e - Osmania University

INTERNAL ASSIGNMENT QUESTIONS
M.A./M.Com./M.Sc. I YEAR
(2014-2015)
PROF. G. RAM REDDY CENTRE FOR DISTANCE EDUCATION
(RECOGNISED BY THE DISTANCE EDUCATION BUREAU, UGC, NEW DELHI)
OSMANIA UNIVERSITY
(A University with Potential for Excellence and Re-Accredited by NAAC with “A” Grade)
DIRECTOR
Prof. H.VENKATESHWARLU
Hyderabad – 7 , Telangana State
PROF.G.RAM REDDY CENTRE FOR DISTANCE EDUCATION
OSMANIA UNIVERSITY, HYDERABAD – 500 007
Dear Students,
Every student of M.A./M.Com./M.Sc. I year has to write and submit Assignment for
each paper compulsorily. Each assignment carries 20 marks. University Examinations will
be held for 80 marks. The concerned faculty evaluates these assignment scripts. The marks
awarded to you will be forwarded to the Controller of Examination, OU for inclusion in the
University Examination marks. If you fail to submit Internal Assignments before the stipulated
date, the internal marks will not be added to University examination marks under any
circumstances. The assignment marks will not be accepted after the stipulated date,
You are required to pay Rs.300/- fee towards Internal Assignment marks through DD (in favour of
Director, PGRRCDE, OU) and submit the same along with assignment at the concerned counter on
or before 23rd May, 2015 and obtain proper submission receipt.
ASSIGNMENT WITHOUT THE DD WILL NOT BE ACCEPTED
Assignments on Printed / Photocopy / Typed papers will not be accepted and will not be
valued at any cost. Only hand written Assignments will be accepted and valued.
Methodology for writing the Assignments:
1.
2.
3.
4.
5.
First read the subject matter in the course material that is supplied to you.
If possible read the subject matter in the books suggested for further reading.
You are welcome to use the PGRRCDE Library on all working days including Sunday for
collecting information on the topic of your assignments.
(10.30 am to 5.00 pm).
Give a final reading to the answer you have written and see whether you can delete
unimportant or repetitive words.
The cover page of the each theory assignments must have information as given in
FORMAT below.
FORMAT
a. NAME OF THE STUDENT
:
b. ENROLLMENT NUMBER
:
c. Course/Subject :M.COM./ M.Sc/M.A. : _______________________________________
d. NAME OF THE PAPER
: _______________________________________
e. DATE OF SUBMISSION
: _______________________________________
f. D.D Details
:No:____________Date ---------------------Amount:----------------Bank______________Branch_________
6.
7.
8.
Write the above said details clearly on every subject assignments paper, otherwise your
paper will not be valued.
Tag all the assignments paper wise and submit assignment number wise.
Submit the assignments on or before 23-05-2015 at the concerned counter at PGRRCDE,
OU on any working day and obtain receipt.
Prof.B.APPA RAO
JOINT DIRECTOR
Prof.H.VENKATESHWARLU
DIRECTOR
Note: Those who have paid Examination Fee only Eligible to Submit Assignments.
P G R R C D E Osmania university
Assignment
M SC Mathematics (Previous)
Paper title: ALGEBRA
Paper: I
Max.marks:20.
Note: The assignment consists of two parts PART-A and PART-B carrying
10marks each. Answer all the questions from both PART-A and
PART-B.
PART-A (5X2=10 Marks)
(50 Words)
1.
2.
3.
4.
5.
Show that every finite group has a composition series.
State and prove Cayley’s theorem.
State and prove Gauss lemma.
Show that the polynomial
1 is irreducible over Q.
S State and prove Eisenstein criterion.
Part-B (2X5=10 Marks)
(150 Words)
6. State and prove Cauchy’s theorem for abelian groups.
7. If R is a commutative ring with unity and M is an ideal of R then show that
M is a maximal ideal of R if and only if R/M is a field.
