MSTAT Question Papers

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(MSTAT Question Papers)

1. Suppose V is the space of all n×n matrices with real elements. Define
T : V ! V by T (A) = AB − BA, A 2 V , where B 2 V is a fixed
matrix. Show that for any B 2 V
(a) T is linear;
(b) T is not one-one;
(c) T is not onto.
2. Let f be a real valued function satisfying
|f (x) − f (a)| _ C |x − a| ,
for some γ > 0 and C > 0.
(a) If γ = 1, show that f is continuous at a;
(b) If γ > 1, show that f is differentiable at a.
3. Suppose integers are formed by taking one or more digits from the
following
2, 2, 3, 3, 4, 5, 5, 5, 6, 7.
For example, 355 is a possible choice while 44 is not. Find the number
of distinct integers that can be formed in which
(a) the digits are non-decreasing;
(b) the digits are strictly increasing.
3
4. Consider n independent observations {(xi, yi) : 1 _ i _ n} from the
model
Y = α + βx + ǫ,
where ǫ is normal with mean 0 and variance σ2. Let ˆα, ˆ β and ˆσ2 be the
maximum likelihood estimators of α, β and σ2, respectively. Let v11,
v22 and v12 be the estimated values of Var(ˆα), Var( ˆ β) and Cov(ˆα, ˆ β),
respectively.
(a) What is the estimated mean of Y when x = x0? Estimate the
mean squared error of this estimator.
(b) What is the predicted value of Y when x = x0? Estimate the
mean squared error of this predictor.
5. A box has an unknown number of tickets serially numbered 1, 2, . . . ,N.
Two tickets are drawn using simple random sampling without replace-
ment (SRSWOR) from the box. If X and Y are the numbers on these
two tickets and Z = max(X, Y ), show that
(a) Z is not unbiased for N;
(b) aX+bY +c is unbiased for N if and only is a+b = 2 and c = −1.
6. Suppose X1,X2 and X3 are three independent and identically dis-
tributed Bernoulli random variables with parameter p, 0 < p < 1.
Verify if the following statistics are sufficient for p:
(a) X1 + 2X2 + X3;
(b) 2X1 + 3X2 + 4X3.
4
7. Suppose X1 and X2 are two independent and identically distributed
random variables with Normal (θ, 1) distribution. Further, consider a
Bernoulli random variable V with P[V = 1] = 1/4, which is indepen-
dent of X1 and X2. Define X3 as
X3 = 8<
:
X1 when V = 0,
X2 when V = 1.
For testing H0 : θ = 0 against H1 : θ = 1 consider the test:
Reject H0 if (X1 + X2 + X3) /3 > c.
Find c such that the test has size 0.05.
8. Suppose X1 is a standard normal random variable. Define
X2 = 8<
:
−X1 if |X1| < 1,
X1 otherwise.
(a) Show that X2 is also a standard normal random variable.
(b) Obtain the cumulative distribution function of X1+X2 in terms of
the cumulative distribution function of a standard normal random
variable.
9. Envelopes are on sale for Rs. 30 each. Each envelope contains exactly
one coupon, which can be one of four types with equal probability.
Suppose you keep on buying envelopes and stop when you collect all
the four types of coupons. What will be your expected expenditure?
5
10. There are 10 empty boxes numbered 1, 2, . . . , 10 placed sequentially on
a circle as shown in the figure.
We perform 100 independent trials. At each trial, one box is selected
with probability 1/10 and one ball is placed in each of the two neigh-
bouring boxes of the selected one.
Define Xk to be the number of balls in the kth box at the end of 100
trials.
(a) Find E[Xk] for 1 _ k _ 10.
(b) Find Cov (Xk,X5) for 1 _ k _ 10.









MSTAT syllabus:

Syllabus for Mathematics
Combinatorics; Elements of set theory. Permutations and combinations.
Binomial and multinomial theorem. Theory of equations. Inequalities.
Linear Algebra: Vectors and vector spaces. Matrices. Determinants. So-
lution of linear equations. Trigonometry. Co-ordinate geometry.
Complex Numbers: Geometry of complex numbers and De Moivres the-
orem.
Calculus: Convergence of sequences and series. Functions. Limits and con-
tinuity of functions of one or more variables. Power series. Differentiation.
Leibnitz formula. Applications of differential calculus, maxima and minima.
Taylor’s theorem. Differentiation of functions of several variables. Indefinite
integral. Fundamental theorem of calculus. Riemann integration and prop-
erties. Improper integrals. Double and multiple integrals and applications.

Attached Files Available for Download

	MSTAT Question Papers.pdf (199.4 KB, 57 views)
	MSTAT Question Papers-2.pdf (837.7 KB, 30 views)
	MSTAT syllabus.pdf (55.5 KB, 33 views)

Here is the Indian Statistical Institute University MSTAT Maths Syllabus:

Permutations and Combinations. Binomial and multinomial theorem.
Theory of equations. Inequalities.
Determinants, matrices, solution of linear equations and vector spaces.
Trigonometry, Coordinate geometry of two and three dimensions.
Geometry of complex numbers and De Moivre’s theorem. Elements of
set theory.
Convergence of sequences and series. Power series. Functions, limits and
continuity of functions of one or more variables.
Differentiation, Leibnitz formula, maxima and minima, Taylor’s theorem.
Differentiation of functions of several variables. Applications of differential
calculus.
Indefinite integral, Fundamental theorem of calculus, Riemann integra-
tion and properties. Improper integrals. Double and multiple integrals and
applications.

For detailed syllabus & question Paper, here is attachment:

Attached Files Available for Download

	ISI MSTAT Question Paper.pdf (50.5 KB, 34 views)
	ISI MSTAT Question Paper 1.pdf (80.9 KB, 40 views)

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