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Geethanjali College of Engineering and Technology

Cheeryal (V), Keesara (M), Ranga Reddy District – 501 301 (T.S)

PROBABILITY AND STATISTICS

COURSE FILE

DEPARTMENT OF

COMPUTER SCIENCE & ENGINEERING

(2016-2017)

Faculty In charge HOD-CSE

A. RAMESH, N. NAGI REDDY Dr. S Nagender Kumar

Contents

S.No

Topic

Page. No.

1

Cover Page

3

2

Syllabus copy

4

3

Vision of the Department

5

4

Mission of the Department

5

5

PEOs and POs

6

6

Course objectives and outcomes

7

7

Course mapping with POs

8

8

Brief notes on the importance of the course and how it fits into the curriculum

9

9

Prerequisites if any

10

10

Instructional Learning Outcomes

10

11

Class Time Table

11

12

Individual time Table

15

13

Lecture schedule with methodology being used/adopted

16

14

Detailed notes

28

15

Additional topics

28

16

University Question papers of previous years

29

17

Question Bank

45

18

Assignment Questions

56

19

Unit wise Quiz Questions and long answer questions

61

20

Tutorial problems

66

21

Known gaps ,if any and inclusion of the same in lecture schedule

69

22

Discussion topics , if any

70

23

References, Journals, websites and E-links if any

70

24

Quality Measurement Sheets

71

A

Course End Survey

71

B

Teaching Evaluation

71

25

Student List

71

26

Group-Wise students list for discussion topic

71

Course coordinator Program Coordinator HOD

GEETHANJALI COLLEGE OF ENGINEERING AND TECHNOLOGY

DEPARTMENT OF SCIENCE AND HUMANITIES

Name of the Subject : probability & statistics

JNTU CODE: 113AN Programme : UG

Branch: CSE- A,B,C&D

Version No : 01

Year : I I Year Updated on : 04-05-2016

Semester: I No. of pages : 71

Prepared by : 1) Name : Mr. A. Ramesh

2) Design : Asst. Professor

3)Sign :

4) Date :04-05-2016

Verified by : 1) Name :Dr.V.S.Triveni

2) Sign :

3) Design : Professor

4) Date : 10-05-2016

* For Q.C Only.

1) Name :

2) Sign :

3) Design :

4) Date :

Approved by : (HOD ) 1) Name : Dr. G. Neeraja Rani

2) Sign :

3) Date :

2.Syllabus

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

II Year B.Tech CSE-I Sem L T/P/D C

4 -/-/- 4

Syllabus:

Single Random variables and probability distributions: Random variables – Discrete and continuous. Probability distributions, mass function/ density function of a probability distribution . Mathematical Expectation, Moment about origin, Central moments Moment generating function of probability distribution. Binomial, Poisson & normal distributions and their properties. Moment generating functions of the above three distributions, and hence finding the mean and variance.

UNIT-II

Multiple Random variables, Correlation & Regression: Joint probability distributions- Joint probability mass / density function, Marginal probability mass / density functions, Covariance of two random variables, Correlation -Coefficient of correlation, The rank correlation.

Regression- Regression Coefficient, The lines of regression and multiple correlation & regression.

UNIT-III

Sampling Distributions and Testing of Hypothesis

Sampling: Definitions of population, sampling, statistic, parameter. Types of sampling, Expected values of Sample mean and variance, sampling distribution, Standard error, Sampling distribution of means and sampling distribution of variance.

Parameter estimations – likelihood estimate, interval estimations.

Testing of hypothesis: Null hypothesis, Alternate hypothesis, type I, & type II errors – critical region, confidence interval, Level of significance. One sided test, two sided test.

Large sample tests:

(i) Test of Equality of means of two samples equality of sample mean and population mean (cases of known variance & unknown variance, equal and unequal variances)

(ii) Tests of significance of difference between sample S.D and population S.D.

(iii) Tests of significance difference between sample proportion and population proportion & difference between two sample proportions.

Small sample tests:

Student t-distribution, its properties; Test of significance difference between sample mean and population mean, difference between means of two small samples. F- distribution and it’s properties. Test of equality of two population variances. Chi-square distribution , it’s properties, Chi-square test of goodness of fit.

UNIT-IV

Queuing Theory: Structure of a queuing system, Operating Characteristics of queuing system, Transient and steady states, Terminology of Queuing systems, Arrival and service processes- Pure Birth-Death process. Deterministic queuing models- M/M/1 Model of infinite queue, M/M/1 model of finite queue.

UNIT-V

Stochastic processes: Introduction to Stochastic Processes –Classification of Random processes, Methods of description of random processes, Stationary and non-stationary random process, Average values of single random process and two or more random processes. Markov process, Markov chain, classification of states –Examples of Markov Chains, Stochastic Matrix.

TEXT BOOKS:

1) Higher Engineering Mathematics by Dr. B.S. Grewal, Khanna Publishers

2) Probability and Statistics for Engineers and Scientists by Sheldon M.Ross, Academic Press

3) Operations Research by S.D. Sarma,

REFERENCE BOOKS:

1. Mathematics for Engineers by K.B.Datta and M.A S.Srinivas,Cengage Publications

2. Probability and Statistics by T.K.V.Iyengar & B.Krishna Gandhi Et

3. Fundamentals of Mathematical Statistics by S C Gupta and V.K.Kapoor

4. Probability and Statistics for Engineers and Scientists by Jay l.Devore.

3.Vision of the Department:

To produce globally competent and socially responsible computer science engineers contributing to the advancement of engineering and technology which involves creativity and innovation by providing excellent learning environment with world class facilities.

4.Mission of the Department:

1. To be a center of excellence in instruction, innovation in research and scholarship, and service to the stake holders, the profession, and the public.

2. To prepare graduates to enter a rapidly changing field as a competent computer science engineer.

3. To prepare graduate capable in all phases of software development, possess a firm understanding of hardware technologies, have the strong mathematical background necessary for scientific computing, and be sufficiently well versed in general theory to allow growth within the discipline as it advances.

4. To prepare graduates to assume leadership roles by possessing good communication skills, the ability to work effectively as team members, and an appreciation for their social and ethical responsibility in a global setting.

5.Program Educational Objectives (PEOs):

1. To provide graduates with a good foundation in mathematics, sciences and engineering fundamentals required to solve engineering problems that will facilitate them to find employment in industry and / or to pursue postgraduate studies with an appreciation for lifelong learning.

2. To provide graduates with analytical and problem solving skills to design algorithms, other hardware / software systems, and inculcate professional ethics, inter-personal skills to work in a multi-cultural team.

3. To facilitate graduates to get familiarized with the art software / hardware tools, imbibing creativity and innovation that would enable them to develop cutting-edge technologies of multi-disciplinary nature for societal development.

PROGRAM OUTCOMES (PO)

1. An ability to apply knowledge of mathematics, science and engineering to develop and analyze computing systems.

2. an ability to analyze a problem and identify and define the computing requirements appropriate for its solution under given constraints.