PGRRCDE, OSMANIA UNIVERSITY, HYDERABAD‐ 7 M.sc ( Mathematics) PREVIOUS, INTERNAL ASSIGNMENT PAPER‐II ( REAL ANALYSIS) Answer All Questions Max marks: 20 PART‐A 5X2=10M 1) Give an example to show that the arbitrary intersection of open sets need not be open in a Metric space. 2) Give an example to show that a continuous function need not be uniformly continuous. 3) State the Fundamental Theorem of Integral Calculus. 4) State Weierstass’s M‐test. 5) State Inverse Function Theorem. PART‐B 2X5=10M 1) Let ( X , d ) and (Y , ρ ) be any two Metric spaces. Then show that a mapping f : X → Y is continuous if and only if f −1 (V ) is open in X for every open set V in Y . 2) Let γ be a curve on [a, b] such that γ ′ is continuous on [a, b] . Then show b
that γ is rectifiable and ∧ (γ ) = ∫ γ ′(t ) dt . a
PGRRCDE, OSMANIA UNIVERSITY, HYDERABAD‐ 7 M.sc ( Mathematics) PREVIOUS, INTERNAL ASSIGNMENT PAPER‐IV ( Elementary Number Theory) Answer All Questions Max marks: 20 PART‐A 5X2=10M 1) Express GC D of 56, 72 as 56 x + 72 y where x, y are integers. 2) Find φ (1600 ) , σ (1600 ) . 3) Solve the congruence 5 x ≡ 2(mod 26) . 4) Evaluate (
219
). 383
5) Find p(11) , p(12) where p(n) denotes the number of partitions of n. PART‐B 2X5=10M 1) a) Let n be a fixed positive integer and S be the set of all positive integers which are less than ‘ n ’ and relatively prime to it. What is the sum of all the elements of S . b) If p is a prime and if r ∈ Ν is such that 1 ≤ r < p , then show that p divides pC . r
2) Show that the quadratic congruence x 2 + 1 ≡ 0(mod p ) where p is an odd prime has a solution if and only p ≡ 1(mod 4) . P G R R C D E Osmania university
Assignment
M SC Mathematics (Previous)
Paper title: Mathematical methods
Paper: V
Max.marks:20.
Note: The assignment consists of two parts PART-A and PART-B carrying
10marks each. Answer all the questions from both PART-A and
PART-B.
PART-A (5X2=10 Marks)
(50 Words)
1. Explain F’robenius method of series solution.
2. Define Regular,Singular,and Ordinary points and hence the same to the
equation 2
0
3. Show that (i)
(ii)
4. Show that (i)
5. Express
2
cos
sin .
1
(ii)
1
1.
1 in terms of Legendre polynomials.
Part-B (2X5=10 Marks)
(150 Words)
6. State and prove orthogonal property of Legendre polynomials.
7. State and prove Rod rigues formula for Legendre polynomials.
FACULTY OF SCIENCE
M.Sc. (Previous) CDE : ASSIGNMENT - 2015
SUBJECT : STATISTICS
PAPER- III: DISTRIBUTION THEORY& MULTIVARIATE ANALYSIS
MAX. MARKS: 20
Name of the Student : ___________________________________ Roll No: ___________
SECTION-A (10X ½ = 5 Marks)
Give the correct choice of the answer like ‘a’ or ‘b’ etc in the brackets provided against the question.
1. Match the distribution with the property
a) log normal
i) M.D. = 4/5 σ
b) Pareto
ii) (1/c) !
c) Normal
iii) Mean = ak/(a-1)
d) Weibul
iv) Median = exp (µ)
a) a-i, b-ii, c-iii, d-iv b) a-ii, b-i, c-iv, d-iii c) a-i, b-ii, c-iv, d-iii
d) None of these
[
]1
2. Match the distribution with the property
a) log normal
i) Mean < Median > Mode
b) Pareto
ii) Mean > Median > Mode
c) Normal
iii) Mean < Median< Mode
d) Weibul
iv) Mean = Median = Mode
a) a-iii, b-ii, c-iv, d-i b) a-ii, b-i, c-iv, d-iii c) a-i, b-ii, c-iv, d-iii
d) None of these
[ ]2
3. The distributions belong to the exponential family among : i) Binomial ii) Negative Binomial iii)HyperGeometric iv) Uniform
v) Gamma
a) i, ii, iii, iv, v
b) i, ii, iv, v, vii
vi) Cauchy vii) Normal viii) Beta , is
c) i, ii, v, vii,viii
d) None of these
]3
[
4. Which of the following distribution is said to be the distribution of income
(a) Pareto
(b) log normal
(c) Normal
(d) None of these
[ ]4
5. If X1, & X2 are independent and follows Lognormal with ( μ 1 , σ 1 2 ) & ( μ 2 , σ 2 2 ) then X1X2 ____
(a) LN ( μ 1 + μ 2 , σ 12 + σ 22 )
(b) N ( μ 1 + μ 2 , σ 12 + σ 22 )
(c) LN ( μ 1 − μ 2 , σ 12 + σ 22 )
(d) N ( μ 1 − μ 2 , σ 12 + σ 22 )
6. Identify the correct MGF for the distribution
a) Binomial
i) (1-qeit)-1
b) Gamma
ii) (q + p eit )n
c) Cauchy
iii) exp( -| t | )
d) Laplace
iv) (1+t2)-1
a) a-i, b-ii, c-iii, d-iv b) a-ii, b-i, c-iv, d-iii
c) a-i, b-ii, c-iv, d-iii
d) None of these
7. . If A ∼WP(Σ, m) B∼ WP(Σ, n) (m, n ≥ p) are independent then |A| / |A+B| follows
a) Normal
b) WP(Σ, m+n)
c) Wilks Λ
d) None of these
8. If X ∼ NP ( μ, Σ) then the distribution of sample mean vector follows
a) NP (nμ, nΣ)
b) NP (μ/n, Σ/n)
c) NP (μ, Σ/n) d) None of these
9. If X ∼ NP ( μ, Σ), and consider a linear transformation Y(1) =X(1)+M.X(2), Y(2)
Covariance (Y(1), Y(2)) then the value for M is ______________.
