Exam Details
| Subject | statistics | |
| Paper | paper 1 | |
| Exam / Course | indian forest service | |
| Department | ||
| Organization | union public service commission | |
| Position | ||
| Exam Date | 2015 | |
| City, State | central government, |
Question Paper
STATISTICS
Paper I
Time Allowed: Three Hours
maximum Marks 200
QUESTION PAPER SPECIFIC INSTRUCTIONS
Please read each of the following instructions carefully before attempting questions.
There are EIGHT questions in an, out of which FIVE are to be attempted.
Question No. I _and 5 are compulsory. Out of the remaining SIX questions, THREE are to be
attempted selecting at least ONE question from each of the two Sections A and B.
Attempts of questions shall be counted in sequential order. Unless struck off, attempt of a question shall be counted even if attempted partly. Any page or portion of the page left blank in the answer book must be clearly struck off.
All questions carry equal .marks. The number of marks Carrie by .a question/part is indicated against it. Answers must be written in ENGLISH only. Unless otherwise mentioned, symbols and notations have their usual standard meanings.
Assume suitable data, if necessary and indicate the same clearly.
SECTION
1. Answer all of the following.: 8x5=40
Define conditional probability of event A given B. Show that for any three events B and C with
If the random variable X has the probability density function
e-xt/j x 0 O·
fJ
,fJ •
f
X
otherwise,
obtain the distribution of the random variable
r
Define the moment generating function of a random variable. Obtain the mgf
of X where follows Poisson distribution with parameter Ai ... n).
is:l
Assume are independent.
I
•
Prove The minimum variance unbiased estimator of a parameter is unique.
Define an unbiased test: Show that a most powerful test of a simple hypothesis
against a simple alternative is necessarily unbiased.
1
c-geq-o-tuua
An ecologist wishes to mark off a circular sampling region having radius l O mts. However, due to reasons of topography etc, radius of actual region marked a
random variable R with probability density function
.. 9:-Sr..ll
..
f
4
R
otherwise:
What is the expected area of the. resulting circular region
10 Consider the random variable X with probability density function {3af3
Verify that fx is a pdf. Show that the distribution is positively skewed.
10
2.( Examine
whether the WLLN holds for independent random variables with
distribution
l .
1
P[Xn
P[X11 n2 n
n
....
.
10
Define characteristic function of a random variable. Show that for-two independent random variables X and Y the characteristic functions lf/ X and If/ y satisfy the relation lj/ x IJI If!
X
Y Z
where 1/f
is the characteristic function of Z X Y. 10
z Let X1 X2, .... Xn be a random sample from an exponential distribution with pdf
X 0 Q·
ix e
otherwise. Find MVB estimator of e. · 10 l 2
Explaining· SPRT, develop it for testing H0 =-against H1 e -for the 3 3distribution
fx
X 0 8
otherwise_
10
Given a random sample X1, X2,function
... .., Xn from a distribution with probability density
fx X 0,. 8 0.
Show that there exists no UMP test for testing H0 80 against H1 8 fJ0• 10 Define a maximum likelihood estimator of a parameter. Obtain MLE of cr2 based on · a random sample of size n from a cr2) population when
µ is known, and is unknown. Examine whether estimators obtained are unbiased. 10
c-geq-o-tuua
2
J
Show that under squared error loss function posterior. mean is the Bayes' . ... estimator of a parameter .
•
·Let X p). Assuming n known, obtain the Bayes' estimator of p under SEL when a priori distribution of p is n 0 p 1. I4.(b) State and prove the Rao.Blackwell theorem. Explain its usefulness in finding a 0
UMVU estimator of an unknown parameter. 1 04.(c) Show that for Cauchy distribution with location parameter µ and scale parameter er,
having pdf
1
1UY
the three quartiles of the distribution are given by
Q1
Q2
Q3
µ+a. 10 Define con·vergence in probability and convergence in distribution. Show that the former implies the latter. When is the converse true Substantiate your answer.
·..z • 1 O
SECTION
5. Answer all of the following 8x5=4ff · Suppose that Y1 Y2 and Y3 are independent random observations with
2
02 02 83 =03 -01 and i 3._ Examine whether 82 is estimable.
Obtain the best estimator for 81 382 283•5.(b) The all..le of a pea section can be AA, Aa and aa with probabilities pand p
12 3respectively. Let X1, X2 and X3
categories in a sample of size 10denote respectively the frequencies of these three . Write down the joint probability mass function of (X1 X2, X3) and calculate th c..rre..ation coefficient between X1 and X2.
