SHIVAJI UNIVERSITY, KOLHAPUR
CBCS SYLLABUS WITH EFFECT FROM JUNE, 2019
B. Sc. Part-II: SEMESTER III
SUBJECT: STATISTCS -V
DSC - 7C: Probability Distributions–I
Theory: 36 Hrs. Marks-50 (Credit 2)
OBJECTIVES:
The main objective of this course is to acquaint students with the basic concepts of discrete
distributions defined on countably infinite sample space, continuous univariate and bivariate
distributions, transformation of univariate continuous random variable.
By the end of the course students are expected to be able to:
a) understand concept of discrete and continuous probability distributions with real life
situations.
b) distinguish between discrete and continuous distributions.
c) find the various measures of random variable and probabilities using its probability
distribution.
d) know the relations among the different distributions.
e) understand the concept of transformation of univariate and bivariate continuous random
variable.
CONTENTS:
Unit-1: (18 hrs.)
1.1 Discrete Distributions: Poisson, Geometric and Negative Binomial Distribution:
Definition of random variable (defined on countably infinite sample space). Poisson
Distribution: Definition of Poisson with parameter λ. Mean, variance, probability
generating function (p. g. f.). Recurrence relation for successive Probabilities,
Additive property of Poisson distribution. Poisson distribution as a limiting case of
Binomial distribution, examples. Geometric Distribution: Definition of Geometric
with parameter p. Mean, Variance, distribution function, p. g. f., Lack of memory
property, examples. Negative Binomial Distribution: Definition of Negative Binomial
with parameters (k, p), Geometric distribution is a particular case of Negative
Binomial distribution, Mean, Variance, p. g. f., Recurrence relation for successive
probabilities, examples.
1.2: Continuous Univariate Distributions: Definition of the continuous sample space
with illustrations, Definition of continuous random variable (r. v.), probability density
function (p. d. f.), cumulative distribution function (c. d. f.) and its properties.
Expectation of r. v., expectation of function of r. v., mean, median, mode, quartiles,
variance, harmonic mean, raw and central moments, skewness and kurtosis,
examples. Moments generating function (m. g. f.): definition and properties
(i) Standardization property MX (0) = 1, (ii) Effect of change of origin and scale,
(iii) Uniqueness property of m. g. f., if exists, (statement only). Generation of raw
and central moments. Cumulant generating function (c. g. f.): definition, relations
between cumulants and central moments (up to order four). Examples.
Unit-2: (18 hrs.)
2.1 Continuous Bivariate Distributions: Definition of bivariate continuous r. v. (X, Y),
Joint p. d. f., c. d. f with properties, marginal and conditional distribution,
independence of r. vs., evaluation of probabilities of various regions bounded by
straight lines. Expectation of function of r. vs., means, variances, covariance,
correlation coefficient, conditional expectation, regression as conditional expectation
if it is linear function of other variable and conditional variance, proof of
i) E (X ± Y) = E(X) ± E(Y), ii) E [E(X/Y)] = E(X). If X and Y are independent r. vs.
then (i) E (XY) = E(X) E(Y), (ii) M x + y (t) = Mx (t) × My (t). Examples.
2.2: Transformations of continuous random variable: Transformation of univariate
continuous r. v.: Distribution of Y = g(X), where g is monotonic or non-monotonic
functions using (i) Jacobian of transformation, (ii) Distribution function and (iii) m.g.f.
methods. Transformation of continuous bivariate r. vs.: Distribution of bivariate r. vs.
using Jacobin of transformation. Examples and problems.
References and Recommended Readings
1. Parimal Mukhopadhyaya: An Introduction to the Theory of Probability. World
Scientific Publishing.
2. Hogg R. V. and Criag A.T.: Introduction to Mathematical Stat ist ics (Third
edition), Macmillan Publishing, New York.
3. Gupta S. C. & Kapoor V.K.: Fundamentals of Mathematical Statistics. Sultan Chand
& sons, New Delhi.
4. Goon, A.M., Gupta M.K. and Dasgupta B: Fundamentals of Statistics Vol. I and
Vol. II World Press, Calcutta.
5. Dr. Kore B. G. and Dr. Dixit P. G.: “Probability Distributions-I”, Nirali Prakashan,
Pune.
6. Mood A.M., Graybill F.A.: Introduction to theory of Statistics. (Chapter II, IV, V,
VII) and Boes D.C. Tata, McGraw Hill, New Delhi. (Third Edition)
7. Walpole R.E. & Mayer R.H.: Probability & Statistics. (Chapter 4, 5, 6, 8, 10)
MacMillan Publishing Co. Inc, New York.
