Solve the following two-person zero sum game using graphical procedure

1
3 5
-1 6
4 1
2 2
-5 0

1.(b) If the demand for a certain product ha a rectangular distribution between 4000 and 5000, then find the optimal order quantity if the storage cost is RS 1 per unit and the shortage cost is RS 7 Per unit.

1.(c) A supermarket has two salesgirls at the counter If the service time for each customer is exponential with mean 4 minutes and if people arrive in a poisson fashion at RS 10 per hour then find the expected percentage of idle time for each salesgirl.

1.(d) Find out ML estimators of the parameters in the density

c/b x c 0
based on the following censored scheme

X(1), X(2),...,X(n-r)
Where n and r are respectively the sample size and number of censored observations.

1.(e) Defining a location-scale family, outline the procedure for obtaining the coefficient of BLUE's.

2.(a) A computer contains 10000 resistors. When any resistor fails, it is replaced. The cost of replacing a resistor individually is Rs 1 only. If all the resistors are replaced at the same time, the cost per resistor comes down to 35 paise. The percent of surviving, at the end of t-th month is

t 0 1 2 3 4 5 6 100 97 90 70 32 15 0
Derive the optimum replacement plan.

2.(b) A random sample of 25 observations drawn from Weibull model (with shape parameter 2 and scale parameter is given. Obtain the moment estimates of the parameters :

1.8487 0·3761 0·7500 3·0530 1.3545
1·8802 1·5700 1·7708 1·3592 3.0466
1·7961 1·5319 0·5903 0·6288 0·6461
1·6560 1·7172 1·9310 1·0509 1·6173
1.3162 0·7705 1·8889 1.8889 4·1505

3.(a) Discuss the sources of demographic data in India and and point out the use and limitation of these data

3.(b) Explain the stationary and stable population model. Discuss the situation when stationary and stable populations are identical.

3.(c) Explain briefly the uses of life table· In usual notation, prove that

dLx/dx -dx dTx/dx -lx

3.(d) Define crude death rate specific death rate and standardized death rate,Interpret these rates.

3.(e) State the genera1 procedure and steps for the construction of life tables.

4.(a) Explain an abridged life table and discuss its different columns. Discuss the King's method for its construction

4.(b) What do you mean by fertility? Define crude birthrate,general fertility rate and age specific fertility rate.How are these rates computed in practice?

5.(a) Define hazard function and survival function. Obtain the same for an exponential distribution.

5.(b) If and are the density,distribution and hazard functions of a random variable then show that

h(t)
Also establish a suitable relationship between and reliability function.

5.(c) If Px is the probability that a person aged x will survive to age x then show that nPx nqx=1, where

npx lx n/lx

5.(d) Discuss some specific situations in which it would be difficult or inefficient to perform clinical trials.

5.(e) What are the principles for ethical clinical trials?

6.(a) Define Type-I and Type-II censoring schemes. Obtain the maximum likelihood stimator of parameter 8 in exponential distribution under the above censoring schemes. Also obtain
the Fisher information for the
parameter.

6.(b) Discuss various phases involved in a clinical trial. Also discuss the pros and cons of each phase.

7.(a) Distinguish between process control and product control. Discuss the situations where they are used.

7.(b) Discuss various sources of assignable causes and random causes of variations. Also state how thy are detected in a manufacturing process.

7.(c) Define the terms ASN, AOQ, ATI and ARL for an acceptance sampling, plan.

7.(d) What is an acceptance sampling plan? Discuss a single sampling plan where the sampling is carried out using a binomial model. Find Pa if n 10, c=3 and p=O·05.

7.(e) State the importance of exponentially weighted moving average charts. How are these used in practical situations?

8.(a) Discuss the importance of control charts for variables.

8.(b) Obtain the control limits for and charts when standards are known and unknown.

8.(c) In usual notation, show that

C2 sqrt(2/n) sqrt(n/2)/sqrt(n-1/2)

where C2 is a constant used for constructing control limits.

8.(d) Discuss in detail the double SaIllpling plan n1, c1, n2, c2) stating the assumptions followed in both the stages. Hence or otherwise, obtain the ASN function of this sampling procedure.

9.(a) Define p-variate normal distribution and obtain its characteristic function. Assume mean vector M and dispersion
matrix .

9.(b) Define Hotelling's T^2 and mention Its application.Show that

T^2 mp/m-p 1 Fp,m-p l

9.(c) Explain the importance of principal components and discuss the method for extraction of principal component.

9.(d) Define Wishart distribution and obtain its characteristic function.

9.(e) Define canonical variates and canonical correlation Give your interpretation.

10.(a) state the chief properties of Wishart distribution. If

Ai 2,...
then show that

A E 1 to k)

has the Wishart distribution with parameters E and where

n E 1 to k)

assuming the independence of A·l •

10.(b) Define multiple correlation coefficient and obtain its non-null distribution.

11.(a) What are the basic principles of design of experiments? Explain them and their roles.

11.(b) Define completely randomized design, and mention its merits and demerits Also give its statistical analysis.

11.(c) Define Latin square design with its applications, advantages and disadvantages.

11.(d) Explain analysis of covariance with its application Give an illustration for the use of analysis of covariance in identifying the response variable and concomitant variable(x)

11.(e) Explain split plot design and give some situations where Split-plot design can be suitably adopted.

12.(a) What are factorial experiments? The following table gives the layout and the results of 2^3 factorial design laid out in four replicates. The purpose of the experiment is to determine the effect of different kind of fertilizer, Nitrogen Potash and Phosphate on potato Crop yield:

Block 1
nk kp p np
291 391 312 373

Block 2
kp p k nk
407 324 272 306

Block 3
p 1 np kp
323 87 324 423
nk k n nkp
334 279 128 471

Block 4
np nk n p
361 272 103 324
k 1 nkp kp
302 131 437 435

Analyse the design and draw conclusions.

12.(b) What is confounding? A 2^3 factorial experiment is conducted in 2 blocks of size 4 each, in 3 replicates.The arrangements are as follows.

Replicate 1 Replicate 2 Replicate 3
Key Block (1),b,ac,abc

Identify the confounded effect in each replicate.

12.(c) Define a randomized Block Design (RBD)specifying the statistical model associated the with the design. Carry out a complete analysis of RBD with one missing observation.

13.(a) Given a five-digit positive integer, write a C program to find the sum of individual gigits

(For example, if the given, number 96785, the required sum is 9 6 7 8 5

13.(b) Write C program to pick up the largest tender from a set of tenders assuming that Tender-id and Tender-value for each tender are specified.

13.(c) Express in infix and postfix forms.

13.(d) Write a program in R to plot Q-Q plot assuming that the two samples (Xi 1,2 ...,25 and Yi, .,250) are drawn from normal population.

13.(e) Given n positive integerst write R code to find their ranks.

14.(a) Illustrate call by value and call by reference, and advantages as well as disadvantages of each method. Write a C function to evaluate the series without using a built-in function
sin(x) x ... 00

14.(b) Given X where M and a 2 are specified, write a C program to compute

P(X integral Density of 2)dx (limits t to 00)