Let X be N3 with r2 4 0].0 0 1Which pairs of variables are independent Find correlation coefficient between
.
X1 and X2•5.(d) From a bivariate normal N2 distribution, a random sample of size 3 is drawn.
The un..iascd estimates of µ . and are X S l Compute
Hot..lling's T2 and test the null hypothesis H0
.
5.( Consider two blocks with two plots each and yields as below
Block 1 Y1 1 Y12Block 2
Y23 Y24
Show that the treatment contrasts T1 • r1 -T3 and T2 are not estimabl...
3 c-geq-o-tuua
For a linear model, obtain the least squares estimate of the vector of parameters
and show that it .is unbiased. .
.._.
Define multiple correlation coefficient R1.23 ··-P between x1 and x2, x3, .•.. xP.
Show that 1-Ri.23 ·.... p . where R11 is the minor of the element in the first row
and first cplumn of the dbrrelation rn..trix R . 10 · 6.( Define a p-variate normal distribution. When is it said to .be singular If
. 1 l
1
X -N3 cd and L 1 find the distribution of the
vector (X1
-X2 X2 -X3 .
10
·Explain the problem of discriminant analysis. Describe the allocation rule
under Fisher discriminant function. Show that it minimises the misclassification
probability. 1 0
What is cluster sampling Considering clusters of unequal sizes, suggest an
unbiased estimator of the population mean and obtain its variance. 1 0
Consider all possible samples of size 2 from a• population of size namely
.
1
s1 s2 ands3 with probabilities P(s1 3.
3
Define the estimator
Yi I
y2 if
4
1 ·1
-y3
s3 1s se 1ected
-y2
4 2
Examine whether t is unbiased. Find its variance. 1 0 Construct an LSD of size 5. Delete one column from the LSD. Prove that the . resulting design is a symmetrical BIBD with parameters v b r k 4 and 3. 10 . Why is a split-plot design needed Give two examples to motivate the use of such a design. Present the relevant ANOVA table. I 0 Why were randomised response techniques devised Describe Warner technique for estimating the population proportion relating to the sensitive character. 1 o· Define an orthogonal contrast. Explain Tukey's test for testing its significance. 10 s .. Explain the need for factotj.al experiments. Develop a method to estimate all main effects of a 23 factorial experiment. Give its ANOVA table. l 0 Define Mahalanobis D2 statistic. Show that it is invariant. Give art appHcation · of this statistic. 10
1·
r ·
.
c-geq-o-tuua
4
Paper I
Time Allowed: Three Hours
maximum Marks 200
QUESTION PAPER SPECIFIC INSTRUCTIONS
Please read each of the following instructions carefully before attempting questions.
There are EIGHT questions in an, out of which FIVE are to be attempted.
Question No. I _and 5 are compulsory. Out of the remaining SIX questions, THREE are to be
attempted selecting at least ONE question from each of the two Sections A and B.
Attempts of questions shall be counted in sequential order. Unless struck off, attempt of a question shall be counted even if attempted partly. Any page or portion of the page left blank in the answer book must be clearly struck off.
All questions carry equal .marks. The number of marks Carrie by .a question/part is indicated against it. Answers must be written in ENGLISH only. Unless otherwise mentioned, symbols and notations have their usual standard meanings.
Assume suitable data, if necessary and indicate the same clearly.
SECTION
1. Answer all of the following.: 8x5=40
Define conditional probability of event A given B. Show that for any three events B and C with
If the random variable X has the probability density function
e-xt/j x 0 O·
fJ
,fJ •
f
X
otherwise,
obtain the distribution of the random variable
r
Define the moment generating function of a random variable. Obtain the mgf
of X where follows Poisson distribution with parameter Ai ... n).
is:l
Assume are independent.
I
•
Prove The minimum variance unbiased estimator of a parameter is unique.
Define an unbiased test: Show that a most powerful test of a simple hypothesis
against a simple alternative is necessarily unbiased.
1
c-geq-o-tuua
An ecologist wishes to mark off a circular sampling region having radius l O mts. However, due to reasons of topography etc, radius of actual region marked a
random variable R with probability density function
.. 9:-Sr..ll
..
f
4
R
otherwise:
What is the expected area of the. resulting circular region
10 Consider the random variable X with probability density function {3af3
Verify that fx is a pdf. Show that the distribution is positively skewed.
10
2.( Examine
whether the WLLN holds for independent random variables with
distribution
l .
1
P[Xn
P[X11 n2 n
n
....
.
10
Define characteristic function of a random variable. Show that for-two independent random variables X and Y the characteristic functions lf/ X and If/ y satisfy the relation lj/ x IJI If!