************************
SHIVAJI UNIVERSITY, KOLHAPUR
CBCS SYLLABUS WITH EFFECT FROM JUNE, 2019
B. Sc. Part-II: SEMESTER III
SUBJECT: STATISTCS -VI
DSC - 8C: Statistical Methods-I
Theory: 36 Hrs. Marks-50 (Credit 2)
OBJECTIVES:
The main objective of this course is to acquaint students with the basic concepts of Multiple
Linear Regression, Multiple and Partial Correlation, Sampling Theory and Demography.
By the end of the course students are expected to be able to be:
a) understand the concept of Multiple Linear Regression.
b) understand the concept of Multiple Correlations and Partial Correlation.
c) know the concept of sampling theory.
d) understand the need of vital statistics and concept of mortality and fertility.
CONTENTS:
Unit 1: (18 hrs.)
1.1 Multiple Linear Regression (for trivariate data only): Concept of multiple linear
regression, Plane of regression, Yule’s notation, correlation matrix. Fitting of regression
plane by method of least squares, definition of partial regression coefficients and their
interpretation. Residual: definition, order, properties, derivation of mean and variance,
Covariance between residuals.
1.2 Multiple and Partial Correlation (for trivariate data only): Concept of multiple
correlations. Definition of multiple correlation coefficient Ri.jk, derivation of formula
for multiple correlation coefficient. Properties of multiple correlation coefficient;
i) 0 Ri.jk 1, (ii) Ri.jk > |rij|, (iii) Ri.jk > |rik| i = j =k = 1, 2, 3. i j, i k. Interpretation
of Ri.jk =1, Ri.jk = 0, coefficient of multiple determination R2
1.23. Concept of partial
correlation. Definition of partial correlation coefficient rij.k, derivation of formula for
rij.k. Properties of partial correlation coefficient; (i) -1 rij.k 1, (ii) bij.k × bji.k = r2
ij.k.
Examples and problems.
Unit-2: (18 hrs.)
2.1 Sampling Theory: Concept of distinguishable elementary units, sampling units,
sampling frame, random sampling and non-random sampling. Advantages of
sampling method over census method, objectives of a sample survey. Designing a
questionnaire, Characteristics of a good questionnaire, Concept of sampling and nonsampling
errors. Handling of non- response cases.
Simple random sampling: Simple random sampling from finite population of size N
with replacement (SRSWR) and without replacement (SRSWOR): Definitions,
population mean and population total as parameters. Following results with proof.
i. In SRSWOR, the probability of a specified unit being selected in sample at any
given draw is equal to 1/N.
ii. In SRSWOR, the probability of a specific unit included in the sample is n/N.
iii. In SRSWOR, the probability of drawing a sample of size ‘n’ from a population
of size N units is
1
(
𝑁
𝑛
)
.
iv. In SRSWR, the probability of a specific unit included in the sample is 1 −
(1 −
1
𝑁
)
𝑛
.
v. In SRSWR, the probability of drawing a sample of size ‘n’ from a population
of size N units is
1
𝑁𝑛.
2.2 Demography: Introduction and need of vital statistics. Mortality rates: Crude death
rate (CDR), Specific Death Rate (SDR), Standardized Death Rate (STDR). Fertility
Rates: Crude Birth Rate (CBR), Age Specific Fertility Rate (ASFR), General Fertility
Rate (GFR), Total Fertility Rate (TFR). Reproduction Rate: Gross Reproduction rate
(GRR), Net Reproduction Rate (NRR).
References and Recommended Readings
1. Cochran, W.G: Sampling Techniques, Wiley Eastern Ltd., New Delhi.
2. Des Raj: Sampling Theory.
3. Gupta S. C. and Kapoor V. K., “Fundamentals of Applied Statistics”, Sultan and
Chand, (2010).
4. Dr. Kore B. G. and Dr. Dixit P. G.: “Statistical Methods-I”, Nirali Prakashan, Pune.
5. Mukhopadhay, Parimal: Theory and Methods of Survey Sampling, Prentice Hall.
6 . Srivastav D. S: A Text book of Demography.
7. Sukhatme, P.V. and Sukhatme, B.V.: Sampling Theory of Surveys with Applications,
Indian Society of Agricultural Statistics, New Delhi.