a) Σ12Σ-122Σ21
b) -Σ12Σ-122Σ21
c) -Σ12Σ-122 d) None of these
10. If A ∼WP(Σ, n) then E[A] =
a) Σ
b) Σ/n
c) nΣ
d) None of these
[
]5
[
]6
[
]7
[
]8
(2)
=X with
[
]9
[
]
SECTION-B (10 x ½ = 5)
Fill in the blanks.
11. If X follows Standard Normal then X2 follows __________________ variate.
12. The sum of two independent Exponential variates is _________________________ .
13. If X & Y follows Chi-square with n1 and n2 d.f. independently then the ratio of two chi-squares
X/Y is _____________________________ variate.
− ( x+ y )
; x ≥ 0, y ≥ 0 then the marginal density of y is f(y) = ____
14. If f (x,y) = e
15. The If the Normal density function f(x)= ½exp{-½{(x1-1)2 + (x2-2)2}} then mean vector μ and
covariance matrix Σ of the distribution are _________________________
16. In CGF of two-variate normal distribution, the coefficients of (t12/2!) and (t22/2!)are _______
_________________________________________________________________________ .
17. The proportion of the total variance explained by the ith principal component is ____________
18. The main purpose of canonical correlation analysis is to study _____________
____________________________________________________.
19. In MDS, the measure stress is defined as ___________________________________
20. The Multi Dimensional Scaling is defined as ________________________________________
_____________________________________________________________________
SECTION-C (10x1=10)
Answer the following questions in brief.
1. Derive the MGF of standard Weibul distribution.
2. If X follows Lognormal distribution then derive the distribution of 1/X.
3. State the properties of Laplace distribution.
4. State the importance of Cauchy distribution
5. Define the Mixture distribution and illustrate with a suitable example.
6. State the applications of Wishart distribution
7. Explain the steps involved in the construction of Path diagram
8. State the Assumptions on Orthogonal Factor Model.
9. How the factor model can be obtained from Principal components model
10. Draw the Dendogram for the following distance matrix after clustering using Single linkage
method
D = 0 8 9
8 0 10
9 10 0
_______
FACULTY OF SCIENCE
M.Sc. (Previous) CDE : ASSIGNMENT - 2015
SUBJECT : STATISTICS
Paper-I : MATHEMATICAL ANALYSIS & LINEAR ALGEBRA
Name of the Student : ___________________________________ Roll No: _________
I.
Give the correct choice of the Answer like ‘a’ or ‘b’ etc in the brackets provided against the
question. Each question carries half Mark.
1.
If f(z) = 0, then ‘z’ is known as __________ of f(z)
a) Singular point
2.
n
∑ Δf
k =1
k
(
b) >
c) <
d) >
(
)
(
)
Which of the following conditions are equivalent to Riemann – Stieltjes
a) f ε R(α) on [a,b]
b) I ( f , α ) = I ( f , α ) c) neither (a) nor (b) d) both (a) and (b)
In bivariate case, if partial derivatives of a function f(x, y) with respect to ‘x’ and ‘y’ exist at
b), then f(x, y) _______ at (a, b).
a) is continuous
b) is discontinuous c) need not be continuous
d) need not be discontinuous
5.