X
Y Z
where 1/f
is the characteristic function of Z X Y. 10
z Let X1 X2, .... Xn be a random sample from an exponential distribution with pdf
X 0 Q·
ix e
otherwise. Find MVB estimator of e. · 10 l 2
Explaining· SPRT, develop it for testing H0 =-against H1 e -for the 3 3distribution
fx
X 0 8
otherwise_
10
Given a random sample X1, X2,function
... .., Xn from a distribution with probability density
fx X 0,. 8 0.
Show that there exists no UMP test for testing H0 80 against H1 8 fJ0• 10 Define a maximum likelihood estimator of a parameter. Obtain MLE of cr2 based on · a random sample of size n from a cr2) population when
µ is known, and is unknown. Examine whether estimators obtained are unbiased. 10
c-geq-o-tuua
2
J
Show that under squared error loss function posterior. mean is the Bayes' . ... estimator of a parameter .
•
·Let X p). Assuming n known, obtain the Bayes' estimator of p under SEL when a priori distribution of p is n 0 p 1. I4.(b) State and prove the Rao.Blackwell theorem. Explain its usefulness in finding a 0
UMVU estimator of an unknown parameter. 1 04.(c) Show that for Cauchy distribution with location parameter µ and scale parameter er,
having pdf
1
1UY
the three quartiles of the distribution are given by
Q1
Q2
Q3
µ+a. 10 Define con·vergence in probability and convergence in distribution. Show that the former implies the latter. When is the converse true Substantiate your answer.
·..z • 1 O
SECTION
5. Answer all of the following 8x5=4ff · Suppose that Y1 Y2 and Y3 are independent random observations with
2
02 02 83 =03 -01 and i 3._ Examine whether 82 is estimable.
Obtain the best estimator for 81 382 283•5.(b) The all..le of a pea section can be AA, Aa and aa with probabilities pand p
12 3respectively. Let X1, X2 and X3
categories in a sample of size 10denote respectively the frequencies of these three . Write down the joint probability mass function of (X1 X2, X3) and calculate th c..rre..ation coefficient between X1 and X2.
Let X be N3 with r2 4 0].0 0 1Which pairs of variables are independent Find correlation coefficient between
.
X1 and X2•5.(d) From a bivariate normal N2 distribution, a random sample of size 3 is drawn.
The un..iascd estimates of µ . and are X S l Compute
Hot..lling's T2 and test the null hypothesis H0
.
5.( Consider two blocks with two plots each and yields as below
Block 1 Y1 1 Y12Block 2
Y23 Y24
Show that the treatment contrasts T1 • r1 -T3 and T2 are not estimabl...
3 c-geq-o-tuua
For a linear model, obtain the least squares estimate of the vector of parameters
and show that it .is unbiased. .
.._.
Define multiple correlation coefficient R1.23 ··-P between x1 and x2, x3, .•.. xP.
Show that 1-Ri.23 ·.... p . where R11 is the minor of the element in the first row
and first cplumn of the dbrrelation rn..trix R . 10 · 6.( Define a p-variate normal distribution. When is it said to .be singular If
. 1 l
1
X -N3 cd and L 1 find the distribution of the
vector (X1
-X2 X2 -X3 .
10
·Explain the problem of discriminant analysis. Describe the allocation rule
under Fisher discriminant function. Show that it minimises the misclassification
probability. 1 0
What is cluster sampling Considering clusters of unequal sizes, suggest an
unbiased estimator of the population mean and obtain its variance. 1 0
Consider all possible samples of size 2 from a• population of size namely
.
1
s1 s2 ands3 with probabilities P(s1 3.
3
Define the estimator
Yi I
y2 if
4
1 ·1
-y3
s3 1s se 1ected
-y2
4 2
Examine whether t is unbiased. Find its variance. 1 0 Construct an LSD of size 5. Delete one column from the LSD. Prove that the . resulting design is a symmetrical BIBD with parameters v b r k 4 and 3. 10 . Why is a split-plot design needed Give two examples to motivate the use of such a design. Present the relevant ANOVA table. I 0 Why were randomised response techniques devised Describe Warner technique for estimating the population proportion relating to the sensitive character. 1 o· Define an orthogonal contrast. Explain Tukey's test for testing its significance. 10 s .. Explain the need for factotj.al experiments. Develop a method to estimate all main effects of a 23 factorial experiment. Give its ANOVA table. l 0 Define Mahalanobis D2 statistic. Show that it is invariant. Give art appHcation · of this statistic. 10
1·
r ·
.
c-geq-o-tuua
4