## TYBAstatisticssyllabus 1 Syllabus Mumbai University by munotes

## Page 2

UNIVERSITY OF MUMBAI

Syllabus for the T.Y.B.A.

Programme: B.A.

Sem. V & Sem. VI

Course: STATISTICS

(As per Credit Based and Choice System

with effect from the academic year 2018 –2019)

## Page 3

T.Y.B.A. STATISTICS Syllabus

Restruc tured for Credit Based and Grading System

To be implemented from the Academic year 2018 -2019

T.Y.B.A. STATISTICS

Students who have opted for

ONE paper at F.Y.B.A.Statistics and TWO papers at S.Y.B.A.Statistics

will opt for

THREE papers at T.Y.B.A. S tatistics (3Units)

## Page 4

T.Y.B.A. STATISTICS Syllabus Credit Based and Grading System

To be implemented from the Academic year 2018 -2019

SEMESTER V

Theory

Course UNIT TOPICS Credits L / Unit

UASTA 501 I Univariate Random Variables (Discrete)

Biva riate probability distributions (Discrete)

3

15

II Standard Discrete Probability Distributions:

Uniform, Bernoulli, Binomial , Poisson and

Geometric distribution

15

III 1.Negative Binomial and hyper geometric

distributions.

2. Statistic al computing using R –software

15

UASTA 502 I Concepts of Sampling and Role of sampling in

Research Methodology

3 15

II Simple random sampling

15

III 1. Stratified Random Sampling

2. Applications of R software

15

UASTA 503 I Mortality Tables

3 15

II Compound Interest and Annuities Certain

15

III Assurance Benefits

15

UASTA P5

Practical of

Course UASTA 501,

Course UASTA 502,

Course UASTA 503

3 3 lecture

periods

per

course

per week

## Page 5

Semester VI

Theory

T.Y.B.A. STATI STICS Syllabus Credit Based and Grading System

To be implemented from the Academic year 2018 -2019

Course UNIT TOPICS Credits L / Unit

UASTA 601 I Univariate Random Variables (Continuous)

Bivariate probability distributions

(Continuous)

3

15

II Standard Continuous Probability Distributions

15

III Exact Sampling Distributions

15

UASTA 602 I Analysis of Variance

3 15

II Design of Experiments,

Completely Randomized Design

Randomized Block Design and

Latin Square design

15

III 1. Missing plot and Efficiency of all three

designs

2. Applications of R software

15

UASTA 603 I Simulation

3 15

II Linear Regression Model

15

III 1. Concepts of Autocorrelation,

Heteroscedasticity, Multicollinearity.

Concept of Logistic r egression.

2. Applications of R software

15

UASTA P6

Practical of

Course UASTA 601,

Course UASTA 602,

Course UASTA 603

3 3 lecture

periods

per

course

per week

## Page 6

Semester V

Course

code Title Credits

UASTA

501 PROBABILITY DISTRIBUTIONS and R SOFT WARE 3 credits

(45 lectures)

Unit 1 Univariate Random variables(Discrete):

Moment Generating function, Cumulant Generating function - Their

important properties. Relationship between moments and cumulants and

their uses.

Transformation of random variable (only statement and application)

Characteristic Function :

Definition and properties (without Proof)

Examples of obtaining raw moments and central moments up to order four

using M.G.F. and C.G.F. for discrete distributions.

Bivariate Probability distribut ion(Discrete):

Joint probability mass function. Its properties. Marginal and Conditional

distributions. Independence of Random variables. Conditional Expectation

and Variance.

Transformation of Random variables and Jacobian of transformation with

illustra tions 15 Lectures

Unit 2 Standard Discrete Probability distributions:

Uniform, Bernoulli, Binomial , Poisson and Geometric distribution

The following aspects of the above distribution (wherever applicable) to be

discussed :

Mean , variance Measures of Sk ewness and Kurtosis based on moments

using M.G.F and C.G.F. Nature of Probability distribution with change in the

values of parameters ,Mode and Additive property.

If X follows Binomial then the distribution of n -x. Recurrence relation for

moments with pr oof.

If X and Y are two independent Binomial variables , conditional distribution

of X given X + Y with proof

If X and Y are two independent Poisson variables, then the conditional

distribution of X given X + Y with proof.

