Exam Details
| Subject | statistics | |
| Paper | paper 4 | |
| Exam / Course | indian economic service and indian statistical service examination (ies/iss) | |
| Department | ||
| Organization | union public service commission | |
| Position | ||
| Exam Date | 2014 | |
| City, State | central government, |
Question Paper
1. Answer all of the following
Define
persistent
transient
ergodic states.
Show that if i j and if state i is persistent, then state j is also persistent.
Define random walk and explain gambler's ruin problem.
Using graphical method, solve the following problems
Maximize Z =6x1 4x2
subject to
2x1 4x2 4
4x1-8x2 16
xl, x2 0
Maximize Z 10x1 6x2
subject to
5x1 +3x2 30
xl 2X2 18
xl, x2 0
A shop produces three items in lots. The demand rate for each item is constant and can be assumed to be deterministic. No back orders are allowed. The pertinent data for the items is given as follows:
Item I II III
Carrying cost (Rs per unit per year) 20 20 20
Setup cost (Rs per setup) 50 40 60
Cost per unit Rs 6 7 5
Yearly demand (units) 10000 12000 7500
Determine approximately the economic order quantity for three items subject to the condition that the total value of average inventory levels of these items does not exceed" 1,000.
Explain the procedure of generating random numbers from N(J.L. a 2 (where J.L and are specified) using Box-Muller
formula.
Prove that a discrete parameter stochastic process n is called a martingale if 00 and E{Xn Xn, Xo} Xn.
Define branching process and state the properties of its generating function.
Explain the additive property of Poisson process and also the difference of two independent Poisson processes.
Describe birth and death processes, and show that it is a particular case of Poisson process.
XYZ Tobacco Company purchases tobacco and stores in warehouses located in the following four cities
Warehouse Location Capacity (in tonnes)
City A 90
City B 50
City C 80
City D 60
The warehouses supply tobacco to cigarette companies in three cities that have the following demand
Cigarette Company Demand (in tonnes)
Bharat 120
Janata 100
Re Lamp 110
Because of railroad construction, shipments are temporarily prohibited from warehouse at city A to Bharat Cigarette Company.
Find the optimum distribution for XYZ Tobacco Company.
Are there any multiple optimum solutions? If yes, identify them.
Write the dual of the given transportation problem and use it for checking the optimum solution.
Some of the spare parts of a ship cost Rs 50,000. These parts can only be ordered together with the ship. If not ordered at the time the ship is constructed, these parts are not available as and when needed. Suppose that a loss of Rs 4,50,000 is incurred for each spare part that is needed when none is available in the stock and the probability distribution that the spares will be needed for replacement during lifetime is
Spares required 0 1 2 3 4 5
Probability 0·90 0·04 0·025 0·02 0.01 0·05 0
Find the optimum order quantity.
Machine A costs r 45,000 and the operating costs are estimated at Rs 1,000 for the first year increasing by Rs 10,000 per year in the second and subsequent years. Machine Beasts Rs 50,000 and the operating costs are Rs 2,000 for the first year, increasing by Rs 4,000 per year in the second and subsequent years. If we now have a machine of type should we replace it with If so, when? Assume that both machines have no resale value and future costs are not discounted.
A bakery keeps stock of a popular brand of cakes. Previous experience shows that the daily demand pattern for the item with associated probabilities is as given below
Daily demand (in numbers) 0 10 20 30 40 50
Probability 0·01 0-20 0-15 0.50 0·12 0.02
Use the following sequence of random numbers to simulate the demand for the next 10 days
25 39 65 76 12 05 73 89 19 49
Estimate the daily average demand for the cakes on the basis of simulated data.
Can you estimate the daily average demand analytically? If so, find the bias in estimating daily average demand.
Obtain Chapman-Kolmogorov equation with transition probabilities for a Markov chain.
Discuss MIMI! queuing model with steady-state behaviour.
Use simplex method to solve the following problem
A company makes two kinds of leather belts. Belt A is of high quality and belt B is of low quality. The respective profits are 4 and f 3 per belt. The production of each of type A requires as much time as belt of type B and if all belts were of type the company could make 1000 belts per day. The supply of leather is sufficient for only 800 belts per day (both A and B put together). Belt A requires fancy buckles and only 400 per day are available. There are only 700 buckles a day available for belt B. What should be the daily production of each type of belt to maximize the total profit?