3. An ability to perform experiments to analyze and interpret data for different applications.

4. An ability to design, implement and evaluate computer-based systems, processes, components or programs to meet desired needs within realistic constraints of time and space.

5. An ability to use current techniques, skills and modern engineering tools necessary to practice as a CSE professional.

6. An ability to recognize the importance of professional, ethical, legal, security and social issues and addressing these issues as a professional.

7. An ability to analyze the local and global impact of systems /processes /applications /technologies on individuals, organizations, society and environment.

8. An ability to function in multidisciplinary teams.

9. An ability to communicate effectively with a range of audiences.

10. Demonstrate knowledge and understanding of the engineering, management and economic principles and apply them to manage projects as a member and leader in a team.

11. A recognition of the need for and an ability to engage in life-long learning and continuing professional development

12. Knowledge of contemporary issues.

13. An ability to apply design and development principles in producing software systems of varying complexity using various project management tools.

14. An ability to identify, formulate and solve innovative engineering problems.

6.Course Objectives and Outcomes:

Course Objectives:

The aim of this course is,

1. Understand a random variable that describes randomness or an uncertainty in certain realistic situation. It can be of either discrete or continuous type.

2. In the discrete case, study of the binomial and the Poisson random variables and the Normal random variable for the continuous case predominantly describe important probability distributions. Important statistical properties for these random variables provide very good insight and are essential for industrial applications.

3. Most of the random situations are described as functions of many single random variables. In this unit, the objective is to learn functions of many random variables through joint distributions.

4. The types of sampling, Sampling distribution of means, Sampling distribution of variance, Estimations of statistical parameters, Testing of hypothesis of few unknown statistical parameters.

5. The mechanism of queuing system, the characteristics of queue, the mean arrival and service rates, the expected queue length, the waiting line.

6. The random processes, the classification of random processes, Markov chain, classification of states.

7. Stochastic matrix (transition probability matrix), Limiting probabilities, Application of Markov chains.

Course Outcomes:

At the end of this course, the students are able to

1. CO1: Explain the concept of random variables and probability distributions and applythe same to solve simple engineering problems.

2. CO2: Explain the concept of correlation coefficient, regression coefficient and regression lines and apply the same to solve simple engineering problems to analyze the situations properly and to take a right decision to choose the right one.

3. CO 3: Explain the concept of hypothesis testing for small as well as for large samples and apply the same to solve simple engineering problems.

4. CO4: Explain the models of queuing system and apply the same to solve some real problems and simple engineering problems.

5. CO5: Explain the concept of Markov Chains and Markov process and apply the same to solve simple engineering problems and also to find solutions to real world problems.

7.Mapping of Course to PEOs and Pos:

Course Component

POs

PEOs

Mathematics

i,ii,iv,v,vii, viii,xi,xii, xiii,xivv

PEO1, PEO2

Course Mapping With PEOs And Pos:

Mapping of Course with Programme Educational Objectives:

S.No

Course component

code

course

Semester

PEO 1

PEO 2

PEO 3

1

Mathematics

113AN

P&S

I

√

√

POs

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Mathematics

P&S

CO 1

√

√

√

√

CO 2

√

√

√

CO 3

√

√

CO 4

√

√

√

√

CO 5

√

√

√

8.Brief notes on the Importance of the course and how to fits into the curriculum:

1. Most of the random situations are described as functions of many single random variables. The importance is to learn functions of many random variables through joint distributions.

2. The types of sampling, Sampling distribution of means ,Sampling distribution of variance, Estimations of

3. statistical parameters, Testing of hypothesis of few unknown statistical parameters.

4. The mechanism of queuing system ,The characteristics of queue, The mean arrival and service rates

5. The expected queue length, The waiting line

6. The random processes, The classification of random processes and Markov process.

7. It improves the analytic and logical skills of the student. They will be in a position to guess the result of their project results.

9.Prerequisites:

The student should have the fundamental knowledge about

1. Sets and functions

2. Permutations

3. Combinations

4. Probability

5. Statistics

6. Intermediate mathematics.

10.Instructional Learning outcomes:

Unit wise Learning Outcomes:

UNIT I: Single random variables and probability distributions

After the completion of this unit, the students should be able to:

1. Explain the difference between discrete random variable and continuous random variables

2. Apply the distributions in certain realistic situations

UNIT II: Multiple random variables, correlation & Regression


1. Explain the closeness of the relationship between the variables.

2. Explain how to find the value of one variable, when the value of the other variable is given.

3. Explain the difference between correlation coefficient and regression coefficient.

UNIT III: Sampling Distributions and Testing of Hypothesis


1. Explain mean and proportions for large samples as well as for small samples

2. Explain the concept of confidence interval for large as well as for small samples

3. Apply different types of tests for large and small samples

UNIT IV: Queuing Theory


1. Explain the terminology of queuing system

2. Apply the models of queuing system in real life situations

3. How best the Queuing models simplify the problematic part

UNIT V: Stochastic Processes


1. Explain random process

2. Explain Markov chains and Markov process and he will know applications to real life problem

11.Class Time Tables II A,B,C & D sec

Geethanjali College of Engineering & Technology

Department of Computer Science & Engineering

Year/Sem/Sec: II-B. Tech I-Semester A-Section

Room No:

A.Y : 2016 -17

WEF:

Class Teacher:

Time

09.30-10.20

10.20-11.10

11.10-12.00

12.00-12.50

12.50-1.30

1.30-2.20

2.20-3.10

3.10-4.00

Period

1

2

3

4

LUNCH

5

6

7

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

S.No

Subject(T/P)

Faculty Name

Contact No

1


2

MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE

3

DIGITAL LOGIC DESIGN

4

ELECTRONIC DEVICES AND CIRCUITS

5

BASIC ELECTRICAL ENGINEERING

6

DATA STRUCTURES

7

ELECTRICAL AND ELECTRONICS LAB

8

DATA STRUCTURES LAB

9

SEMINAR

10

*-Tutorial Hour/Discussion Hour

TT. Cord:___________ HOD:________ Dean Academics:-_______ Principal:___________________



Year/Sem/Sec: II-B.Tech I-Semester B-Section

Room No:

A. Y : 2016--167 WEF:

Class Teacher:

Time

09.30-10.20

10.20-11.10

11.10-12.00

12.00-12.50

12.50-1.30

1.30-2.20

2.20-3.10

3.10-4.00

Period

1

2

3

4

LUNCH

5

6

7

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

S.No

Subject(T/P)

Faculty Name

Contact No

1

2

3

4

5

6

7

8

9

10

TT. Coord:___________ HOD:________ Dean Academics:-_______ Principal:___________________



Year/Sem/Sec: II-B. Tech I-Semester C-Section

Room No:

A.Y : 2016 -17

WEF:

Class Teacher:

Time

09.30-10.20

10.20-11.10

11.10-12.00

12.00-12.50

12.50-1.30

1.30-2.20

2.20-3.10

3.10-4.00

Period

1

2

3

4

LUNCH

5

6

7

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

S.No

Subject(T/P)

