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CON 159 7
CON 159 8
Paper II Descriptive type on Statistics
A Probability and Sampling
1. State and prove Basic inequality.
Show that
3
2
P(0 if X has probability density function 0
2
e

2
x
x
x
x
f
and otherwise.
2. Describe linear systematic sampling. If N nk and Yi i for i ....., N.
Derive variance of Ysys. Show that in this case systematic sampling is more
precise than simple random sampling.
In nursaries that produce young trees for sale, it is advisable to estimate, in
early spring, how many healthy young trees are likely to be on hand. A
study of sampling methods for the estimation of total number of seedlings
was undertaken. The data that follow were obtained from a bed of silver
maple seedlings 1 ft. wide and 400 ft. long. The sampling unit was 1 ft. of
the length of bed, so that N 400. By complete enumeration of the bed it
was found that Y 1.96, S2 80, these being the true population values.
With simple random sampling how many units must be taken to estimate Y
within 10% of the true value with a confidence coefficient of 95%. (Zα 1.96).
3. Define ratio estimate. Show that it is biased. Explain how to obtain unbiased
ratio type estimator.
A population consists of 5 units, with response variable Y taking values
and auxiliary variable X taking values Consider all simple
random samples without replacement of size 3 and obtain Bias
of Y ˆ
R ˆ
R denotes the ratio estimate of population mean.
B Linear Models and Economics Statistics
4. Consider the usual model Y X β e where e N In 0,σ2 . Suppose X is a n × p
matrix of rank p and let A be a p × k matrix of rank k. Derive the appropriate test
statistic to test β 0 . Also find its distribution. Comment on the case k 1.
Let y1 ß1 ß2 e1, y2 ß3 e2 and y3 ß1 ß2 e3 where e1, e2 and e3 are
σ 2). Derive a test to test the 2ß1 ß2 0.
5. Consider the two-way classification model—
yij α i ßj eij i ......p, j ....., q
State the assumptions required for the analysis and dervie the tests for the
hypotheses H α 1 α 2 ...... α p and H ß1 ß2 ....... ßq .
CON 159—3 9
Assume that an experiment was run on three ovens, each at four temperatures
to ascertain the strength of the final product. The observations are given in
the following table. Obtain ANOVA and write your conclusions.
Oven 1 2 3
Temperature
1 3 4 3
2 6 6 8
3 3 3 5
4 4 3 7
Fα 4.76 Fα 5.14 and α 0.05.
6. Discuss different problems that arise in the construction of cost of living index
numbers.
Derive Fisher's formula for cost of living index numbers. Show that it satisfies
Time reversal test and Factor reversal test.
Using the data given below find Fisher's Price Index Numbers and show that
it satisfies time reversal test—
Commodity 1999 2002
Quantity Price Quantity Price
Rice 50 32 50 30
Barley 35 30 40 25
Maize 55 16 50 18
C Statistical Inference
7. Let 1
1

Yi are independent identically distributed rvs with uniform
Show that the sample median and Y 2n+1
2 1
1


Y n
n
i
i the
sample mean are consistent estimators. Which estimators will you prefer Why
Let X1 X2 ......, Xn be iid rvs with exponential distribution, ƒ where
ƒ θ
1 x
θ
x θ 0. Prove that the UMVUE of ƒ is given
by h x t the conditional pdf of X1 given T where T