Let the value of the money be assumed to be depreciated 10% per year and suppose that machine A is replaced after every three years whereas machine B is replaced every six years. The yearly costs (in Rs of both the machines are given as under
Year 1 2 3 4 5 6
Machine A 1000 200 400 1,000 200 400
Machine B 1,700 100 200 300 400 500
Determine which machine should be purchased.
Briefly outline the uses of life tables.
Mention how logistic curve fitting IS used for population projection.
Outline and describe the inter-censal and post-censal estimates.
Write short notes on
Magnetic Ink Character Recognition
Plotters
Use complement to
subtract 3 from
subtract from
add and
add 5 and 4.
Compute
GFR
SFR
TFR
gross production rate
from the data given below:
Age group of child- bearing females 15-19 20-24 25-29 30-34
No. of women 16·0 16·4 15·8 15·2
Total births 260 2244 1894 1320
Age group of childbearing females 35-39 40-44 45-49
No. of women 14·8 15·0 14.5
Total births 916 280 145
Assume that the proportion of female births is 46.2%
Outline the uses of Makeham and Gompertz curves in life tables.
Describe stable and stationary populations.
Explain various mortality rates and standardized death rates.
Write binary multiplication algorithm using register(s) and accumulator.
Discuss the following types of operating system:
Batch
Multiprogramming
Time-sharing
Real-time
Given Yi where i n. Write a flow chart to find a and b in fitting Y a bX.
Discuss error detecting and error correcting codes, and illustrate single error detecting code in detail.
State the general procedure and steps for the construction of life tables.
From the data given below, calculate the gross reproduction rate and net reproduction rate
Age group No. of children born to 1000 women passing through the age group Mortality rate (per 1000)
16-20 150 120
21-25 1500 180
26-30 2000 150
31-35 800 200
36-40 500 220
41-45 200 230
46-50 100 250
Sex ratio being males females:; 52 48.
Illustrate mail merge application in detail describing the common features available in any word processing package.
Discuss various features of graphics wizard in any spreadsheet package in connection with statistical data processing.
Define
persistent
transient
ergodic states.
Show that if i j and if state i is persistent, then state j is also persistent.
Define random walk and explain gambler's ruin problem.
Using graphical method, solve the following problems
Maximize Z =6x1 4x2
subject to
2x1 4x2 4
4x1-8x2 16
xl, x2 0
Maximize Z 10x1 6x2
subject to
5x1 +3x2 30
xl 2X2 18
xl, x2 0
A shop produces three items in lots. The demand rate for each item is constant and can be assumed to be deterministic. No back orders are allowed. The pertinent data for the items is given as follows:
Item I II III
Carrying cost (Rs per unit per year) 20 20 20
Setup cost (Rs per setup) 50 40 60
Cost per unit Rs 6 7 5
Yearly demand (units) 10000 12000 7500
Determine approximately the economic order quantity for three items subject to the condition that the total value of average inventory levels of these items does not exceed" 1,000.
Explain the procedure of generating random numbers from N(J.L. a 2 (where J.L and are specified) using Box-Muller
formula.
Prove that a discrete parameter stochastic process n is called a martingale if 00 and E{Xn Xn, Xo} Xn.
Define branching process and state the properties of its generating function.
Explain the additive property of Poisson process and also the difference of two independent Poisson processes.
Describe birth and death processes, and show that it is a particular case of Poisson process.
XYZ Tobacco Company purchases tobacco and stores in warehouses located in the following four cities
Warehouse Location Capacity (in tonnes)
City A 90
City B 50
City C 80
City D 60
The warehouses supply tobacco to cigarette companies in three cities that have the following demand
Cigarette Company Demand (in tonnes)
Bharat 120
Janata 100
Re Lamp 110
Because of railroad construction, shipments are temporarily prohibited from warehouse at city A to Bharat Cigarette Company.
Find the optimum distribution for XYZ Tobacco Company.
Are there any multiple optimum solutions? If yes, identify them.