************************
SHIVAJI UNIVERSITY, KOLHAPUR
CBCS SYLLABUS WITH EFFECT FROM JUNE, 2019
B. Sc. Part-II: SEMESTER IV
SUBJECT: STATISTCS -VII
DSC-7D: Probability Distributions-II
Theory: 36 Hrs. Marks-50 (Credit 2)
OBJECTIVES:
The main objective of this course is to acquaint students with the Uniform, Exponential,
Gamma and Beta, Normal distributions and Exact Sampling distributions.
By the end of the course students are expected to be able to:
a) know some standard continuous probability distributions with real life situations.
b) distinguish between various continuous distributions.
c) find the various measures of continuous random variable and probabilities using its
probability distribution.
d) understand the relations among the different distributions.
e) understand the Chi-Square, t and F distributions with their applications and inter
relations.
CONTENTS:
Unit-1: (18 hrs.)
1.1 Uniform and Exponential Distribution: Uniform distribution: Definition of Uniform
distribution over (a, b), c.d.f., m.g.f., mean, variance, moments. Distribution of (i)
(X-a) / (b-a), ii) (b-X) / (b-a), (iii) Y = F(x) where F(x) is c.d.f. of any continuous r.v.
Exponential distribution: p.d.f. (one parameter),
f ( x ) = θ e
− θ x
, x ≥ 0 , θ > 0
= 0 , o .w
c.d.f., m.g.f., c.g.f., mean, variance, C.V., moments, Cumulants, median, quartiles, lack
of memory property, and distribution of – (1 / θ) logX where X~U (0, 1).
1.2: Gamma and Beta Distributions: Gamma distribution: Gamma distribution with
scale parameter θ and shape parameter n, special case θ = 1, n =1, m.g.f., c.g.f., mean,
mode, variance, moments, cumulants, β1, β2, γ1 and γ2 coefficients, additive property:
distribution of sum of i.i.d. exponential variates. Beta distribution of first kind: Beta
distribution of first kind with parameters m & n. mean, mode, variance, symmetric
when m = n, Uniform distribution as a particular case when m = n = 1, distribution of
(1-X). Beta distribution of second kind: Beta distribution of second kind with
parameters m & n. mean, mode, variance, relation between beta distribution of first
kind and second kind, distribution of X+Y, X/Y and X/(X+Y) where X and Y are
independent gamma variate.
Unit-2: (18 hrs.)
2.1 Normal distribution: Normal distribution with parameters μ & σ2, Definition of
standard normal distribution, properties of normal curve, m.g.f., c.g.f., mean, variance,
median, mode, mean deviation, moments, cumulants, measures of skewness &
kurtosis, distribution of linear combination of variates. Distribution of X2 if
X ~ N (0, 1).
2.2: Exact Sampling Distributions: Chi-Square distribution: Definition of chi square,
derivation of p.d.f. of chi square distribution with n degrees of freedom using m.g.f.,
c.g.f., mean, variance, moments, cumulants, mode, skewness and kurtosis, additive
property.
Student’s t- distribution: Definition of student’s t variate. Derivation of p.d.f., mean,
mode, variance, moments, β1, β2, γ1 and γ2 coefficients.
Snedecor’s F distribution: Definition of F variate, derivation of p.d.f., mean, variance
and mode. Distribution of 1/F. Inter relation between t, F and χ2 (Without Proof).
References and Recommended Readings
1. Trivedi R. S.: Probability and Statistics with Reliability and Computer Science
Application, Prentice – Hall of India Pvt. Ltd., New Delhi.
2. Parimal Mukhopadhyaya: An Introduction to the Theory of Probability. World
Scientific Publishing.
3. Hogg R.V. and Criag A.T.: Introduction to Mathematical Stat ist ics (Third
edition), Macmillan Publishing, New York.
4. Goon, A.M., Gupta M.K. and Dasgupta B: Fundamentals of Statistics Vol. I and
Vol. II World Press, Calcutta.
5. Gupta S. C. & Kapoor V.K.: Fundamentals of Mathematical Statistics. Sultan Chand
& sons, New Delhi.
6. Gupta S. C. & Kapoor V.K.: Applied Statistics. Sultan Chand & sons, New Delhi.
7. Dr. Kore B. G., Dr. Dixit P. G. and Mr. P. S. Kapre: “Probability Distributions-II”,
5
Nirali Prakashan, Pune.
