consider a population of 30 people classi_ed according to age and sex as in
the following table.
Age Group
Sex _ 40 > 40 Total
Male 4 6 10
Female 10 10 20
Total 14 16 30
(a) Describe a sampling scheme of drawing a random sample of size 3 from
this population so that both sexes and both age groups are represented.
(b) For your scheme, compute the _rst order inclusion probabilities (_i's).

Sample Questions : RSII
1. Let V1; V2; V3; X1;X2;X3, be independently and identically distributed N(0; 1)
variables. Find the distribution of
T = V1X1 + V2X2 + V3X3 pV 2
1 + V 2
2 + V 2
3
:
Hence _nd the distribution of
S = V1X1 + V2X2 + V3X3
V 2
1 + V 2
2 + V 2
3
:
2. Let X _ Np(0;_1) and Y _ Np(0;_2), where _1 and _2 are positive
de_nite. If P(XTAX _ 1) _ P(Y TAY _ 1) for all positive semide_nite A
of rank 1, show that _2 _ _1 is positive semide_nite.
3. Express the following as the probability of an event and evaluate without
integrating by parts: Z 1
_1
_(x)
1
p2__
e_ x2
2_2 dx;
where _(_) is the standard normal cumulative distribution function.
4. Suppose that X1;X2;X3; _ _ _ ; are i.i.d. random variables with EX1 = 0,
EX2
1 = 1, EX4
1 < 1. Show that
n_1=2[
n X1
(Xi _ Xn)2 _ n]
8
converges in law to a normal distribution with zero mean, as n ! 1. Here
Xn = Pn
1 Xi=n.
5. Let X1;X2; _ _ _ ;Xn be i.i.d. observations from a continuous distribution and
R1; _ _ _ ;Rn be the ranks of the observations. Find Cov(Rn_1;Rn).
6. Let fXng be a sequence of independent random variables. Show that for each
real number _; P(Xn ! _) = 0 or 1 by explicitly applying the Borel-Cantelli
lemmas.
7. Let (Xi; Yi); i = 1; 2; _ _ _ ; n be a random sample on (X; Y ) which is a bivariate
vector of the continuous type. Let T = Max.(X; Y ) and Ri be the rank of
Ti among T1; T2; _ _ _ ; Tn. Find the mean and variance of the statistic S =
Pn
i=1(n+1_Ri)_i assuming that (X; Y ) and (Y;X) are identically distributed,
where _ = IfX<Y g is the indicator function.
8. A company desires to operate S identical machines. These machines are sub-
ject to failure according to a given probability law. To replace these failed
machines the company orders new machines at the beginning of each week to
make up the total S. It takes one week for each new order to be delivered.
Let Xn be the number of machines in working order at the beginning of the
nth week, and let Yn denote the number of machines that fail during the nth
week.
(a) Establish the recursive formula Xn+1 = S_Yn, and show that Xn; n _ 1
constitutes a Markov Chain.
(b) Suppose that the failure rate is uniform i.e.,
P[Yn = j j Xn = i] =
1
i + 1; j = 0; 1; _ _ _ ; i:
Find the transition matrix of the Chain, its stationary distribution, and
expected number of machines in operation in the steady state.
9. Consider a Markov chain with state space S = f1; 2; 3; 4; 5g and transition
probability matrix
P = 0BBBB@
1
2 0 1
2 0 0
1
3
1
3
1
3 0 0
1
2 0 1
2 0 0
0 0 0 1
2
1
2
0 0 0 0 1
1CCCCA
(a) Identify the closed sets of S.
(b) Identify the transient and absorbing states in S.
(c) Examine the asymptotic behavior of p(n)
ij as n ! 1.
9
10. (a) In a forest there are 50 tigers and an unknown number L of lions. Assume
that the 50+L animals randomly move in the forest. A naturalist sights
5 di_erent lions and 15 di_erent tigers in the course of a trip in the forest.
Estimate, stating assumptions and with theoretical support, the number
of lions in the forest.
(b) A bank decided to examine a sample of vouchers before conducting a
thorough audit. From a very large number of accumulated vouchers,
samples were drawn at random. The _rst defective voucher was obtained
in the 23rd draw, and the second defective voucher in the 57th draw.
Estimate, giving reasons, the proportion of vouchers that are defective.
11. Let X and Y be two random variables such that
P(X > x; Y > y) = exp[__1x _ _2y _ _12 max(x; y)]:
(a) Find the marginal distribution of X.