Geometric Distribution -

Definit ion in terms of no. of failures and no. of trials. Lack of memory

property with proof.

If X and Y are two independent and identically distributed Geometric

variables, conditional distribution of X given X + Y with proof. Real life

situations of Geometric d istribution. 15 Lectures

Unit 3 Part 1: ( 9L)

Negative Binomial 15 Lectures

## Page 7

Course

code Title Credits

UASTA

501 PROBABILITY DISTRIBUTIONS and R SOFT WARE 3 credits

(45 lectures)

The following aspects of the above distribution (wherever applicable) to be

discussed :

Mean , variance, Measures of Skewness and Kurtosis based on moments

using M.G.F and C.G.F. Nature of Probability distribution with change in the

values of parameters, Mode and Additive property. Recurrence relation for

central moments , Variance,µ 3,µ4 using recurrence relation for central

moments, Recurrence relation for probabilities . Fitting of distribu tion.

Real life situations of Negative Binomial distributions.

Hyper geometric distribution

Definition, Mean, Variance, Limiting distribution of Hyper geometric

distribution (with proof)

For two i.i.d. Binomial variables X and Y ,conditional distribution o f X

given X+Y (with proof). Real life situations of Hyper geometric

distributions.

Part 2: Statistical computing using R –software :

Fundamentals of R - I (4L):

Entering data using c function, Creating a vector using scan function, Simple

calculations, ve ctor operations, simple functions such as log ,prod, cumsum,

length, sqrt, min, summary, round, seq, rep, sort .

Creating a data frame, Importing data from MS -Excel file, Using read.table

command.

Frequency distribution:

Construction of frequency distribu tion using table and cut functions for

discrete and continuous distributions, bivariate distribution, less than and

more than cumulative frequencies, relative frequency.

Plotting Diagrams and graphs using Diagrams:

Bar diagrams, Pie diagram, histogram, fr equency curve , ogives, box plot .

Fundamentals of R – II (2L):

Finding and plotting of probability and cumulative probability for standard

discrete distributions such as binomial, Poisson, geometric.

REFERENCES:

1. Mood A. M., GraybillF.A. Boy es D. C.: Introduction to the theory of statistics, Third

Edition; McGraw -Hill Book Company.

2. Hogg R .V., Craig A.T.: Introduction to Mathematical Statistics, Fourth Edition; Collier

McMillan Publishers.

3. Hogg R.V., Tanis, E. A.: Probability and Statistical Inference, Third Edition; Collier McMillan

Publishers.

## Page 8

4. Miller I., Miller M.: .John E. Freund’s Mathematical Statistics; Sixth Edition; Pearson

Education Inc.

5. Hoel P.G.; Fourth Edition Introduction to Mathematical Statistics; John Wiley & Sons Inc.

6. Gupta S. C., Kapoor V.K.: Fundamentals of Mathematical Statistics; Eighth Edition; Sultan

Chand & Sons.

7. Kapur J. N., Saxena H.C.: Mathematical Statistics, Fifteenth Edition; S. Chand & Company

Ltd.

8. Medhi J. : Statistical Methods: An Introductory Text; Second editio n; Wiley Eastern Ltd.

9. Goon A.M., Gupta M.K., Das Gupta B.: An Outline of Statistical Theory Vol. 1; Third

Edition; The World Press Pvt. Ltd.

10. R for Statistics by Julie josse, Maela Kloareg, CRC Press, Taylor and Francis group

11. Statistics using R by Sharad D Gore, Sudha G Purohit, Shailaja R Deshmukh, Norosa

Publishing house.

==========

## Page 9

Semester V

Course

code Title Credits

UASTA

502 THEORY OF SAMPLING and R SOFTWARE 3 credits

(45 lectures)

Unit 1 Concepts of Sampling and Role of sampling in Research Met hodology:

Basic definitions involved in sample survey and population survey.

Objectives of a sample survey. Designing a questionnaire, characteristics of

a good questionnaire (Questions with codes & scores are to be discussed).

Reliability and validity tes ting by using

(i) Test – Retest method

(ii) Internal Consistency:

(A) Kuder Richardson Coefficient (KR -20)

(B) Cronbach’s Coefficient Alpha.

Planning, execution and analysis of a sample survey, practical problems at

each of these stages. Sampling and non -sampling errors with illustrations.

