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.
*****************