8. Mood A.M., Graybill F.A.: Introduction to theory of Statistics. (Chapter II, IV,
V, VII) and Boes D.C. Tata, McGraw Hill, New Delhi. (Third Edition)
9. Walpole R.E. & Mayer R.H.: Probability & Statistics. (Chapter 4, 5, 6, 8, 10)
MacMillan Publishing Co. Inc, New York
************************
SHIVAJI UNIVERSITY, KOLHAPUR
CBCS SYLLABUS WITH EFFECT FROM JUNE, 2019
B. Sc. Part-II: SEMESTER IV
SUBJECT: STATISTICS - VIII
DSC-8D: Statistical Methods-II
Theory: 36 Hrs. Marks-50 (Credit 2)
OBJECTIVES:
The main objective of this course is to acquaint students with the concepts of Time Series,
Statistical Quality Control, Testing of Hypothesis.
By the end of the course students are expected to be able to:
a) know the concept and use of time series.
b) understand the meaning, purpose and use of Statistical Quality Control, construction and
working of control charts for variables and attributes
c) apply the small sample tests and large sample tests in various situations.
CONTENTS:
Unit-1: (18 hrs.)
1.1 Time Series: Meaning and need of time series analysis, components of time series; (i)
Secular trend (ii) Seasonal Variation (iii) Cyclical Variation (iv) Irregular Variation,
Additive and Multiplicative model, utility of time series. Measurement of trend: (i)
Moving averages method (ii) Progressive average method (iii) Least square method.
(iv) Measurement of seasonal indices by simple average method.
1.2 Statistical Quality Control: Meaning and purpose of S.Q.C., Process control, Product
control, chance causes, assignable causes, Shewhart’s control chart- construction &
working, lack of control situation. Control charts for variables - control chart for mean,
control chart for range, construction and working of mean & range charts for unknown
standards, revised control limits. Control charts for Attributes – Defects, defectives,
fraction defective, control chart for fraction defective (p-chart) for fixed sample size
and unknown standards, construction and working of chart. Control charts for number
of defects (C-chart), for unknown standards, construction and working of C-chart.
Unit 2: (18 hrs.)
2.1 Testing of Hypothesis - I: Notion of Population, Sample, Parameter, Statistic,
Sampling distribution of Statistic, hypothesis, Simple and composite hypothesis, Null
and alternative hypothesis, type I and type II errors, Critical region, level of
significance. one and two tailed test, power of test. Large Sample Tests: General
procedure of testing of hypothesis. a) Tests for means: i) testing of population mean;
H0: μ = μ0, ii) testing equality of population means; H0: μ1 = μ2 b) Tests for
Proportion: i) testing of population Proportion; H0: P = P0 ii) testing equality of
population Proportion; H0:P1 = P2 c) test for population correlation: i) H0: ρ = ρ0
ii) H0: ρ1 = ρ2 (by Z-transformation)
2.2: Testing of Hypothesis - II (Small sample tests): Definition of Fisher’s t- variate,
t - test: a) test for means: i) H0: μ = μ0, ii) H0: μ1 = μ2, (σ1
2 = σ2
2), iii) Paired t- test,
χ2 - test: i) test for population variance H0: σ2 = σ0
2 (Mean known and unknown), ii)
test for goodness of fit, iii) test for independence of attributes; a) m x n contingency
table, b) 2 x 2 contingency table, test statistic with proof. Yate’s correction for
continuity.
F – test: test for equality of two population variances H0: σ1
2 = σ2
2.
References and Recommended Readings
1. Barlow R. E. and Proschan Frank, “Statistical Theory of Reliability and Life
Testing”, Holt Rinebart and Winston Inc., New Yark.
2. Chatfield C. (2004), “The Analysis of Time Series –An Introduction”, Chapman &
Hall.
3. Gupta S. C. & Kapoor V. K., “Fundamentals of Applied Statistics”, Sultan Chand &
Sons, New Delhi.
4. Kendall M.G. (1978), “Time Series”, Charles Griffin.
5. Dr. Kore B. G. and Dr. Dixit P. G.: “Statistical Methods-II”, Nirali Prakashan, Pune.
6. Montgomery D. C. (2009). “Introduction to quality Control”, Jon Wiley and sons.
7. Sinha S. K., “Reliability and Life Testing”, Second Edition, Wiley Eastern Publishers,
New Delhi.
8. Snedecor G.W. and Cochoran W. G. “Statistical Methods”, Lowa State University
Press.
