Exam Details
| Subject | statistics | |
| Paper | paper 1 | |
| Exam / Course | indian forest service | |
| Department | ||
| Organization | union public service commission | |
| Position | ||
| Exam Date | 2014 | |
| City, State | central government, |
Question Paper
SECTION-·A
Q. 1. Answer the following
Q. If B and C are events in a sample space, show that
P(A u 8 u S 8X5=40
Bl'b2 and 83 are mutually exclusive events with 1/3 and P(A I 8j)
j l. 2. 3. Evaluate 8
Q.l(b) is a bivariate random ve.ctor with P(X Y exp Y .. 0.
Find the margfoal and .lnditional distributions. Also find E(Y X). 8
Q. 1 Obtain the distribution fur which the characteristic function is
if x and Y are independent random variables having the characteristic function state the distribution of Y).
Q.1(d) Obtain l00(1-alpha)% confidence intervel for the ratio of variances based on two samples from
N(µ1,sigma12) and N(µ2,sigma22) when both .µ1 and µ2 are known and unknown.
Q. Find the maximum likelihood estimate of a in the density
0 x 1
0 otherwise. 8
Q. Identify a distribution, each, for which mean variance and mean variance.
Among the 120 applicants. for a job only 80 are actually qualified. If five of the applicants are randorltly selected for an in-depth interview, find the probability that at
least two of the five will be from among those qualified for the job. 10
Q. If X and Yn Y discuss about the convergence of (Xn Yn) and Xn Yn. 10
Q. X is a random variable having the binomial distribution with parameters n and 0. Show
that the distribution of
X-n0/route
approaches the ·standard normal distribution as n ->sigma.
Q. A fair coin tossed uhtil a head appears. If X denotes the number of tosses required, find
the p.d.f. of probability generat..ng function of X and the mean and variance of X. 10
Q. For what value of there exists an UMP test for testing
Ho µ µo, lamda =lamda0 when a sample is drawn from a distribution with density
f(x;mew,lamda)=1/lamda e-(x-mew)/lamda,mew<x<infinity
Q. If T is unb1ased for et show that T2 is biased for 02
If T is consistent tfor show that T2 is also consistent for 02• 10
Q. Find the maximum likelihood estimate of 0 based on a random sample from a distribution
with density
Q. Write short notes on
Run test
Kolmogorov-Smironov test
with illustrative examples. 10
Q.4(a) If the random variable X has· an exponential distributiont show that the relationship
P(X
holds for all real s 0. Discuss about the utility of this n:sult in life time data analysis.
Q. Examine the validity of the statement "tne initial probability vector,and the t.p.m. uniquely
determine a Markov chain". Explain how you will classify the states in a Markov chain. illustrate by an example. 10
Q. If X1, X2, Xn is a random sample from a distribution with a discrete probability function
x 0 ...
0 p 1
0 otherwise,
find a sufficient statistic for 8. (Assume p is known) 10
Q.4(d) Find the MVB (minimum variance bound) estimator for the parameter of Poisson distribution and obtain the value of MVB. 10
SECTION.B
Q.5.Answer all of the following
8x5=40
Q. Three independent observations follow the Gauss·Markoff linear model
Find all error functions and the error degrees of freedom. 8
Q. Give an example to show that the normality of the conditional probability density function
does not imply the· bivariate density to be normal. 8
Q. Define cluster sampling for proportions and write down the situations where it. is used. Also find the unbiased estimator of population pr9portion. . 8
Q. Distinguish between fixed effect, and random effect models. Find the expected value of mean squares, for two.way classified. data with one observation per cell underthe random effect model. 8
Q. Develop Fisher's inequality for a BIBD. 8
Q. Consider two indepEndent experiments ·leading to the observation of a 4·vector and a 2-vector respectively. is subject to the Gauss.Markoff model x(l>beta, sigma2i4)
with <img images/1285-6a1.jpg'>
and iS subject to the model (y(2),x(2)beta,sigma2i2) with both
1 0 1 1 1 0 1),beta=(beta1,beta1,beta3,beta4), and sigma2>0 being same in both cases.
Find the BLUEs beta1(1) beta1(2) and nbeta1 of beta1 on the basis of alone alone and
taken together respectively.
Find the best linear combination of b191) andb1(2)and compare its precision with that
of 10
Q. A car manufacturing bompany plans to test the average life of each of the four brands of tyres. The co..pany {uses aU the four brands on randomly selected caii. The records showing the lives (h..ndreds of miles) of tyres are given in die following table below
brand1:20 23 18 17
Brand2 :19 15 17 20 16
Brand3 :21 19 20 17
Brand4 :15 17 16 18
Test the hypothesis tHat the average life for each brand of tyres is the same. Assume 1 level .of significance.(F3,14,.01 5.56 and F4,13,0.1 5.21)
Q. Let Y be a univariate random variable (r.v.) and X be a p pinto1 random Vector. Let alpha y alpha)
alpha=1... N be thy data available on X and Y. • I
Define the sample principal component of X and discuss their uses.