Faculty Name

Contact No

1


2

MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE

3

DIGITAL LOGIC DESIGN

4

ELECTRONIC DEVICES AND CIRCUITS

5

BASIC ELECTRICAL ENGINEERING

6

DATA STRUCTURES

7

ELECTRICAL AND ELECTRONICS LAB

8

DATA STRUCTURES LAB

9

SEMINAR

10

*-Tutorial Hour/Discussion Hour




Year/Sem/Sec: II-B.Tech I-Semester D-Section

Room No:

A. Y : 2016--167 WEF:

Class Teacher:

Time

09.30-10.20

10.20-11.10

11.10-12.00

12.00-12.50

12.50-1.30

1.30-2.20

2.20-3.10

3.10-4.00

Period

1

2

3

4

LUNCH

5

6

7

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

S.No

Subject(T/P)

Faculty Name

Contact No

1

2

3

4

5

6

7

8

9

10


12.Individual Time Table:

Time table CSE-A Name of the faculty:

Time

9.30-10.20

10.20-11.10

11.10-12.00

12.00-12.50

Lunch Break

1.30-2.20

2.20-3.10

3.10-4.00

Period

1

2

3

4

5

6

7

Mon

Tues

Wed

Thurs

Fri

Sat

Time table CSE-B Name of the faculty:

Time

9.30-10.20

10.20-11.10

11.10-12.00

12.00-12.50

Lunch Break

1.30-2.20

2.20-3.10

3.10-4.00

Period

1

2

3

4

5

6

7

Mon

Tues

Wed

Thurs

Fri

Sat

Time table CSE-C Name of the faculty:

Time

9.30-10.20

10.20-11.10

11.10-12.00

12.00-12.50

Lunch Break

1.30-2.20

2.20-3.10

3.10-4.00

Period

1

2

3

4

5

6

7

Mon

Tues

Wed

Thurs

Fri

Sat

Time table CSE-D Name of the faculty:

Time

9.30-10.20

10.20-11.10

11.10-12.00

12.00-12.50

Lunch Break

1.30-2.20

2.20-3.10

3.10-4.00

Period

1

2

3

4

5

6

7

Mon

Tues

Wed

Thurs

Fri

Sat

13. Lecture Schedule with methodology being used/adopted

Lecture Schedule for CSE-A Section:

S. No

Unit No

Total no. of Periods

Topics to be covered

Regular / Additional

Teaching aids used

LCD/OHP/BB

Date

1

I

1

Introduction to Probability

regular

BB

2

1

Introduction to Random variables

regular

BB

3

1

Discrete Random variables- probability distributions, mass function

regular

BB

4

1

Problems on D.R.V

regular

BB

5

1

continuous Random variables- probability distributions, density function, mathematical expectation

regular

BB

6

1

Problems on C.R. V

regular

BB

7

1

Moment about origin, central moments

regular

BB

8

1

Moment generating function of probability distribution

regular

BB

9

1

Binomial distribution- Generating function, mean and variance

regular

BB

10

1

Problems on Binomial distribution

regular

BB

11

1

Poisson distribution- Generating function, mean and variance

regular

BB

12

1

Problems on Poisson distribution

regular

BB

13

1

Normal distribution- Generating function, mean and variance

regular

BB

14

1

Normal distribution- median, mode

regular

BB

15

1

Standard normal variate- mean, variance

regular

BB

16

1

Areas under normal curves

regular

BB

17

II

1

Introduction to joint probability distributions

regular

BB

18

1

Joint probability mass / density function

regular

BB

19

1

Marginal probability mass / density function

regular

BB

20

1

Covariance of two random variables

regular

BB

21

II

1

Correlation – coefficient of correlation

regular

BB

22

1

Problems on correlation

regular

BB

23

1

Problems on correlation for rounded mean

regular

BB

24

1

Rank correlation- non repeated ranks

additional

BB

25

1

Rank correlation- repeated ranks

regular

BB

26

1

Multiple correlation

regular

BB

27

1

Lines of regression

regular

BB

28

1

Finding out correlation coefficient using regression lines

regular

BB

29

1

Multiple regression

regular

BB

30

III

1

Basic definitions on sampling

regular

BB

31

1

Expected values of sample mean and variance

regular

BB

32

1

Sampling distribution- Sampling distribution of mean and variances

regular

BB

33

III

1

Introduction to estimation – likelihood estimation

regular

BB

34

1

Interval estimation- confidence interval

regular

BB

35

1

Problems

regular

BB

36

1

Testing of hypothesis- procedure

regular

BB

37

1

Large samples – Z- test for single mean

regular

BB

38

1

Z- test for difference between two means

regular

BB

39

1

Test for standard deviation

regular

BB

40

1

Test for single proportion

regular

BB

41

1

Test for difference between two proportions

regular

BB

42

1

Small samples- t- distribution and its properties

regular

BB

43

1

t- test for single mean

regular

BB

44

1

Paired t- test

regular

BB

45

1

t- test for difference between two means

regular

BB

46

1

F- distribution and its properties

regular

BB

47

III

1

F- test for variances

regular

BB

48

1

Chi square distribution and its properties

regular

BB

49

1

Chi square test for goodness of fit

regular

BB

50

IV

1

Introduction to queuing theory- terminology

regular

BB

51

1

States of queuing theory, arrival and service processes

regular

BB

52

1

Pure birth-death process

regular

BB

53

1

Problems on M/M/1/∞

regular

BB

54

1

Problems on M/M/1/N

regular

BB

55

V

1

Introduction to Stochastic processes- Classification and methods of random processes

regular

BB

56

1

Stationary and non stationary random processes

regular

BB

57

1

Average values of single and two or more random processes

regular

BB

58

1

Markov process

regular

BB

59

1

Markov chain

regular

BB

60

1

Classification of states

regular

BB

61

1

Stochastic matrix

regular

BB

62

1

Summary And Revision

regular

BB

Lecture Schedule for CSE-B Section:

S. No

Unit No




Teaching aids used

LCD/OHP/BB

Date

1

I

1


regular

BB

2

1


regular

BB

3

1


regular

BB

4

1

Problems on D.R.V

regular

BB

5

1


regular

BB

6

1

Problems on C.R. V

regular

BB

7

1


regular

BB

8

1


regular

BB

9

1


regular

BB

10

1


regular

BB

11

1


regular

BB

12

1


regular

BB

13

1


regular

BB

14

1


regular

BB

15

1


regular

BB

16

1


regular

BB

17

II

1


regular

BB

18

1


regular

BB

19

1


regular

BB

20

1


regular

BB

21

II

1


regular

BB

22

1


regular

BB

23

1


regular

BB

24

1


additional

BB

25

1


regular

BB

26

1


regular

BB

27

1

Lines of regression

regular

BB

28

1


regular

BB

29

1

Multiple regression

regular

BB

30

III

1


regular

BB

31

1


regular

BB

32

1


regular

BB

33

III

1


regular

BB

34

1


regular

BB

35

1

Problems

regular

BB

36

1


regular

BB

37

1


regular

BB

38

1


regular

BB

39

1


regular

BB

40

1


regular

BB

41

1


regular

BB

42

1


regular

BB

43

1


regular

BB

44

1

Paired t- test

regular

BB

45

1


regular

BB

46

1


regular

BB

47

III

1


regular

BB

48

1


regular

BB

49

1


regular

BB

50

IV

1


regular

BB

51

1


regular

BB

52

1


regular

BB

53

1


regular

BB

54

1

Problems on M/M/1/N

regular

BB

55

V

1


regular

BB

56

1


regular

BB

57

1


regular

BB

58

1

Markov process

regular

BB

59

1

Markov chain

regular

BB

60

1


regular

BB

61

1

Stochastic matrix

regular

BB

62

1


regular

BB

Lecture Schedule for CSE-C Section:

S. No

Unit No




Teaching aids used

LCD/OHP/BB

Date

1

I

1


regular

BB

2

1


regular

BB

3

1


regular

BB

4

1

Problems on D.R.V

regular

BB

5

1


regular

BB

6

1

Problems on C.R. V

regular

BB

7

1


regular

BB

8

1


regular

BB

9

1


regular

BB

10

1


regular

BB

11

1


regular

BB

12

1


regular

BB

13

1


regular

BB

14

1


regular

BB

15

1


regular

BB

16

1


regular

BB

17

II

1


regular

BB

18

1


regular

BB

19

1


regular

BB

20

1


regular

BB

21

II

1


regular

BB

22

1


regular

BB

23

1


regular

BB

24

1


additional

BB

25

1


regular

BB

26

1


regular

BB

27

1

Lines of regression

regular

BB

28

1


regular

BB

29

1

Multiple regression

regular

BB

30

III

1


regular

BB

31

1


regular

BB

32

1


regular

BB

33

III

1


regular

BB

34

1


regular

BB

35

1

Problems

regular

BB

36

1


regular

BB

37

1


regular

BB

38

1


regular

BB

39

1


regular

BB

40

1


regular

BB

41

1


regular

BB

42

1


regular

BB

43

1


regular

BB

44

1

Paired t- test

regular

BB

45

1


regular

BB

46

1


regular

BB

47

III

1


regular

BB

48

1


regular

BB

49

1


regular

BB

50

IV

1


regular

BB

51

1


regular

BB

52

1


regular

BB

53

1


regular

BB

54

1

Problems on M/M/1/N

regular

BB

55

V

1


regular

BB

56

1


regular

BB

57

1


regular

BB

58

1

Markov process

regular

BB

59

1

Markov chain

regular

BB

60

1


regular

BB

61

1

Stochastic matrix

regular

BB

62

1


regular

BB

Lecture Schedule for CSE-D Section:

S. No

Unit No




Teaching aids used

LCD/OHP/BB

Date

1

I

1


regular

BB

2

1


regular

BB

3

1


regular

BB

4

1

Problems on D.R.V

regular

BB

5

1


regular

BB

6

1

Problems on C.R. V

regular

BB

7

1


regular

BB

8

1


regular

BB

9

1


regular

BB

10

1


regular

BB

11

1


regular

BB

12

1


regular

BB

13

1


regular

BB

14

1


regular

BB

15

1


regular

BB

16

1


regular

BB

17

II

1


regular

BB

18

1


regular

BB

19

1


regular

BB

20

1


regular

BB

21

II

1


regular

BB

22

1


regular

BB

23

1


regular

BB

24

1


additional

BB

25

1


regular

BB

26

1


regular

BB

27

1

Lines of regression

regular

BB

28

1


regular

BB

29

1

Multiple regression

regular

BB

30

III

1


regular

BB

31

1


regular

BB

32

1


regular

BB

I-MID EXAM

Regression- Regression coefficient

33

III

1


regular

BB

31/08/2015

34

1


regular

BB

01/09/2015

35

1

Problems

regular

BB

02/09/2015

36

1


regular

BB

04/09/2015

37

1


regular

BB

07/09/2015

38

1


regular

BB

08/09/2015

39

1


regular

BB

09/09/2015

40

1


regular

BB

11/09/2015

41

1


regular

BB

14/09/2015

42

1


regular

BB

15/09/2015

43

1


regular

BB

16/09/2015

44

1

Paired t- test

regular

BB

18/09/2015

45

1


regular

BB

19/09/2015

46

1


regular

BB

21/09/2015

47

III

1


regular

BB

22/09/2015

48

1


regular

BB

23/09/2015

49

1


regular

BB

26/09/2015

50

IV

1


regular

BB

28/09/2015

51

1


regular

BB

29/09/2015

52

1


regular

BB

03/10/2015

53

1


regular

BB

05/10/2015

54

1

Problems on M/M/1/N

regular

BB

06/10/2015

55

V

1


regular

BB

07/10/2015

56

1


regular

BB

09/10/2015

57

1


regular

BB

12/10/2015

58

1

Markov process

regular

BB

14/10/2015

59

1

Markov chain

regular

BB

16/10/2015

60

1


regular

BB

17/10/2015

61

1

Stochastic matrix

regular

PPT

19/10/2015

62

1


regular

BB

23/10/2015

14.Detailed Notes : Hardcopy available

15.Additional topics :

1. Fundamentals on Probability

2. Mutually Exclusive Events

3. Mutually Exhaustive Events

4. Conditional Probability

5. Addition theorem on two and three events

6. Baye’s theorem, Total probability Theorem

16.University Question papers of previous years:

GEETHANJALI COLLEGE OF ENGINEERING & TECHNOLOGY

Cheeryal (v), Keesara (M), R.R.Dist.-501301.

R13 Model Question Paper-I

B.Tech. II-year I-Semester,

Subject: Probability and Statistics

Time: 3 hours Max.Marks:75

_______________________________________________________________

Note: This question paper contains two parts A and B. Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 units. Answer any one full question from each unit. Each Question carries 10 marks and may have a, b, c as sub questions.

Part-A (25 Marks)

1. a) Define continuous random variable? (2M)

b) Derive mean of a Poisson distribution? (3M)

c) Define joint probability mass function? (2M)

d) If Then find cov (x,y)? (3M)

e) Define confidence interval? (2M)

f) How many different samples of size 2 can be chosen from a finite population of size 25? (3M)

g) What is pure birth and death process? (2M)

h) A machine repairing shop gets on average 16 machines per day ( of eight hours) for repair and the arrival pattern is Poisson. Find arrival rate per hour? (3M)

i) Explain about Stationary Processes. (2M)

j) Explain the process of finding expected duration of the game? (3M)

Part – B (50 Marks)

2. A random variable x has the following probability distribution.

x

1

2

3

4

5

6

7

8

P(x=x)

k

2k

3k

4k

5k

6k

7k

8k

Find the value of (i)K (ii) p(x 2) (iii) p(2 x ≤ 5). (10M)

(OR)

3. (i) The mean and variance of binomial distribution are 4 and 4/3 respectively. Find p(x>1).

