IJ
IB..JGT..J·TUB I
945
STATISTICS
Paper II

ITime Allowed .. Three HoursI IMaximum Marks: 2001
INSTRUCTIONS
Candidates should attempt questions 1 and 5
which are compulsory, and any THREE ofthe

remaining questions, selecting at least ONE
question from each Section.
The number ofmarks carried by each question

is indicated at the end ofthe question.
Assume suitable data, ifconsidered necessary,
and indicate the same clearly.
Answers must be written in ENGLISH.
Unless otherwise indicated, symbols and
notations have their usual meanings.

SECTION A
(Industrial Statistics and Optimization Techniques>

1. Attempt any four of the following 4xlO=40
Define a Markov chain. Obtain Chapman -Kolmogorov equation for higher order transition probabilities in a Markov chain.
B-JGT-J-TUB 1 lContd.]

Define hazard rate In life testing models. When are censored observations needed in experiments Describe Type I and Type II censoring.


For a (oo/FIFO) queueing system with arrival rate A. and service rate state the expression for the probability that there are n customers In the system. Hence obtain the expressions for average number of customers in the queue and in the system.


If a linear programming problem has an optimal solution, then prove that it attains its optimum at an extreme point of the convex set generated by the set of all feasible solutions to the linear programming problem.
Write briefly on the SPSS software package.
2. How do single and double sampling plans for
attributes differ with respect to their ASN and
ATI curves? Explain the situation when a double
sampling plan should be preferred to a single
sampling plan. 12
Suppose that three electronic devices have a
failure law given by an exponential distribution
with parameters (12 and (13 respectively.
Suppose also that these three devices function
independently and are connected in parallel to
form a single system.

Obtain an expression for the reliability of the system.
Find the mean time to failure of the system. 14 B-JGT-J-TUB 2 (Contd.]


What do you understand by the term "Sensitivity
analysis" Discuss the changing cost coefficients
In a linear programming problem without
affecting the optimal solution of the problem. 14
3. Define Monte -Carlo simulation technique and
mention its steps. Describe how you would
generate random numbers for the probability
distribution exp x O. 14
assignable and chance causes of quality
variation. When is a production process said to be
under statistical control? Describe chart
and mention its advantages and disadvantages
over chart. 12
What do you mean by probabilistic inventory
control models Deduce an expreSSlOn for
optimum inventory level when the demand is
stochastic and is instantaneous and
replenishment unit is discrete. 14
4. Describe a two-person zero sum game and in this
context explain the terms pay-off matrix,
saddle point, mixed strategies. Let
be a real-valued function and be defined
whenever i E j e B. Suppose both
maxmin and minmax exist. Then
i j j i
prove that a necessary and sufficient condition
that max mm fOJ mIn max is that
i j J 1
function f possesses a saddle point. 14

E.xplain.clearly individual and group replacement policies. The following mortality rates have been observed for certain type of electric bulbs
Week Percentage failing
1 10
2 25
3 50
4 80
5 100

The total number of bulbs is 1000. The individual replacement cost is Rs. 10 while cost for replacement of all the bulbs is Rs. 2·50 per bulb. Calculate the optimal group replacement interval. Find the value of the rectangular game and optimal strategies of both the players whose pay-off matrix below is 12 14
Player B
B1 B2 Ba B4 B5
Al 9 3 1 8 0
7 4 4 6 7
Player A As 2 4 3 3 8
A4 5 6 2 ·2 1

B-JGT-J-TUB 4 [Contd.]

SECTION B
(Quantitative Economics and Official Statistics)

5. Attempt any four of the following: 4><1.0=40

Explain the mathematical tests f0t: a 19ood' index number. Define Fisher's price index number and . examine whether it satisfies these tests.


Discuss the consequences of multicollinearity in the data. Also suggest remedial measures to overcome the undesirable consequences.


Write a note on the present official statistical system in India relating to agriculture.


Cd) What are reproduction rates Explain the meaning of Gross Reproduction Rate (G.R.H.) and Net Reproduction Rate (N.R.R.)
What is the problem of scaling Describe z-scaling and t-scaling.
6. Explain what is meant by consumer price index number and mention its uses. Outline any two methods of constructing such an index number. 12

What do you mean by 'seasonal index' in the analysis of a time series Discuss the different steps involved in the computation of the seasonal indices in the link relatives method. 14


Explain the life table and the meaning of various columns of a life table. Determine the values of ISl' l82 and l83' given that lso =13987, d =2018. QSl =0-15649 and Pa2 =0·83042. 14


ao
7. State the functions of Central Statistical
'Organization (C.S.O.), National Sample
Survey Organization (N.S.S.O.) and Labour
Bureau. 12
Discuss autocorrelation. Show that the series
determined by
ut+1 Ia 1
where ut has zero mean, has an autocorrelation
given by
Pk a k if the successive values of v t are
independent,
Pk


if vt itself obeys a relation of the form
vt+1 b vt Ib where successive values of wt are independent. 14
Discuss the problem of estimation of structural parameters in simultaneous equation system. For the following system of equations:
Ct Zt Ut Yt Zt obtain indirect least squares estimators of the parameters. 14
B·JGT-J-TUB
6 lContd.]
8. Explain how you would fit a Gompertz curve to
the data for the purpose of population projection. 14
Distinguish between T-scores and standard
scores. Discuss the method of converting raw
scores into T-scores. 12
Define crude and standardized death rates. How
is standardized death rate superior to crude
death rate Discuss briefly the direct and
indirect methods of finding standardized death
rates. 14