)
______ M for all partitions on [a, b], where Δf k = f ( x k − f k −1 ) .
a) <
4.
d) Pole
A function ‘f’ is said to be a function of Bounded variation if there exists a positive number M such
that
3.
b) Analytical point c) Saddle point
A series of the form ao + a1(z – a) + a2(z – a)2 + . . . . . . . =
∞
∑a
n =0
n
(a,
(
)
( z − a) n is called _________ series
of (z – a)
a) Tailor
6.
7
8.
9.
b) Power
For any matrix A, (A/ A)+ is
a) (A/ )+ A+ b) A+ (A/ )+
c) Laurent
c) A A+
d) Analytical
d) A+ A
For a characteristic root of a matrix there corresponds
a) One characteristic vector
b) two characteristic vectors
c) Different characteristic vector
d) none of the above
If the system AX = b is consistent then the solution Xo = A+b is unique if
a) A+ = Ab) AA+= b c) A+A = b d) A+A = I
(
)
(
)
(
)
(
)
(
)
(
)
For any generalized inverse A- of an mxn matrix A and the matrices A-A and A A- are
a) Idempotent
b) orthogonal c) I
d) none of the above
10. If A is null matrix of order mxn then A+ is a null matrix of order
a) mxn
b) nxm c) mxm
d) nxn
II.
Fill in the blanks. Each question carries half Mark.
11.
A function ‘f’ is said to tend to a limit ‘Ç’ as ‘x’ tends to ‘h’ if the left limit and right limit exist and
are _______________________.
12.
If a function ‘f’ is monotonic on [a, b] then ‘f’ is a function of __________________________.
13.
Contour integral is also known as __________________________ integral.
14.
15.
If partial derivatives of a function exist, then the function _____________________ continuous.
If f(z) is not defined at z = a, but lim f ( z ) exists, the z → a is called _______________________
z→ a
singularity.
16. The necessary and sufficient condition for a linear transformation X = PY to preserve length is
that__________________
17. The MP inverse of the transpose of A is the transpose of the _______________
18. In the Hermite form of a
_________________________
matrix,
the
diagonal
elements
of
the
matrix
are
19. A solution to the system AX = b exists iff the rank of the coefficient matrix A is equal to the
rank of the_______________
20.
Let Abe an (nxn) matrix. The linear homogenous system AX = 0 has a solution other than
X = 0, iff_______________
III. Write short answers to the following. Each question carries TWO Marks.
21.
State Cauchy Residue theorem.
22.
Define removable discontinuity.
23.
Riemann-Steiltjes (R-S) Integral and its linear properties
24.
State additive property and sufficient conditions of a function of bounded variation.
25.
Define stationary point, extreme point, saddle point. Derive them with an example.
26. Compute the norm of [3,-1,-2,1].
27. Find a vector orthogonal to [2,2,0].
28. Define index and signature of a real Quadratic form.
29. Define algebraic and geometric multiplicity of characteristic roots.
30. Define Moore Penrose inverse and write any two properties of the same.
FACULTY OF SCIENCE
M.Sc. (Previous) CDE : ASSIGNMENT - 2015
SUBJECT : STATISTICS
PAPER- II: PROBABILITY THEORY
Name of the Student : ___________________________________ Roll No: ___________
N.B.: Answer all questions.
(a) Give the correct choice of the answer like ‘a’ or ‘b’ etc in the brackets provided against the
question, Each question carries ½ mark:
1. If {X n ; n ≥ 1} is a sequence of random variables such that P[ X n = 1 ] = 1/n and P[ X n = 0 ] = 1-1/n ;
∀ n ≥ 1 then the sequence {X n } converges in
(
)
(a) probability
(b) quadratic mean
(c) both
(d) None
−x
2. The distribution function of a random variable X is given by F(x) = 1− (1+ x)e ; for x ≥ 0 and
F(x)=0;otherwise.Then, the probability density function
(1 + x)e− x
(a)
(b) ( x − 1)e
−x
(e−x + x +1)
(c)
(d)
e−x
3. If A,B,C ∈ ζ and A and B imply C, then
(a) p(C ) ≥ p( A ) + P( B )
(b) p(C ) ≤ p ( A ) − P( B )
(c) p(C ) ≥ p( A ) − P( B )
(d) p(C / ) ≤ p ( A / ) + P( B / )
/
/
/
/
/
/
/
/
e − λ (e
(a)
−1)
5. If β n = E X
(a)
6.