(ii)Let x denote the number of heads is a single toss of 4 fair coins. determine

(a) p(x2)

(b) p(1≤x 3)(10M)

4. Ten participants in a contest are ranked by two judges as follows

X

1

6

5

10

3

2

4

9

7

8

y

6

4

9

8

1

2

3

10

5

7

Calculate the rank correlation coefficient?(10M)

(OR)

5. If is an angle between two regression lines show that . Explain the significance when r = 0 and r = 1. (10M)

6. An unbiased coin is thrown n times. It is desired that the relative frequency of the appearance of heads should lie between 0.49 and 0.51. Find the smallest value of n that will ensure this result with 90% confidence? (10M)

(OR)

7. Two horses A and B were tested according to the time in seconds to run a particular race with the following results. Test whether you can discriminate between two horses

(10M)

Horse A

28

30

32

33

33

29

34

Horse B

29

30

30

24

27

29

--

8. At the election commission office, for the Voter’s identity Card, a Photographer takes passport size photo at an average rate of 24 photos per hour. The photographer must wait until the voter blinks or scowls, so the time to take a photo is exponentially distributed. Customers arrive at Poisson distributed average rate of 20 voters per hour. Find i) what the utilization of Photographer is. (ii) How much time, the voter has to spend at the election commission office on an average to get the service. (10M)

(OR)

9. a) Explain about queuing theory characteristics? (10M)

b) Define preemptive discipline and non-preemptive priority? (10M)

10. The transition probability matrix of a markov chain is given by verify Whether the matrix is irreducible or not?(10M)

(OR)

11. Find the nature of states of the markov chain and explain with transition probability matrix (10M)



R13 Model Question Paper-II




________________________________________________________________


Part-A (25 Marks)

1. a) Define continuous random variable? (2M)

b) Derive mean of a Poisson distribution? (3M)

b) Define joint probability mass function? (2M)

c) If Then find cov (x,y)? (3M)

d) Define confidence interval? (2M)

e) How many different samples of size 2 can be chosen from a finite population of size 25? (3M)

f) What is pure birth and death process? (2M)

g) A machine repairing shop gets on average 16 machines per day ( of eight hours) for repair and the arrival pattern is Poisson. Find arrival rate per hour? (3M)

h) Explain about Stationary Processes. (2M)

i) Explain the process of finding expected duration of the game? (3M)


2. (i) Using recurrence formula find the probabilities when x= 0,1,2,3,4 and 5: if the mean of Poisson distribution is 3? (ii) Out of 800 families with 5 children each, how many would you expect to have (i) 3 boys (ii) either 2 or 3 boys

(10M)

(OR)

3. Fit a normal curve to the following distribution(10M)

x

2

4

6

8

10

f

1

4

6

4

1

4. The correlation table given below shows that the ages of husband and wife of 53 married couples living together on the census night of 1991.Calculate the coefficient of correlation between the age of the husband and that of the wife.

(10M)

Age of husband

Age of wife

Total

15-25

25-35

35-45

45-55

55-65

65-75

15-25

1

1

-

-

-

-

2

25-35

2

12

1

-

-

-

15

35-45

-

4

10

1

-

-

15

45-55

-

-

3

6

1

-

10

55-65

-

-

-

2

4

2

8

65-75

-

-

-

-

1

2

3

Total

3

17

14

9

6

4

53

(OR)

5. Find the regression line of x on y and y on x for the following data(10M)

X

10

12

13

16

17

20

25

Y

10

22

24

27

29

33

37

6. A certain stimulus administered to each of 12 patients resulted in the following increases of blood pressure.5,2,8,-1,3,0,-2,1,5,0,4,6. Can it be concluded that the stimulus will in general be accompanied by an increase in blood pressure?(10M)

(OR)

7. The measurements of the output of two units have given the following results. Assuming that both samples have been level whether the two populations have the same variance. (10M)

Unit-A

14.1

10.1

14.7

13.7

14.0

Unit-B

14.0

14.5

13.7

12.7

14.1

(OR)

8. A fast food restaurant has one drive is window. It is estimated that cars arrive according to a Poisson distribution at the rate of 2 every 5 minutes and that there is enough space to accommodate a line of 10 cars. Other arriving cars cannot wait outside this space, if necessary. If takes 1.5 minutes on the average to fill an order, but the service time actually varies according to an exponential distribution. Determine the following.

a)The probability that the facility is idle.

b)The expected number of customers waiting to be served.(10M)

(OR)

9. A car park contains 5 cars. The arrival of cars is Poisson with a mean rate of 10 per hour. The length of time each car spends in the car park has negative exponential distribution with mean 2 hours. How many cars are in the car park on average and what is the probability of a newly arriving customer finding the car park full and having to park his car elsewhere?(10M)

10. is this matrix stochastic?(10M)

(OR)

11. Find the nature of states of the Markova chain with transition probability matrix

(10M)



R13 Model Question Paper-III




________________________________________________________________


Part-A (25 Marks)

1

i. Define discrete random variable? (2M)

ii. Derive mean of aNormal distribution? (3M)

iii. Define marginal probability mass function? (2M)

iv. The two regression equations of the variables x and y are (3M)

and . Then find mean of y?

v. Define population, sample and sampling with examples? (3M)

vi. For an F- distribution find with and ? (2M)

vii. Explain the terms of queuing theory and give two examples. (2M)

viii. Show that Average queue length is (3M)

ix. Explain about Markov Processes. (2M)

x. Let then find the expected duration of the game. (3M)


1. If f(x) is the distribution function x given by

Determine (i) f(x) (ii) k (iii) mean (10M)

(OR)

2. If the masses of 300 students are normally distributed with mean 68 kgs and standard deviation 3 kgs how many students have masses (i)Greater than 72 kgs (ii) Less than or equal to 64kgs (iii) Between 65 and 71 kgs inclusive ?(10M)

3. Psychological tests of intelligence and of engineering ability were applied to 10 students. Hence is a record of ungrouped data showing intelligence ratio (I.R.) and engineering ratio (E.R.) Calculate the coefficient of correlation. (10M)

Student

A

B

C

D

E

F

G

H

I

J

I.R.

105

104

102

101

100

99

98

96

93

92

E.R.

101

103

100

98

95

96

104

92

97

94

(OR)

4. In the following table are recorded data showing the test scores made by salesmen on an intelligence test and their weekly sales.

Sales men

1

2

3

4

5

6

7

8

9

10

Test Scores

40

70

50

60

80

50

90

40

60

60

Sales(‘000)

2.5

6.0

4.5

5.0

4.5

2.0

5.5

3.0

4.5

3.0

Calculate the regression line of sales on test scores and estimate the most

probable weekly sales volume if a sales man makes a score of 70? (10M)

5. A 11 students were given a test in statistics they were given a month’s further tuition and a second test of equal difficulty was held at the end of it. Do the marks give evidence that the students have benefited by extra coaching? (10M)

Boys

1

2

3

4

5

6

7

8

9

10

11

Marks I test

23

20

19

21

18

20

18

17

23

16

19

Marks II test

24

19

22

18

20

22

20

20

23

20

17

(OR)

6. A sample of 900 members is found to have a mean of 3.4 cm. Can it be reasonably regarded as a truly random sample from a large population with mean 3.25 cm and S.D. 1.61cm? (10M)

7. A manager of a Local hamburger restaurant is preparing to open a new fast food restaurant called Hasty Burgers. Based on the arrival rates at existing outlets. Manager expects customers to arrive at the drive in window according to a Poisson distribution with a mean of 20 customers per hour. The service rate is flexible however, the service times are expected to follow an exponential distribution the drive in window in single ever operation.