Write the dual of the given transportation problem and use it for checking the optimum solution.
Some of the spare parts of a ship cost Rs 50,000. These parts can only be ordered together with the ship. If not ordered at the time the ship is constructed, these parts are not available as and when needed. Suppose that a loss of Rs 4,50,000 is incurred for each spare part that is needed when none is available in the stock and the probability distribution that the spares will be needed for replacement during lifetime is
Spares required 0 1 2 3 4 5
Probability 0·90 0·04 0·025 0·02 0.01 0·05 0
Find the optimum order quantity.
Machine A costs r 45,000 and the operating costs are estimated at Rs 1,000 for the first year increasing by Rs 10,000 per year in the second and subsequent years. Machine Beasts Rs 50,000 and the operating costs are Rs 2,000 for the first year, increasing by Rs 4,000 per year in the second and subsequent years. If we now have a machine of type should we replace it with If so, when? Assume that both machines have no resale value and future costs are not discounted.
A bakery keeps stock of a popular brand of cakes. Previous experience shows that the daily demand pattern for the item with associated probabilities is as given below
Daily demand (in numbers) 0 10 20 30 40 50
Probability 0·01 0-20 0-15 0.50 0·12 0.02
Use the following sequence of random numbers to simulate the demand for the next 10 days
25 39 65 76 12 05 73 89 19 49
Estimate the daily average demand for the cakes on the basis of simulated data.
Can you estimate the daily average demand analytically? If so, find the bias in estimating daily average demand.
Obtain Chapman-Kolmogorov equation with transition probabilities for a Markov chain.
Discuss MIMI! queuing model with steady-state behaviour.
Use simplex method to solve the following problem
A company makes two kinds of leather belts. Belt A is of high quality and belt B is of low quality. The respective profits are 4 and f 3 per belt. The production of each of type A requires as much time as belt of type B and if all belts were of type the company could make 1000 belts per day. The supply of leather is sufficient for only 800 belts per day (both A and B put together). Belt A requires fancy buckles and only 400 per day are available. There are only 700 buckles a day available for belt B. What should be the daily production of each type of belt to maximize the total profit?
Let the value of the money be assumed to be depreciated 10% per year and suppose that machine A is replaced after every three years whereas machine B is replaced every six years. The yearly costs (in Rs of both the machines are given as under
Year 1 2 3 4 5 6
Machine A 1000 200 400 1,000 200 400
Machine B 1,700 100 200 300 400 500
Determine which machine should be purchased.
Briefly outline the uses of life tables.
Mention how logistic curve fitting IS used for population projection.
Outline and describe the inter-censal and post-censal estimates.
Write short notes on
Magnetic Ink Character Recognition
Plotters
Use complement to
subtract 3 from
subtract from
add and
add 5 and 4.
Compute
GFR
SFR
TFR
gross production rate
from the data given below:
Age group of child- bearing females 15-19 20-24 25-29 30-34
No. of women 16·0 16·4 15·8 15·2
Total births 260 2244 1894 1320
Age group of childbearing females 35-39 40-44 45-49
No. of women 14·8 15·0 14.5
Total births 916 280 145
Assume that the proportion of female births is 46.2%
Outline the uses of Makeham and Gompertz curves in life tables.
Describe stable and stationary populations.
Explain various mortality rates and standardized death rates.
Write binary multiplication algorithm using register(s) and accumulator.
Discuss the following types of operating system:
Batch
Multiprogramming
Time-sharing
Real-time
Given Yi where i n. Write a flow chart to find a and b in fitting Y a bX.
Discuss error detecting and error correcting codes, and illustrate single error detecting code in detail.
State the general procedure and steps for the construction of life tables.
From the data given below, calculate the gross reproduction rate and net reproduction rate
Age group No. of children born to 1000 women passing through the age group Mortality rate (per 1000)
16-20 150 120
21-25 1500 180
26-30 2000 150
31-35 800 200
36-40 500 220
41-45 200 230
46-50 100 250
Sex ratio being males females:; 52 48.
Illustrate mail merge application in detail describing the common features available in any word processing package.
Discuss various features of graphics wizard in any spreadsheet package in connection with statistical data processing.