Compare the multiple correlations between Y and X and between Y and U where
U is a p x 1 sample -principal component of X.
variaqce-covariance matrix for X and Y can be taken for ..e above study) 10
Q. Consider three sampl.. observations from a trivariate normal distribution sigma) where
· µ 1 Let the observations be X1, X2, X3• Write Y1 X1 X2, Y2 X2+ X3 and
y3 X3+x1.Then obtain
the joint distribution of Y1, Y 2 and Y3•
the marginal distribution of Y 3.
the conditional .mean and dispersion of y1 y2 gi..en Y3
the conditional mean and variance of Ygiven Y2 3 and Y3 =4.
the conditional mean and dispersion of (y1 y2) given y3=3.
Q. When is an Incomplete Block Design said to be connected Show that a connected incomplete block design is balanced if and only if all the non-zero characteristic roots of
c-matrix are equal. 10
Q. Write down the need of factorial designs and explain the analysis of a 23 factorial design, clearly stating the assumption used.10
Show that the variance of mean of systematk sample is Var(y sys) s2/n where p is the correlation between pairs of units that are in the same systematic sample. 10
Q. Explain the need of probability proportion to size sampling. and obtain the variance of
Horvitz-Thompson .estimator. 10
Q. Show that the prediction of X1 by means of a linear equation in X2, X3• ......... Xp-1 ir1 will
be improved by including XP as well (as an independent variable) iff p 1p.2.3..(p-1)is non
zero. (Necessary result needs to be proved). 10
Q.8(b) The following summarized data refer to a sample of ..pproximately 2-year old boys from Sikkim. For low land children of the same age; the height, chest and MUAC (mid uppper
arm circumference) means are considered to be 90, 58 and 16 cm. respectively. Test the
hypothesis that the Sikkim children also have the same means.
<img src='./qimages/1285-8b.jpg'>
(f.0.1,3.3=29.5)
Q. What are the situations for which two phase sampling is used Find the unbiased estimator
of population total for ·such a sampling scheme and obtain its variance. · 10
Q. Discuss the classification of Lattice designs, and the allocation of treatments within the
complete block in .each replication. 10 ·
C-HENT-N-RUZUA
Q. 1. Answer the following
Q. If B and C are events in a sample space, show that
P(A u 8 u S 8X5=40
Bl'b2 and 83 are mutually exclusive events with 1/3 and P(A I 8j)
j l. 2. 3. Evaluate 8
Q.l(b) is a bivariate random ve.ctor with P(X Y exp Y .. 0.
Find the margfoal and .lnditional distributions. Also find E(Y X). 8
Q. 1 Obtain the distribution fur which the characteristic function is
if x and Y are independent random variables having the characteristic function state the distribution of Y).
Q.1(d) Obtain l00(1-alpha)% confidence intervel for the ratio of variances based on two samples from
N(µ1,sigma12) and N(µ2,sigma22) when both .µ1 and µ2 are known and unknown.
Q. Find the maximum likelihood estimate of a in the density
0 x 1
0 otherwise. 8
Q. Identify a distribution, each, for which mean variance and mean variance.
Among the 120 applicants. for a job only 80 are actually qualified. If five of the applicants are randorltly selected for an in-depth interview, find the probability that at
least two of the five will be from among those qualified for the job. 10
Q. If X and Yn Y discuss about the convergence of (Xn Yn) and Xn Yn. 10
Q. X is a random variable having the binomial distribution with parameters n and 0. Show
that the distribution of
X-n0/route
approaches the ·standard normal distribution as n ->sigma.
Q. A fair coin tossed uhtil a head appears. If X denotes the number of tosses required, find
the p.d.f. of probability generat..ng function of X and the mean and variance of X. 10
Q. For what value of there exists an UMP test for testing
Ho µ µo, lamda =lamda0 when a sample is drawn from a distribution with density
f(x;mew,lamda)=1/lamda e-(x-mew)/lamda,mew<x<infinity
Q. If T is unb1ased for et show that T2 is biased for 02
If T is consistent tfor show that T2 is also consistent for 02• 10
Q. Find the maximum likelihood estimate of 0 based on a random sample from a distribution
with density
Q. Write short notes on
Run test
Kolmogorov-Smironov test
with illustrative examples. 10
Q.4(a) If the random variable X has· an exponential distributiont show that the relationship
P(X
holds for all real s 0. Discuss about the utility of this n:sult in life time data analysis.