(a) What service rate is needed to keep the average number of customers is the service system to 4.

(b) For the service rate in part (a). What is the probability that more than 4 customers are in the line and being served? (10M)

(OR)


(a) The probability that the facility is idle.

(b) The expected number of customers waiting to be served. (10M)

9. Explain classification of states and chains of markov process. (10M)

(OR)

10. If the transition probability matrix of market shares of three brands A,B and C

is Find (a) The market shares in second and third periods

(b) The limiting probabilities (10M)

17.Question Bank:

Unit – I

Short answer Questions

1. Define probability.

1. What are the axioms of probability?

1. Define random experiment.

1. Define random variable and types of random variable.

1. Define an Event.

1. Define Sample space.

1. Define discrete random variable.

1. Define continuous random variable.

1. Derive probability distribution for D.R.V,?

1. Derive probability distribution for C.R.V,?

1. Define Binomial Distribution.

1. Derive probability function of a Binomial distribution?

1. Derive mean of a Binomial distribution?

1. Derive variance of a Binomial distribution?

1. Define Moment generating function.

1. Find the moment generating function for Binomial distribution.

1. Define Poisson distribution.

1. Find the moment generating function for Poisson distribution.

1. Derive probability function of a Poisson distribution?

1. Derive mean of a Poisson distribution?

1. Derive variance of a Poisson distribution?

1. Define Normal distribution.

1. Derive probability function of a Normal distribution?

1. Derive mean of a Normal distribution?

1. Derive variance of a Normal distribution?

1. Derive median of a Normal distribution?

1. Derive mode of a Normal distribution?

1. Find the ratio of Mean Deviation, Standard deviation, Standard deviation and quartile deviation.

Long Answer Questions:

1. If f(x) is the distribution function x given by

Determine (i) f(x) (ii) k (iii) mean

1. A random variable x has the following probability distribution.

x

1

2

3

4

5

6

7

8

P(x=x)

k

2k

3k

4k

5k

6k

7k

8k

Find the value of

1. K (ii) p(x 2) (iii) p(2 x ≤ 5).

1. If X and Y are discrete random variables and k is constant then prove that

(i) E(X+K) = E(X) +K (ii) E(X+Y) = E(X) + E(Y)

1. Let F(X) be the distribution function of random variable X given by

Determine (i) c (ii) mean (iii) p(x>1)

1. (i) The mean and variance of binomial distribution are 4 and 4/3 respectively. Find p(x>1).

(ii)Let x denote the number of heads is a single toss of 4 fair coins. determine (a) p(x2) (b) p(1≤x 3)

1. Average number of accidents on any duty on a national highway is 1.6. Determine the probability that the number of accidents are (i)at least one (ii) at most one.

1. (i) Using recurrence formula find the probabilities when x= 0,1,2,3,4 and 5: if the mean of Poisson distribution is 3? (ii) Out of 800 families with 5 children each, how many would you expect to have (i) 3 boys (ii) either 2 or 3 boys

1. If the masses of 300 students are normally distributed with mean 68kgs and standard deviation 3kgs how many students have masses (i)Greater than 72kgs (ii) Less than or equal to 64kgs (iii) Between 65 and 71kgs inclusive ?

1. In a normal distribution, 31% of the items are under 45 and 8% are over 64. Find the mean and S.D. of the distribution?

1. Fit a normal curve to the following distribution

x

2

4

6

8

10

f

1

4

6

4

1

Unit – II

Short Answer Questions:

1. Define Joint probability.

1. Define joint probability mass function.

1. Define joint probability density function.

1. Define marginal probability mass function.

1. Define marginal probability density function.

1. Define correlation.

1. Define Covariance.

1. Explain the rank of correlation.

1. What is meant by regression?

1. Define regression coefficient.

1. Write the relation between coefficient of correlation and regression coefficient?

1. The two regression equations of the variables x and y are and . Then find correlation coefficient between x &y?

1. The two regression equations of the variables x and y are and . Then find mean of x?

1. The two regression equations of the variables x and y are and . Then find mean of y?

1. If Then find cov (x,y)?

Long answer Questions:

1. Ten participants in a contest are ranked by two judges as follows

X

1

6

5

10

3

2

4

9

7

8

y

6

4

9

8

1

2

3

10

5

7

Calculate the rank correlation coefficient?

1. Psychological tests of intelligence and of engineering ability were applied to 10 students. Hence is a record of ungrouped data showing intelligence ratio (I.R.) and engineering ratio (E.R.) Calculate the coefficient of correlation.

Student

A

B

C

D

E

F

G

H

I

J

I.R.

105

104

102

101

100

99

98

96

93

92

E.R.

101

103

100

98

95

96

104

92

97

94

1. The correlation table given below shows that the ages of husband and wife of 53 married couples living together on the census night of 1991. Calculate the coefficient of correlation between the age of the husband and that of the wife.

Age of husband

Age of wife

Total

15-25

25-35

35-45

45-55

55-65

65-75

15-25

1

1

-

-

-

-

2

25-35

2

12

1

-

-

-

15

35-45

-

4

10

1

-

-

15

45-55

-

-

3

6

1

-

10

55-65

-

-

-

2

4

2

8

65-75

-

-

-

-

1

2

3

Total

3

17

14

9

6

4

53

1. In the following table are recorded data showing the test scores made by salesmen on an intelligence test and their weekly sales.

Sales men

1

2

3

4

5

6

7

8

9

10

Test Scores

40

70

50

60

80

50

90

40

60

60

Sales(‘000)

2.5

6.0

4.5

5.0

4.5

2.0

5.5

3.0

4.5

3.0

Calculate the regression line of sales on test scores and estimate the most probable weekly sales volume if a sales man makes a score of 70?

1. If is an angle between two regression lines show that . Explain the significance when r = 0 and r = 1.

1. Find if there is any significance correlation between the heights and weights given below

Heights in inches

57

59

62

63

64

65

55

58

57

Weights in lbs

113

117

126

126

130

129

111

116

112

1. Find Karl Pearson’s coefficient of correlation from the following data

Wages

100

101

102

102

100

99

97

98

96

95

Cost of living

98

99

99

97

95

92

95

94

90

91

1. A random sample of 5 college students is selected and their grades in Mathematics and Statistics are found to be the following. Find the coefficient of correlation between them

1

2

3

4

5

Mathematics

85

60

73

40

90

Statistics

93

75

65

50

80

1. Find the regression line of x on y and y on x for the following data

X

10

12

13

16

17

20

25

Y

10

22

24

27

29

33

37

1. Calculate coefficient of correlation from the following data

X

12

9

8

10

11

13

7

Y

14

8

6

9

11

12

3

Unit – III: Short Answer Questions

1. Define population, sample and sampling with examples?

1. Explain types of sampling?

1. If ‘N’ refers population and ‘n’ refers sample size then mention

i) The number of samples with replacement. ii) The number of samples without replacement.