Study of some surveys illustrating the above ideas, rounds conducted by

National Sample Surveys organization.

15 Lectures

Unit 2 Simple random sampling:

Simple Random Sampling for Variables:

Definition, Sampling wi th & without replacement (WR/WOR).

Lottery method & use of Random numbers to select Simple random sample.

Estimation of population mean & total.

Expectation & Variance of the estimators,

Unbiased estimator of variance of these estimators. (WR/WOR).

Simple Random Sampling for Attributes:

Estimation of population proportion. Expectation & Variance of the

estimators, Unbiased estimator of variance of these estimators.

(WR/WOR).

Estimation of Sample size based on a desired accuracy in case of SRS for

variab les & attributes. (WR/WOR).

15 Lectures

Unit 3 Part 1( 8L)

Stratified random sampling:

Need for Stratification of population with suitable examples. Definition of

Stratified Sample. Advantages of stratified Sampling.

Stratified Random Sampling:

Estimati on of population mean & total in case of Stratified Random

Sampling (WOR within each strata). Expectation & Variance of the

unbiased estimators, Unbiased estimators of variances of these estimators.

Proportional allocation, Optimum allocation with and with out varying costs.

Comparison of Simple Random Sampling, Stratified Random Sampling 15 Lectures

## Page 10

Course

code Title Credits

UASTA

502 THEORY OF SAMPLING and R SOFTWARE 3 credits

(45 lectures)

using Proportional allocation & Neyman allocation.

Part 2(7L)

Applications of R software I:

Calculation of measures of central tendency, absolute and relative measures

of dispersion, measures of skewness and kurtosis for discrete and continuous

frequency distributions.

Applications of R software II:

Listing simple random samples with replacement and without replacements

for

n = 2, n=3. Verifying formulae for mean and var iance.

R software programs for stratified random sampling:

Selection of stratified samples. Calculation of mean and variance for

Population mean and Population total.

REFERENCES:

1. Cochran W.G.:Sampling Techniques; 3rd Edition; Wiley(1978)

2. Murthy M.N.:Sampling Theory and methods; Statistical Publishing Society. (1967)

3. Des Raj:Sampling Theory; McGraw Hill Series in Probability and Statistics. (1968).

4. Sukhatme P.V. and Sukhatme B.V.:Sampling Theory of Surveys with Aplications;

3rd Edition; Iowa State University Press (1984).

5. Gupta S. C. andKapoor V.K.:Fundamentals of Applied Statistics; 3rd Edition; Sultan

Chand and Sons (2001).

6. SinghDaroga, Chaudhary F.S.: Theory and Analysis of Sample Survey Designs:,

Wiley Eastern Ltd. (1986).

7. Sampath S.: Samp ling Theory and Methods, Second Edition (2005),Narosa.

8. MukhopadhyayParimal:Theory and Methods of Survey Sampling, (1998),Prentice

Hall Of India Pvt. Ltd.

9. R for Statistics by Julie josse, Maela Kloareg, CRC Press, Taylor and Francis group

10. Statistics using R by Sharad D Gore, Sudha G Purohit, Shailaja R Deshmukh, Norosa

Publishing house.

## Page 11

==============

Semester V

Course

code Title Credits

UASTA

503 APPLIED STATISTICS - I 3 credits

(45 lectures)

Unit 1 Mortality Tables:

Various Mortality f unctions. Probabilities of living and dying.

The force of mortality.

Estimation of μx from the mortality table. Mortality table as a population model.

Stationary population. Expectation of life and Average life at death.

Central Death rate. 15 Lectures

Unit 2 Compound Interest and Annuities Certain:

Accumulated value and present value, nominal and effective rates of interest.

Discount and discounted value, varying rates of interest.

Equation of value. Equated time of payment.

Present and accumulated values of annuity certain ( immediate and due) with and

without deferment period.

Present and accumulated values of -

i) increasing annuity

ii) increasing annuity when successive installment form

a) arithmetic progression b) geometric progression

Redemption of loan

(Annuity with frequency different from that with which interes t is convertible

– Not to be done) 15 Lectures

Unit 3 Assurance Benefits:

Present value of Assurance benefits in terms of commutation functions of -

i) Pure endowment assurance, ii) Temporary assurance, iii) Endowment assurance,

iv) Whole life as surance, v) Special endowment assurance,

vi) Deferred temporary assurance, vii) Deferred whole life assurance.