(b) Let _12 = 0:5. Find the method-of-moments estimators of _1, _2.
(c) Under independence of X and Y in the above family of distributions and,
based on a sample of n pairs of observations (Xi; Yi) from the resulting
distribution, _nd an estimator of _1=_2.
(Hint: E(_2
_) = 1=(_ _ 2), _ > 2.)
12. Suppose we have a sample of size one from N(_; 1) distribution, where 2 _ _ _ 3. What will be the uniformly minimum variance unbiased estimate for
_? Justify your answer. Show that one can construct another estimator with
less mean squared error.
13. Let X1; : : : ;Xn be i.i.d. Poisson(_), where _ > 0 is unknown. Find the
minimum variance unbiased estimator T of exp(p2_).
14. Let Y(1) < Y(2) < _ _ _ < Y(n) be the ordered random variables of a sample of
size n from the rectangular (0; _) distribution with _ unknown, 0 < _ < 1. By
a careless mistake the observations Y(k+1); _ _ _ ; Y(n) were recorded incorrectly
and so they were discarded subsequently (Here 1 _ k < n).
(a) Show that the conditional distribution of Y(1); _ _ _ ; Y(k_1) given Y(k) is
independent of _.
(b) Hence, or otherwise, obtain the maximum likelihood estimator of _ and
show that it is a function of Y(k).
(c) If k
n ! p as n ! 1, for some 0 < p < 1, what can you say about the
asymptotic distribution of the maximum likelihood estimator of _ ?
15. Let X _ N6(_; _), where _ = (_1; _ _ _ ; _6)T , and _ is unknown. Obtain an
exact test for testing H0 : _l = l_, l = 1; 2; _ _ _ ; 6, _ being unknown, against
H1 : not H0. Obtain the cut-o_ point of this test.
10
16. Let X be a discrete random variable (taking values 1; 2; : : : ; 6) whose distri-
bution depends on an unknown parameter _. The following table gives the
probability distributions of X under _ = _0 and _ = _1.
X 1 2 3 4 5 6
_ = _1 0.15 0.15 0.15 0.15 0.15 0.25
_ = _0 0.15 0.20 0.15 0.10 0.10 0.30
Consider the problem of testing H0 : _ = _0 versus H1 : _ = _1, based on a
single observation X.
(a) Find a Most Powerful test of level 0.3.
(b) Argue that there are in_nitely many Most Powerful tests of level 0.3.
17. The value of Y is estimated from X = x0 and the linear regression of Y on X.
Let this estimated value of Y be y0. Then the value of X corresponding to
Y = y0 is estimated from the linear regression of X on Y . Let this estimated
value of X be x_0 . Compare x0 and x_0 . Interpret your answer.
18. Let X be a random variable having a density 1
_ e_x=_; x; _ > 0. Consider
H0 : _ = 1 vs. H1 : _ = 2. Let !1 and !2 be two critical regions given by
!1 : Pn
1 Xi _ C1 and !2 : ( number of Xi's _ 2) _ C2.
(a) Determine approximately the values of C1 and C2 for large n so that
both tests are of size _.
(b) Show that the powers of both tests tend to 1 as n ! 1.
(c) Which test would require more sample size to achieve the same power ?
Justify your answer.
19. Let X1;X2; : : : ;Xn be i.i.d observations with a common exponential distri-
bution with mean _. Show that there is no uniformly most powerful test for
testing H0 : _ = 1 against HA : _ 6= 1 at a given level 0 < _ < 1 but there
exists a uniformly most powerful unbiased test and derive that test.
20. Let X = (X1;X2;X3;X4) have a multivariate normal distribution with un-
known mean vector _ = (_1; _2; _3; _4) and unknown variance covariance
matrix _ which is nonsingular. Let X1; _ _ _ ;Xn be a random sample of size n
from the population. Develop a test for testing the hypothesis H0 : _1+2_2 =
_2 + 2_3 = _3 + 2_4. State the distribution of the test statistic.
21. In a linear model Y = A_ + _; E(_) = 0; D(_) = _2I; _0 = (_1; _ _ _ ; _p):
Let C1;C2; _ _ _ ;Cp denote the column vectors of the matrix A. Prove that
(i) _1 is estimable if and only if C1 does not belong to the vector space
spanned by C2;C3; _ _ _ ;Cp.
(ii) _1_1 + _2_2; _1 6= 0; _2 6= 0, is estimable if and only if C1 does not
belong to the vector space spanned by _2C2 _ _1C1;C3; _ _ _ ;Cp.
1