1. Define large sample and small sample.

1. Define Parameters and Statistic.

1. Define sample mean , sample variation, Sample Standard deviation.

1. Explain sampling distribution and sampling distribution of statistic.

1. Write the central limit theorem.

1. Define Standard error and probable error of sample mean.

1. Define confidence interval?

1. Define estimate ,estimator and estimation .

1. Define Bayesian estimation.

1. Define statistical inference. and types of problems under statistical inference.

1. Explain briefly types of estimations.

1. Define an unbiased estimator and show that is an unbiased estimator of the population mean μ.

1. Define statistical hypothesis.

1. Define hypothesis testing, null hypothesis and alternate hypothesis.

1. Explain briefly type I error and type II error .

1. Explain i) critical region ii)left tailed test iii)Right tailed test iv) Two tailed est.

1. A sample of size 300 was taken whose variance is 225 and mean is 54. Construct 95% confidence interval for the mean?

1. What is the value of correction factor if n = 5 and N = 200?

1. How many different samples of size 2 can be chosen from a finite population of size 25?

1. What is meant by degree of freedom?

1. Define the statistics of ‘t’ ,‘F’ and distributions and their major applications.

1. For an F- distribution find with and ?


1. Prove that for a random sample of size n,X1,X2,….Xn taken from an infinite population s2 = is not unbiased estimator of the parameter σ2 but is unbiased.

1. (i)A random sample of 100 teachers in a large metropolitan area revealed a mean weekly salary of Rs.487 with a standard deviation Rs.48. with what degree of confidence can we assert that the average weekly salary of all teachers in the metropolitan area is between 478.6 to 495.4.

(ii) Among 900 people is a state 90 are found to be chapatti eaters. Construct 99%

confidence interval for the true proportion

1. Sample of size 2 are taken from the population 4,8,12,16,20,24 without replacement. Find (a) Mean of the population (b) Standard deviation of the population(c) The mean of the sampling distribution of the means (d) the standard deviation of the sampling distributions of means.

1. A sample of 900 members is found to have a mean of 3.4 cm. Can it be reasonably regarded as a truly random sample from a large population with mean 3.25 cm and S.D. 1.61cm?

1. The means of simple samples of sizes 1000 and 2000 are 67.5 and 68.0 cm respectively. Can the samples be regarded as drawn from the same population of S.D. 2.5cm?

1. An unbiased coin is thrown n times. It is desired that the relative frequency of the appearance of heads should lie between 0.49 and 0.51. Find the smallest value of n that will ensure this result with 90% confidence?

1. A certain stimulus administered to each of 12 patients resulted in the following increases of blood pressure.5,2,8,-1,3,0,-2,1,5,0,4,6. Can it be concluded that the stimulus will in general be accompanied by an increase in blood pressure?

1. A 11 students were given a test in statistics they were given a month’s further tuition and a second test of equal difficulty was held at the end of it. Do the marks give evidence that the students have benefited by extra coaching?

Boys

1

2

3

4

5

6

7

8

9

10

11

Marks I test

23

20

19

21

18

20

18

17

23

16

19

Marks II test

24

19

22

18

20

22

20

20

23

20

17

1. Two horses A and B were tested according to the time in seconds to run a particular race with the following results. Test whether you can discriminate between two horses

Horse A

28

30

32

33

33

29

34

Horse B

29

30

30

24

27

29

--

1. The results of polls conducted 2 weeks and 4 weeks before a election are shown in the following table:

Two weeks before election

4 weeks before election

For Democratic candidate

84

66

Undecided

37

43

For Republican candidate

79

91

Use the 0.05 level of significance to test whether there has been a change in opinion during the 2 weeks between the rolls.

1. The measurements of the output of two units have given the following results. Assuming that both samples have been level whether the two populations have the same variance.

Unit-A

14.1

10.1

14.7

13.7

14.0

Unit-B

14.0

14.5

13.7

12.7

14.1

Unit – IV

Short Answer Questions

1. Explain the terms of queuing theory and give two examples.

1. Explain about arrival pattern and service pattern.

1. Explain the characteristics of queuing theory.

1. Explain about The queue discipline and queue behavior.

1. What is pure birth and death process?

1. Probability that there are n customers in the system.

1. Probability that there are n or more customers in the system.

1. Show that Average number of customers in the system is

1. Show that Average queue length is

1. A TV repair man finds that the time spent on his jobs has an exponential distribution with mean 30 minutes. He repairs sets in the order in which they arrive. The arrival of sets is approximately Poisson with an average rate of 10 per eight hour day. Then find service rate per hour and arrival rate per hour.

1. A machine repairing shop gets on average 16 machines per day (of eight hours) for repair and the arrival pattern is Poisson. Find arrival rate per hour?


1. a) Write the operational characteristics of Queuing theory.

b) Assume the goods trains are coming is a yard at the root of 30 trains per day and suppose that inter arrival times follow an exponential distribution the service time for each train is assumed to be exponential with an average of 36 minutes. If the yard can admit a trains at time (there being 10 lines), one of which is reserved for shunting purpose), calculate the probability that the yard is empty and find the average queue length.

2. a) A T.V repair man finds that the time spent on has jobs has an exponential distribution with mean 30 minutes. If he repairs sets in the order in which they came in, and if the arrival of sets in approximately Poisson with an average rate of 10 per-8-hour day. What is repairman’s expected idle time each day? How many jobs are a head of the average set just brought us?

b) Patients arrive at a clinic according to a poisson distribution at a rate of 30 patients per hours, the waiting room does not accommodate more than 14 patients. Examination time per patients is exponential with mean rate 20 per hour.

i) Find effective arrival rate at the clinic

ii) What is the probability that an arriving patients will not wait?

3. A manager of a Local hamburger restaurant is preparing to open a new fast food restaurant called Hasty Burgers. Based on the arrival rates at existing outlets. Manager expects customers to arrive at the drive in window according to a Poisson distribution with a mean of 20 customers per hour. The service rate is flexible however, the service times are expected to follow an exponential distribution the drive in window in single ever operation.

1. What service rate is needed to keep the average number of customers is the service system to 4.

1. For the service rate in part (a). What is the probability that more than 4 customers are in the line and being served?

4. At a certain petrol pump, customers arrive in a poission process with an average time of five minutes between arrivals, the time intervals between serves at the petrol pump follows exponential distribution and the mean time taken to service a unit is two minutes. Find the following:

1. Average time a customer has to wait is the queue.

1. By how much time the flow of the customers be increases to justify the opening of another service point, where the customer has to wait for five minutes for the service.


1. The probability that the facility is idle.

1. The expected number of customers waiting to be served.

6. a) Explain about queuing theory characteristics?