(b)
n
e λ (e
it
+1)
(c)
e λ (e
it
−1)
(d)
e − λ (e
it
(b)
β k1 / k ≥ β kk ++11
(c)
β kk ≤ β kk ++11
(d)
If a sequence of independent random variables is defined as
[
]
P X n = 2n =
(a) 2 n
7. If X
i
1
; n ≥ 1 then E [ X
2
(b) 2 n + 1
~ N (μ ,σ
2
n
(b) i.i.d. Poisson
(
)
(
)
[
]
(c) 0
1
and
2
P X n = −2n =
(
)
(
)
(d) 1
n
∑
X
i =1
i
then V (Sn ) =
(b) n σ 2
(c) σ 2 /n
(d) σ
(a) σ 2
8. The random variables for which Borels strong law of large numbers is defined are
(a) i.i.d. binomial
)
β k1 / k ≤ β kk ++11
] is
); i = 1 , 2 ,... n ; independently and if S n =
(
+1)
< ∞ then for 2 ≤ k ≤ n we have
β k ≤ β k +1
)
/
4. The characteristic function of Poisson distribution is
it
(
(c) i.i.d. Bernoulli
/
n
(
(d) i.i.d. normal
)
9. For the following TPM, when the state space is S={0,1,2}then the absorbed state is
⎡1 / 3
P = ⎢⎢ 0
⎢⎣ 1 / 2
(a) 0
2 /3
0 ⎤
1
0 ⎥⎥
0
1 / 2 ⎥⎦
(b) 1
10. The notation
p
(3)
ij
(c) 2
(
)
(
)
(d) none
in the Markov chain represents
(a) Probability of reaching the state j from state i in 3 steps
(b) Probability of reaching the state i from state j in 3 steps
(c) Probability of not reaching the state j from state i in 3 steps
(d) Probability of not reaching the state i from state j in 3 steps
(b)
Fill up the blanks, Each question carries ½ mark:
1. If φ X (t ) , t ∈ R1 is a characteristic function of a random variable X then the characteristic function of
Z = (X – μ) / σ is ______________________where μ and σ2 are the mean and variances of X.
2. If a sequence of random variables is convergent in probability then
P ( X n − X ≥ ε ) → ______________, as n → ∞ .
3. In terms of Distribution function, P(a ≤X ≤b) = __________________________________________
4. If
{X n ; n ≥ 1} is a sequence of Bernoulli random variables such that P [X n = 1] = p and
⎡ Sn ⎤
⎥ = ____________________________________
⎣n⎦
P [ X n = 0] = q then E ⎢
5. If {X n ; n ≥ 1} is a sequence of large numbers such that Sn =
E [X i ] = μ i and V ( X i ) = σ 2 then
n
∑X ,
i=1
i
Sn p
⎡S ⎤
⎯
⎯→ E ⎢ n ⎥ as n → ∞ is known as
n
⎣n⎦
___________________ WLLN’s.
6. According to Chebychev’s inequality P( X − μ ≥ k ) ≤ ___________________________________
7. Bernoulli process is a Stochastic process in which state space is _______________________and
index set is _____________________.
8. The SLLN’s which is a particular case of Kolmogorov’s SLLN’s is
_____________________________.
9. A positive recurrent and aperiodic state is known as _____________________________________
state.
X +Y
inequality.
10. E
1/ 2
2
≤ E 1/ 2 X
2
2
+ E 1 / 2 Y is known as __________________________________
Each question carries 1 mark
Answer the following questions
1. Define distribution function and Probability density function.
2. State Lyapunovs inequality.
3. Define the mode of convergence in law:4. Obtain the characteristic function of Cauchy distribution.
5. State Slutzkys theorem and Borel 0 – 1 law
6. State Kolmogorovs SLLN’s and Kolmogorovs inequality.
7. State Levy – Lindeberg CLT and Khintchines WLLN’s.
⎡1/ 3 2/ 3 0⎤
8. In the following TPM, identify the closed class, when the state space is {0,1,2} P = ⎢1/ 2 1/ 2 0⎥ .
⎢
⎥
⎢⎣ 0 1 0⎥⎦
1
9. If P Xk = ± log k = ; k=1,2,3,…n then obtain V( X k ).
2
10. Define Initial distribution of a Markov chain.
[
]
FACULTY OF SCIENCE
M.Sc. (Previous) CDE : ASSIGNMENT - 2015
SUBJECT : STATISTICS
PAPER IV: SAMPLING THEORY and ESTIMATION THEORY
Name of the Student : ___________________________________ Roll No: ___________
Answer all questions.