SHIVAJI UNIVERSITY, KOLHAPUR
CBCS SYLLABUS WITH EFFECT FROM JUNE, 2019
B. Sc. Part-II
SUBJECT: STATISTICS - Practical
Practical: 192 Hrs. Marks -100 (Credit 4)
OBJECTIVES:
By the end of the course students are expected to be able to:
a) compute probabilities of standard probability distributions.
b) compute the expected frequency and test the goodness of fit.
c) understand how to obtain random sample from standard probability distribution and
7
sketch of the p. m. f. / p. d. f. for given parameters.
d) fit plane of Multiple regression and compute Multiple and Partial correlation
coefficients.
e) draw random samples by various sampling methods
f) construct various control charts.
g) understand the applications of Poisson, Geometric and Negative Binomial
distributions.
Practical – II (Credit 2)
1. Fitting of Discrete Uniform distribution.
2. Fitting of Binomial distribution.
3. Fitting of Hypergeometric distribution.
4. Fitting of Poisson distribution.
5. Fitting of Geometric distribution.
6. Fitting of Negative Binomial
7. Model sampling from Discrete Uniform distribution.
8. Model sampling from Binomial distribution.
9. Model sampling from Hypergeometric distribution.
10. Model sampling from Poisson distribution.
11. Model sampling from Geometric distribution.
12. Model sampling from Negative Binomial distribution
13. Fitting of Continuous Uniform distribution
14. Fitting of Exponential distribution
15. Fitting of Normal distribution.
16. Model sampling from Continuous Uniform distribution.
17. Model sampling from Exponential distribution.
18. Model sampling from Normal distribution using: (i) Normal table and (ii) Box-Muller
transformation.
19. Fitting of Binomial, Poisson & Negative Binomial distribution using MS-EXCEL.
20. Fitting of Exponential & Normal distribution using MS-EXCEL.
Practical - III (Credit 2)
1. Applications of Poisson distribution.
2. Applications of geometric and negative binomial distributions.
3. Applications of Exponential & Normal distribution
4. Multiple regression.
5. Multiple correlation.
6. Partial correlation.
7. Simple Random Sampling.
8. Demography I (Mortality rates).
9. Demography II (Fertility and Reproduction rates).
10. Time Series. (Trend by moving average & least square methods. Seasonal indices by
simple average method).
11. Construction of R and X̅
charts
12. Construction of P and C charts.
13. Large sample tests for means.
14. Large sample tests for proportions.
15. Tests for population correlation coefficients. (Using Fisher’s Z transformation.)
16. Tests based on Chi square distribution. (Test for population variance, Test for
goodness of fit, Tests for independence.)
8
17. Tests based on t distribution (μ = μ0, μ1 = μ2; paired t test)
18. Tests based on F distribution. (σ1= σ2,)
19. Sketch of discrete distributions: Binomial, Poisson, Geometric and Negative
Binomial distribution for various parameters using MS-EXCEL.
20. Sketch of continuous distributions: Exponential, Gamma and Beta distributions for
various parameters using MS-EXCEL.
Note:
1. For fitting of all distributions, test of goodness of fit is necessary.
2. For model sampling from all distributions, inverse c. d. f. transformation method
has to be used in Practical-II.
3. For experiment no. 1 to 6 in Practical -II, Probabilities has to be calculated by
recurrence relation only.
4. There should be at least FOUR problems in each experiment.
5. Computer printout is to be attached to the journal for the experiment based on
MS-EXCEL.
6. Observation table and/or calculations using statistical formulae should be done by
MS-EXCEL and verify by using library functions for the experiment based on
MS-EXCEL.
7. Student must complete the entire practical to the satisfaction of the teacher concerned.
8. Student must produce the laboratory journal along with the completion certificate
signed by Head of Department, at the time of practical examination.
9. There will be study tour or case study. A report on the same has to be submitted
by every student along with the journal.
Laboratory requirements:
Laboratory should be well equipped with sufficient number of electronic calculators and
computers along with necessary software, printers and UPS.
Nature of Practical Question Paper of B. Sc. Part – II.
a) Each practical paper is of 50 marks, containing four questions each of 20 marks and
students has to solve any two questions. There should be one sub question of 10 marks
based on MS-EXCEL in any one of the four questions.
b) Evaluation of MS-EXCEL based question will be on line and should be demonstrated
by the student to the examiner.
c) 5 marks are reserved for journal and 5 marks are reserved for oral in practical paper-
II examination.
d) 5 marks are reserved for journal and 5 marks are reserved for study tour report / Case
study in practical paper - III examination.
e) Practical examination is of 4 hours duration which includes oral as well as online
demonstration.
f) There should be two subject experts at the time of practical examination.
********************************
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