Present value in terms of commutation functions of Life annuities and Temporary

life

annuities (immediate and due)

Net Level annual premiums for the assurance plans mentioned above. 15 Lectures

## Page 12

Course

code Title Credits

UASTA

503 APPLIED STATISTICS - I 3 credits

(45 lectures)

( i. Increasing whole life assurance

ii. Increasing temporary assurance

iii. Present values of variable and increasing life annuities (immediate and due)

– Not to be done)

References:

1. Neil A.: Life Contingencies, First edition, Heineman educational books, London

2. Dixit S.P., Modi C.S., Joshi R.V.: Mathematical Basis of Life Assurance:, First edition, Insurance

Institute of India

3. Gupta S.C., Kapoor V.K.: Fundamental of Applied Statistics, Fourth edition, Sultan Chand and

Sons, India

4. R for Statistics by Julie josse, Maela Kloareg, CRC Press, Taylor and Francis group

5. Statistics using R by Sharad D Gore, Sudha G Purohit, Shailaja R Deshmukh, Norosa

Publishing house.

## Page 13

Semester VI

Course

code Title Credits

UASTA

601 PROBABILITY and SAMPLING DISTRIBUTIONS 3 credits

(45 lectures)

Unit 1 Univariate Random variables(Continuous):

Moment Generating function, Cumulant Generating function - Their important

properties. (without proof)

Relationship between m oments and cumulants and their uses.

Transformation of random variable(only statement and application)

Bivariate Probability distribution(Continuous):

Joint probability density function. Their properties. Marginal and Conditional

distributions. Independen ce of Random variables. Conditional Expectation and

Variance. Coefficient of Correlation.

Transformation of Random variables and Jacobian of transformation with

illustrations.

Standard Continuous Probability distributions:

Rectangular, Exponential:

The f ollowing aspects of the above distributions(wherever applicable) to be

discussed:

Mean, Median, Mode & Standard deviation. Moment Generating Function,

Additive property, Cumulant Generating Function. Skewness and Kurtosis

(without proof).

***** 15 Lect ures

Unit 2 Standard Continuous Probability distributions:

Triangular, Gamma (with Single & Double parameter), Beta (Type I & TypeII).

The following aspects of the above distributions(wherever applicable) to be

discussed:

Mean, Median, Mode & Standard de viation. Moment Generating Function,

Additive property, Cumulant Generating Function. Skewness and Kurtosis

(without proof). Interrelation between the distributions.

Normal Distribution:

Mean, Median, Mode, Standard deviation, Moment Generating function , Cumulant

Generating function, Moments &Cumulants (up to fourth order). Recurrence

relation for central moments, skewness& kurtosis, Mean absolute deviation.

Distribution of linear function of independent Normal variables. Fitting of Normal

Distribution.

Central Limit theorem for i.i.d. random variables.

Log Normal Distribution: Derivation of mean & variance.

15 Lectures

## Page 14

Course

code Title Credits

UASTA

601 PROBABILITY and SAMPLING DISTRIBUTIONS 3 credits

(45 lectures)

Unit 3 Exact sampling distributions:

Chi-square distribution:

Concept of degrees of freedom. Mean, Median, Mode & Standard devia tion.

Moment generating function, Cumulant generating function. Additive property,

Distribution of the sum of squares of independent Standard Normal variables.

Sampling distributions of sample mean and sample variance and their

Independence for a sam ple drawn from Normal distribution (without proof).

Applications of Chi -Square:

Test of significance for specified value of variance of a Normal population. Test for

goodness of fit & Test for independence of attributes (derivation of test statistics is

not expected), Yates’ correction.

t- distribution:

Mean, Median, Mode & Standard deviation. Distribution of ratio of a Standard

Normal variable to the square root of an independent Chi -square divided by its

degrees of freedom. Asymptotic properties. Studen t’s t.

Applications of t: Confidence interval for: Mean of Normal population,

difference between means of two independent Normal populations having the

same variance.

Test of significance of: mean of a Normal population,

difference in means of two Normal populations

(based on: (i) independent samples with equal variances. (ii) dependent samples).

F-distribution: Mean, Mode & Standard deviation. Distribution of :

Reciprocal of an F variate, Ratio of two independent Chi -squares divided by

their respective degrees of freedom.