(a) Give the correct choice of the answer like ‘a’ or ‘b’ etc in the brackets provided against
the question, Each question carries ½ mark:
1.
2.
SRSWOR is always ___________ efficient than SRSWR.
(
)
a)
less
b)
equally
c)
more
d)
None of the above
In optimum allocation of stratified random sampling, nh is large if stratum variability (
)
3.
Sh is _____________.
a)
large
b)
c)
zero
d)
In ppswr, the unbiased estimator of V( Yˆpps ) is
)
4.
1 n yi ˆ 2
∑ ( − Y pps )
n i =1 p i
1 n yi ˆ 2
c)
∑ ( − Y pps )
n − 1 i =1 p i
Bias of Rˆ is of the order
a)
a)
b)
5.
1
2n
1
n
V( ylrs ) is minimum if bh = Bh = _________
a)
c)
2
S yh
S xh2
S yxh
S xh2
b)
d)
small
None of the above
(
n
y
1
( i − Yˆpps ) 2
∑
n(n − 1) i =1 p i
None of the above
b)
1
n
d)
None of the above
b)
S xh2
2
S yh
d)
None of the above
6. If E( Statistic ) = Parameter then the statistic is known as
(a) Unbiased estimator
(b) Sufficient estimator
(c) Efficient estimator
(d) Consistent estimator
(
)
7. Rao-Blackwell theorem gives
a) MVUE
c) Unbiased and Consistent estimator
(
)
b) UMVUE
d) Sufficient estimator
(
)
(
)
8. If X
i
~ P ( λ ); i = 1 , 2 ,... n ; then the estimator of
2
λ
(
)
2
(b) S
(c) s
(d) Both (a) and (b)
(a) X
9. Let x1, x2, . . . . . . ,xn be a random sample from N(θ,σ2), both parameters unknown. Then the
joint sufficient estimator for (θ,σ2) is i) (Σxi, Σxi2) ii) ( x , s2) iii) (Σxi, (Σxi-θ)2) iv) ( x , (Σxiθ) 2 )
a) (i)
b) (i) and (iii)
c) both (i) and (ii)
d) both (i) and (iv)
(
)
10. Fisher’s information of the random sample X1, X2, . . . . . Xn is written as
d log L
d log L 2
a) E[
]
b) E[
]
dθ
dθ
d log L
d log L 2
c) E[
]=0
d) [E
]
(
dθ
dθ
(b) Fill up the blanks, Each question carries ½ mark:
)
1.
In Stratified Random Sampling, the population is __________________________.
2.
In Systematic sampling, V( y sys ) = ________________________ .
3.
If the total sample size ‘n’ is large then V( YˆRC ) = __________________________ .
4.
In cluster sampling, the relationship between
S
2
, S b2 and S w2 is S2 =
____________.
5.
In two-stage sampling, ‘n’ first stage units and ‘m’ second stage units from each chosen first
r
stage units are selected by SRSWOR then its sampling variance is given by V( y ) =
__________________________.
6. A specified value of the estimator based on the sample is _________________.
7. If X i ~ N ( μ , σ 2 ); i = 1, 2 ,... n ; then among the estimators x and M d of
μ , the more
efficient estimator is _________________________.
n
8. X i ~ b (1, p ); i = 1, 2 ,... n ; then T=
∑
i =1
x i is a sufficient statistic for________.
9. Let x1, x2, . . . . . . ,xn be a random sample from G(α,β) population. Then the sufficient estimator
for (α,β) is _________________________
10. Let x1, x2, . . . . . . ,xn be a random sample with pdf f(x, θ) and T(x) be an unbiased estimate of
τ(θ). Then Cramer-Rao lower bound for varience of T under some regularity conditions is
_______________________________________________
Each question carries 1 mark
Answer the following questions
1. Define ratio estimator and regression estimator.
2. Define ppswor and ppswr.
3. Define cluster sampling and two –stage sampling . Give an example for each.
4. Define non-sampling error.
5. Write about non-sampling bias.
6. Name the methods of constructing nonparametric density estimators.
7. Define the Generalized Jackknife estimator.
8. Define a complete estimator.
9. State Rao – Blackwell theorem.
10. Define a consistent estimator and give an example.