1. Define preemptive discipline and non-preemptive priority?

7. Consider a self service store with one cashier. Assume Poisson arrivals and exponential service time. Suppose that 9 customers arrive on the average of every 5 minutes and the cashier can serve 19 in 5 minutes. Find (i) the average number of customers queuing for service.(ii) The probability of having more than 10 customers in the system. (iii) The probability that the customer has to queue for more than 2 minutes.

8. A car park contains 5 cars. The arrival of cars is Poisson with a mean rate of 10 per hour. The length of time each car spends in the car park has negative exponential distribution with mean 2 hours. How many cars are in the car park on average and what is the probability of a newly arriving customer finding the car park full and having to park his car elsewhere?

9. At the election commission office, for the Voter’s identity Card, a Photographer takes passport size photo at an average rate of 24 photos per hour. The photographer must wait until the voter blinks or scowls, so the time to take a photo is exponentially distributed. Customers arrive at Poisson distributed average rate of 20 voters per hour. Find i) what the utilization of Photographer is. (ii) How much time, the voter has to spend at the election commission office on an average to get the service.

10. Barber A takes 15 minutes to complete one hair cut. Customers arrive in his shop at an average rate of one every 30 minutes, Barber B takes 25 minutes to complete one hair cut and customers arrive at his shop at an average rate of one every 50 minutes. The arrival processes are Poisson and the service times follow an exponential distribution.

Unit – V

Short Answer Questions:

1. What do you mean by stochastic processes and what are the types of stochastic process? Define them.

1. Explain about Markov Processes.

1. Explain about Stationary Processes.

1. Explain about dependent and independent Stochastic Processes.

1. Explain about gamblers Ruin Problem.

1. Show that the probability that the game never ends is zero.

1. Explain the process of finding expected duration of the game?

1. Let then find the expected duration of the game.

1. Calculate the probability of ruin and expected duration of the game, when

1. Explain about Transition matrix.

1. Which of the following matrices are stochastic

1. Define the Stochastic matrix Which of the stochastic matrices are regular.


1. The transition probability matrix of a morkov chain is given by verify

Whether the matrix is irreducible or not?

2. is this matrix stochastic?

3. is this matrix regular?

4. If the transition probability matrix of market shares of three brands A,B and C

is Find (a) The market shares in second and third periods

5. (b)The limiting probabilities.

6. Explain classification of states and chains of markov process.

7. Three boys A,B and C are throwing a ball to each other. A always throws the ball to B and B always throws the ball to C, but C is just as likely to throw the ball to B as to A. Show that the process is Markovian. Find the transition matrix and classify the states. Do all the states ate ergodic?

8. Find the nature of states of the markov chain with transition probability matrix

9. A fair die is tossed repeatedly. If denotes the maximum of the numbers occurring in the first n tosses, find the transition probability matrix P of the markov chain . Find also and .

10. Define Markov chain,regular, ergodic and Stochastic matrices?

11. Check whether the following markov chain is regular and ergodic?

12. A gambler has Rs.2. He bets Rs.1 at a time and wins Rs.1 with the probability 0.5. He stops playing if he looses Rs.2 or wins Rs.4.

(a) What is the transition probability matrix of the related Markov chain?

(b) What is the probability that he has lost his money at the end of 5 plays?

18.Assignment questions:

Unit-I

1. a) Explain, with suitable examples, discrete and continuous random variable.

b) Find the first 3 moments about origin from Moment generating function of the

Binomial distribution.

2. a) If ‘X’ is a continuous random variable whose probability density function is given by

b) A sample of 3 items is selected from a box having 6 items of which 3 are defective

then find the mean of the distribution of defective items.

3. a) If X is the continuous random variable whose probability density function is and Find the values a and b.

b) The mean variance of a binomial distribution are . Find :

4. a) If the weight of 1000 students are normally distributed with mean 75 kgs. And standard deviation 10kgs. How many students have weight greater than 90 kgs.

b) If X and Y are two random variables with joint probability density function Find the value of K.

5. a) Two dice are thrown 4 times. If getting a sum of 7 is a success. Find the probability that getting the success

b) Students of a class were given an examination. Their marks were found to be normally distributed with mean 55 marks and standard deviation 5. Find the number of students who got the marks more than 60 if 500 students wrote the examination.

6. a) Poisson variable has double mode at find the maximum probability and also find

b) If the masses of 300 students are normally distributed with mean 68 kgs and standard deviation 3 kgs how many students have masses greater than 72 kgs.

Unit-2

7. Calculate the coefficient of rank correlation

X

68

64

75

50

64

80

75

40

55

64

Y

62

58

68

45

81

60

68

48

50

70

8. Write the relation between correlation and regression coefficients. Is it possible to have two variables X and Y with regression coefficient as 2.8 and -0.5? Explain.

9. If X and Y are two random variables having joint density function

Find:

10. For the following data, find equations of the two regression lines.

X

1

2

3

4

5

Y

15

25

35

45

55

11. The joint probability density function is given by

Find: a) Marginal probability density function for X

b) Marginal probability density function for Y

c) Conditional P.D.F. of X given Y

d) Conditional P.D.F. of Y given X

e)

12. Calculate the coefficient of correlation between the two variable x and y. Also find the regression coefficients.

X

65

66

67

67

68

69

70

72

Y

67

68

65

68

72

72

69

71

13. The equations of two regression lines obtained in a correlation analysis are

Find the means of x and y.

Unit-3

14. In a city A 20% of a ramdom sample of 900 school boys had a certain physical defect. In another city B 18.5% of a random sample of 1600 school boys had the same defect. Is the difference between the proportions significant?

15. Write the standard error of (i) sample mean (ii) difference of two sample means.

16. Mean of population =0.700, mean of the sample = 0.742, standard deviation of the sample =0.040, sample size=10. Test the null hypothesis for population mean=0.700.

17. A die is thrown 60 times with the following results.

Face

1

2

3

4

5

6

Frequency

8

7

12

8

14

11

Test 5% level of significance if the die is honest.

18. Two horses A and B were tested according to the time (in seconds) to run a particular track with the following results.

Horse A

28

30

32

33

35

29

34

Horse B

29

30

30

24

27

29

Test whether the two horses have the same running capacity.

19. Define type-I and type-II error.

20. A sample of 150 items is taken from a population whose standard deviation is 12. Find the standard error of means.

21. A researcher wants to know the intelligence of students in a school. He selected two groups there 150 students having mean IQ of 75 with a S.D. of 15 in the second group there are 250 students having mean IQ of 70 with S.D. of 20. Is there a significant difference between the means of two groups?

22. Fit a Poisson distribution to the following data and test the goodness of fit.

X

0

1

2

3

4

5

6

Observed frequency

275

72

30

7

5

2

1

23. In a city A 20% of a random sample of 900 school boys had a certain physical defect. In another city B 18.5% of a random sample of 1600 school boys had the same defect. Is the difference between the proportion significant?

24. Two independent samples of 8 and 7 items respectively have the following values.

Sample I

11

11

13

11

15

9

12

14

Sample II

9

11

10

13

9

8

10

-

Is the difference between the means of sample significant?

25. Given below is the number of male births in 1000 families having five children in each family.

Male Children

0

1

2

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