Interrelationship of F with t -distribution, Chi -square distribution & Normal

distribution.

Applications of F: Test for equality of variances of two independent Normal

populations.

15 Lectures

REFERENCES:

1. Mood A. M., Graybill F.A., Boyes D. C.: Introduction to the theory of statistics, Third Edition;

McGraw -Hill Book Company.

2. Hogg R.V., Craig A.T.: Introduction to Mathematical Statistics, Fourth Edition; Collier

McMillan Publishers.

3. Hogg R.V., Tannis, E. A.: Probability and Statistic al Inference, Third Edition; Collier McMillan

Publishers.

4. Miller I., Miller M.: .John E. Freund’s Mathematical Statistics; Sixth Edition; Pearson

Education Inc.

5. Hoel P .G.; Fourth Edition Introduction to Mathematical Statistics; John Wiley & Sons Inc.

## Page 15

6. Gupt aS.C., Kapoor V.K.: Fundamentals of Mathematical Statistics; Eighth Edition; Sultan

Chand & Sons.

7. KapurJ.N., Saxena H.C.: Mathematical Statistics, Fifteenth Edition; S. Chand & Company Ltd.

8. MedhiJ. : Statistical Methods: An Introductory Text; Second editio n; Wiley Eastern Ltd.

9. GoonA.M., GuptaM.K., DasGupta B. :An Outline of Statistical Theory Vol. 1; Third Edition;

The World Press Pvt. Ltd.

## Page 16

Semester VI

Course

code Title Credits

UASTA

602 Analysis of Variance , Designs of Experiments and R software 3 credits

(45 lectures)

Unit 1 Analysis of Variance:

Introduction, Uses, Cochran’s Theorem (Statement only).

One way classification with equal & unequal observations per class, Two way

Classification with one observation per cell.

Mathematical Model, Ass umptions, Expectation of various sums of squares,

F- test, Analysis of variance table.

Least square estimators of the parameters, Variance of the estimators, Estimation

of treatment contrasts, Standard Error and Confidence limits for elementary

treatmen t contrasts.

Concept of ANOCOVA. 15 lectures

Unit 2 Design of Experiments:

Concepts of Experiments, Experimental unit, Treatment, Yield, Block, Replicate,

Experimental Error, Precision.

Principles of Design of Experiments: Replication, Randomization & L ocal

Control.

Efficiency of design D1 with respect to design D2.

Choice of size, shape of plots & blocks in agricultural & non -agricultural

experiments.

Completely Randomized Design (CRD) , Randomized Block Design (RBD)

and Latin Square Design (LSD):

Mathematical Model, Assumptions, Expectation of various sums of squares,

F-test, Analysis of variance table.

Least square estimators of the parameters, Variance of the estimators, Estimation

of treatment contrasts, Standard error and Confidence limits fo r elementary

treatment contrasts. 15 lectures

Unit 3 Part 1 (5L)

Missing plot and Efficiency of all three designs.

Efficiency of RBD relative to a CRD.

Efficiency of LSD relative to RBD, CRD.

Missing plot technique for one missing observation in case of RBD & LSD

Part 2(10L)

Probability using R -software

Finding and plotting of probability and cumulative probability for standard continuous

distributions such as exponential, normal, Chi - square , t, F.

15 lectures

## Page 17

Course

code Title Credits

UASTA

602 Analysis of Variance , Designs of Experiments and R software 3 credits

(45 lectures)

Testing of hypothesis using normal distribution:

Testing for

Single population mean

Two population means

single population proportion

Two population proportions

ANOVA and Designs of Experiments using R software:

Anova for one way and two way classification and LSD

REFERENCES

1. Cochra n W.G. and Cox G.M.: Experimental Designs; Second Edition;John Wiley and Sons.

2. KempthorneOscar :The Design and Analysis of Experiments, John Wiley and Sons.

3. Montgomery Douglas C.:Design and Analysis of Experiments; 6thEdition;John Wiley & Sons.

4. Das M.N.and Giri N.C.: Design and Analysis of Experiments, 2nd Edition; New Age

International (P) Limited;1986.

5. Federer Walter T.:Experimental Design, Theory and Application; Oxford & IBH Publishing Co.

Pvt. Ltd.