n
i
Xi
1

x t
t
n t x
h x t n
n






1
2
0 x t.
CON 159 10
8. Define Similar Test Neyman Structure Test. Prove that every test
having Neyman structure for θ is a similar test where is a boundary
of and for testing H0 θ against H1 θ .
Let X1 X2 ,...... Xn be a random sample drawn from uniform distribution U θ).
Find out a UMP size α test for testing—
H0 θ 0 against H1 θ 0 H0 θ 0 against H1 θ 0 .
9. Define the probability of concordance and probability of discordance
Obtain an unbiased estimate of .
In a flower show the judges agreed that five exhibits were outstanding and
these were numbered arbitrarily from 1 to 5. Three judges each arranged
these five exhibits in order of merits, giving the following rankings—
Judge A 5 3 1 2 4
Judge B 3 1 5 4 2
Judge C 5 2 3 1 4
Compute Kandall's sample tau coefficient T from the three possible pairs of
rankings.
D Stochastic Processes
10. If i j i communicates with j and i is recurrent then show that j is also recurrent.
Classify the states of the Markov Chain with following transition probability
matrix—
0 1 2 3 4 5
0 0 1 0 0 0 0
1 0 0 1 0 0 0
2 0 0 0 1 0 0
3 0.5 0 0 0 0.5 0
4 0 0 0 0 0 1
5 1 0 0 0 0 0
Find the period of the Markov Chain and the stationary distribution.
11. In a Poisson Process n events have occurred during t). Show that probability
distribution of the number of events that have occurred during s t
is binomial with success probability s/t.
Let Xt, be a Markov Process on with generator







λ λ

A
Write down the Kolmogorov forward equation and solve them for transition
probabilities pij for j 2. Find Xt 2 X0 X3t 1
12. A time series model is specified by yt yt-1 0.5 yt-2 et where et is white noise
process with variance σ2 . Determine whether the process is stationary. Also
derive autocorrelation 2 for this process.
CON 159 11
The time series Zt is believed to follow a ARIMA process for some value
of d. The time series Zt obtained by differentiating k times and sample
autocorrelation are shown in the table below for values of k—
k=0 k=1 k=2 k=3
r1 1.00 1.00 0.83 -0.03
r2 1.00 1.00 0.66 -0.12
r3 1.00 1.00 0.54 -0.11
r4 1.00 0.99 0.45 -0.01
r5 1.00 0.99 0.37 -0.03
r6 0.99 0.99 0.30 -0.12
r7 0.99 0.98 0.27 0.03
r8 0.99 0.98 0.24 0.03
r9 0.99 0.97 0.19 0.03
r10 0.99 0.97 0.13 -0.07
Determine values for d and the parameter α in the underlying process.
E Multivariate Analysis
13. Suppose X be Np+q Suppose X
Xp Xq be a partitioned vector. Derive
the marginal distribution of X and conditional distribution of X given
X x
Let X be distributed as N3 where and









1 0 2
0 5 0
4 0 1
Σ .
Obtain the marginal distribution of X1 and conditional distribution of X1 given
that X2 x2 X3 x3
14. Define population principal components. Obtain the correlation coefficient
Yi Xk where Yi denotes the ith principal component and Xk denotes kth
element of vector ......, p X X .
Find the first principal component and the proportion of the total population
variance explained by it when the covariance matrix is—

Hint is one eigenvalue of
2
1
2
1

0
0
2
2 2
2 2 2
2 2
Σ
Σ
σ

σ σ
σ σ σ
σ σ








15. Define Mahalanobis distance a measure of the distance between the two
normal populations.
Suppose 2 i 2 i N X where 2 1





2 3
2 2
2 and Σ .
Compute the Mahalanobis distance and obtain the linear discriminant
function.
What is cluster analysis Using the distance matrix of five items given below—
0 4 6 1 6
4 0 9 7 3
6 9 0 10 5
1 7 10 0 8
6 3 5 8 0
Cluster the items using complete linkage hierarchical method. Draw the
dendrogram.
F Numerical Analysis and Basic Computer Techniques
16. Describe the Newton-Raphson method for unconstrained optimization of an
algebraic function. Does this method always find an optimal solution State
the assumptions of this method.
Make a critical comparison of the method with the bisection method. Illustrate
your answer with an example.
17. Describe the closed quadrature method of numerical integration.
Derive the trapezoidal and Simpson's 1/3rd rules of numerical integration and
make a comparison between these methods in terms of their accuracy. Give
an example to illustrate your answer.
18. Describe any two sorting algorithms.
Discuss the best-case and the worst-case time complexities of the bubble sort
algorithm. Illustrate your answer with an example.
CON 159 12