Q. Examine the validity of the statement "tne initial probability vector,and the t.p.m. uniquely
determine a Markov chain". Explain how you will classify the states in a Markov chain. illustrate by an example. 10
Q. If X1, X2, Xn is a random sample from a distribution with a discrete probability function
x 0 ...
0 p 1
0 otherwise,
find a sufficient statistic for 8. (Assume p is known) 10
Q.4(d) Find the MVB (minimum variance bound) estimator for the parameter of Poisson distribution and obtain the value of MVB. 10
SECTION.B
Q.5.Answer all of the following
8x5=40
Q. Three independent observations follow the Gauss·Markoff linear model
Find all error functions and the error degrees of freedom. 8
Q. Give an example to show that the normality of the conditional probability density function
does not imply the· bivariate density to be normal. 8
Q. Define cluster sampling for proportions and write down the situations where it. is used. Also find the unbiased estimator of population pr9portion. . 8
Q. Distinguish between fixed effect, and random effect models. Find the expected value of mean squares, for two.way classified. data with one observation per cell underthe random effect model. 8
Q. Develop Fisher's inequality for a BIBD. 8
Q. Consider two indepEndent experiments ·leading to the observation of a 4·vector and a 2-vector respectively. is subject to the Gauss.Markoff model x(l>beta, sigma2i4)
with <img images/1285-6a1.jpg'>
and iS subject to the model (y(2),x(2)beta,sigma2i2) with both
1 0 1 1 1 0 1),beta=(beta1,beta1,beta3,beta4), and sigma2>0 being same in both cases.
Find the BLUEs beta1(1) beta1(2) and nbeta1 of beta1 on the basis of alone alone and
taken together respectively.
Find the best linear combination of b191) andb1(2)and compare its precision with that
of 10
Q. A car manufacturing bompany plans to test the average life of each of the four brands of tyres. The co..pany {uses aU the four brands on randomly selected caii. The records showing the lives (h..ndreds of miles) of tyres are given in die following table below
brand1:20 23 18 17
Brand2 :19 15 17 20 16
Brand3 :21 19 20 17
Brand4 :15 17 16 18
Test the hypothesis tHat the average life for each brand of tyres is the same. Assume 1 level .of significance.(F3,14,.01 5.56 and F4,13,0.1 5.21)
Q. Let Y be a univariate random variable (r.v.) and X be a p pinto1 random Vector. Let alpha y alpha)
alpha=1... N be thy data available on X and Y. • I
Define the sample principal component of X and discuss their uses.
Compare the multiple correlations between Y and X and between Y and U where
U is a p x 1 sample -principal component of X.
variaqce-covariance matrix for X and Y can be taken for ..e above study) 10
Q. Consider three sampl.. observations from a trivariate normal distribution sigma) where
· µ 1 Let the observations be X1, X2, X3• Write Y1 X1 X2, Y2 X2+ X3 and
y3 X3+x1.Then obtain
the joint distribution of Y1, Y 2 and Y3•
the marginal distribution of Y 3.
the conditional .mean and dispersion of y1 y2 gi..en Y3
the conditional mean and variance of Ygiven Y2 3 and Y3 =4.
the conditional mean and dispersion of (y1 y2) given y3=3.
Q. When is an Incomplete Block Design said to be connected Show that a connected incomplete block design is balanced if and only if all the non-zero characteristic roots of
c-matrix are equal. 10
Q. Write down the need of factorial designs and explain the analysis of a 23 factorial design, clearly stating the assumption used.10
Show that the variance of mean of systematk sample is Var(y sys) s2/n where p is the correlation between pairs of units that are in the same systematic sample. 10
Q. Explain the need of probability proportion to size sampling. and obtain the variance of
Horvitz-Thompson .estimator. 10
Q. Show that the prediction of X1 by means of a linear equation in X2, X3• ......... Xp-1 ir1 will
be improved by including XP as well (as an independent variable) iff p 1p.2.3..(p-1)is non
zero. (Necessary result needs to be proved). 10
Q.8(b) The following summarized data refer to a sample of ..pproximately 2-year old boys from Sikkim. For low land children of the same age; the height, chest and MUAC (mid uppper
arm circumference) means are considered to be 90, 58 and 16 cm. respectively. Test the
hypothesis that the Sikkim children also have the same means.
<img src='./qimages/1285-8b.jpg'>
(f.0.1,3.3=29.5)
Q. What are the situations for which two phase sampling is used Find the unbiased estimator
of population total for ·such a sampling scheme and obtain its variance. · 10
Q. Discuss the classification of Lattice designs, and the allocation of treatments within the
complete block in .each replication. 10 ·
C-HENT-N-RUZUA