6. Gupta S.C.and Kapoor V.K.: Fundamentals of Applied Sta tistics; 3rd Edition; Sultan Chand and

Sons (2001).

7. Winer B.J.:Statistical Principles in Experimental Design, McGraw Hill Book Company

========

## Page 18

Semester VI

Course

code Title Credits

UASTA

603 APPLIED STATISTICS - II 3 credits

(45 lectures)

Unit 1 Simu lation :

Scope of simulation applications.

Types of simulation, Monte Carlo Techniqueof Simulation.

Elements of discrete event simulation.

Generation of random numbers. Sampling from probability distribution. Inverse

Method.

Generation of random obser vations from -

(i) Uniform distribution, (ii) Exponential distribution,

(iii) Gamma Distribution, (iv) Normal distribution.

Concepts of Inventory problems.

Simulation technique applied to inventory and queuing models. 15 Lectures

Unit 2 Linear Regressi on Model:

Multiple Linear Regression Model with two independent variables.

Assumptions of the model, Derivation of ordinary least square (OLS) estimators of

the regression coefficients, Properties of least square estimators (without proof)

Concept of R2 and adjusted R2.

Procedure of testing of

(i) Overall significance of the model,

(ii) significance of individual coefficients,

(iii) significance of contribution of additional independent variable to the model,

Confidence Intervals for the regression co efficients.

15 Lectures

Unit 3 Part 1(5L):

Concepts of Autocorrelation, Heteroscedasticity, Multicollinearity.

Concept of logistic regression.

Part II(10L):

Testing of hypothesis using Chi, t, and F distributions.

Correlation and Regression analysis using R software:

Scatter diagram, Karl Pearson’s correlation coefficient between two variables,

Multiple Correlation. Linear regression for one explanatory variable, Multiple

regression analysis

15 Lectures

REFERENCES

1. GujrathiDamodar, Sange tha:Basic Econometrics: , Fourth edition, McGraw -Hill Companies

2. Kantiswaroop and Gupta Manmohan:Operations Research, 4th Edition; S Chand & Sons.

3. BrosonRichard :Schaum Series book in O.R., 2nd edition Tata Mcgraw Hill Publishing Company

Ltd.

## Page 19

4. Sasien iMaurice, Yaspan Arthur and Friedman Lawrence: Operations Research: Methods and

Problems,(1959), John Wiley & Sons.

5. Sharma J K.:Mathematical Models in Operations Research, (1989), Tata McGraw Hill Publishing

Company Ltd.

6. Sharma S.D.: Operations Research, 1 1th edition, KedarNath Ram Nath& Company.

7. Taha H. A.: Operations Research:., 6th edition, Prentice Hall of India.

8. Sharma J.K.: Quantitative Techniques For Managerial Decisions, (2001), MacMillan India Ltd.

Assessment of Practical Core Courses Per Semes ter per course

1. Semester work, Documentation, Journal + Viva ------------ 10 Marks.

2. Semester End Practical Examination ----------- 40 Marks

Semester End Examination

Theory: At the end of the semester, Theory examination of three hours duration and 100

marks based on the three units shall be held for each course.

Pattern of Theory question paper at the end of the semester for each course :

There shall be Five Questions of twenty marks each.

Question 1 based on all Three units. Ten sub -questions of two marks each.

Question 2 based on Unit I (Attempt any TWO out of THREE)

Question 3 based on Unit II (Attempt any TWO out of THREE)

Question 4 based on Unit III (Attempt any TWO out of THREE)

Question 5 based on all Three Units combined. (Attempt any TWO out of THREE)

Practicals : At the end of the semester, Practical examination of 2 hours duration and 40

marks shall be held for each course .

Marks for journal and viva in each paper should be given out of 10.

Pattern of Practical question paper at the end of the semester for each course :

There shall be Four Questions of ten marks each. Students should attempt all questions.

Question 1 based on Unit I, Question 2 based on Unit II, Question 3 based on Unit III,

Question 4 based on all Three Units combined.

Students should attempt any two sub questions out of the three in each Question.

## Page 20

Workload:

Theory : 3 lectures per week per course.

Practicals: 3 lecture periods per course per week per batch. All three lecture periods of

the practicals shall be conducted in succession together on a single day. Practicals for R

programming should be conducted on Computer.

*****************