Source document. This page reproduces the syllabus this course was written to, as published — its semesters, credits and paper numbers are that document’s, not this site’s. The course itself is studied on its own, in any order.

Course Information

Title	Theoretical Discrete Distributions
Theory Credits	3 (3 hrs/week)
Practical Credits	1 (2 hrs/week)

Course Outcomes

After successful completion of the course, students will be able to:

1. Deal with the data by the basic discrete distributions such as Uniform and Binomial.
2. Acquaint with the Poisson distribution and its applications.
3. Learn about the Negative Binomial distribution and its real-life applications.
4. Familiarize with handling data by Geometric and Hypergeometric distributions.

Theory — Five Units

Unit 1: Uniform, Bernoulli & Binomial Distributions

Discrete Uniform — definitions, mean, variance. Bernoulli — definitions, mean, variance, MGF. Binomial — definition, moments, MGF, CF, CGF, PGF, additive property if exists, skewness, kurtosis and problems. First two moments through MGF, recurrence relation for probabilities, limiting case of Binomial to Normal.

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Unit 2: Poisson Distribution

Poisson distribution — definition, moments, MGF, CF, CGF, PGF, additive property if exists, skewness, kurtosis, problems. First two moments through MGF, recurrence relation for probabilities. Poisson as a limiting case of Binomial; limiting case of Poisson to Normal.

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Unit 3: Negative Binomial Distribution

Negative Binomial — definition, moments, MGF, CF, CGF, PGF, additive property if exists, skewness, kurtosis, problems. First two moments through MGF, recurrence relation for probabilities. Limiting case to Normal.

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Unit 4: Geometric Distribution

Geometric — definition, moments, MGF, CF, CGF, PGF, additive property if exists, skewness, kurtosis, problems. First two moments through MGF, lack of memory property, recurrence relation for probabilities.

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Unit 5: Hypergeometric Distribution

Hypergeometric — definition, mean and variance, problems. Recurrence relation for probabilities. Limiting case of Hypergeometric to Binomial.

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Practical — List of Experiments (9)

1. Fitting of Binomial distribution — Direct method.
2. Fitting of Binomial distribution — Recurrence relation method.
3. Fitting of Poisson distribution — Direct method.
4. Fitting of Poisson distribution — Recurrence relation method.
5. Fitting of Negative Binomial distribution — Direct method.
6. Fitting of Negative Binomial distribution — Recurrence relation method.
7. Fitting of Geometric distribution — Direct method.
8. Fitting of Geometric distribution — Recurrence relation method.
9. Fitting of Hypergeometric distribution.

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Text Books

1. S. C. Gupta & V. K. Kapoor — Fundamentals of Mathematical Statistics, Sultan Chand & Sons.
2. K. Rohatgi & Ehsanes Saleh — An Introduction to Probability and Statistics, John Wiley & Sons.

References

1. O. P. Gupta — Mathematical Statistics, Kedarnath Ramnath & Co.
2. P. N. Arora & S. Arora — Quantitative Aptitude Statistics — Vol II, S. Chand & Company Ltd.

Suggested Co-Curricular Activities

1. Training of students by related industrial experts.
2. Assignments including technical assignments, if any.
3. Seminars, Group Discussions, Quiz, Debates etc. on related topics.
4. Preparation of audio and videos on tools of diagrammatic and graphical representations.
5. Collection of material / figures / photos of related topics.
6. Invited lectures and presentations of stalwarts on those topics.
7. Visits / field trips of firms, research organizations etc.

Exam Blueprint (recommended)

Unit	Topic	Approx. Weightage
1	Uniform, Bernoulli & Binomial	22 %
2	Poisson Distribution	22 %
3	Negative Binomial	20 %
4	Geometric Distribution	18 %
5	Hypergeometric Distribution	18 %

Quick Reference — All Five Distributions

Distribution	PMF	Mean	Variance	MGF
Uniform (1..N)	1/N	(N+1)/2	(N²−1)/12	—
Bernoulli(p)	pxq1−x	p	pq	q + pet
Binomial(n,p)	C(n,x) pxqn−x	np	npq	(q+pet)n
Poisson(λ)	e−λλx/x!	λ	λ	eλ(et−1)
NB(r,p)	C(x+r−1,x)prqx	rq/p	rq/p²	pr/(1−qet)r
Geometric(p)	qxp	q/p	q/p²	p/(1−qet)
Hypergeometric(N,M,n)	C(M,x)C(N−M,n−x)/C(N,n)	nM/N	npq(N−n)/(N−1)	complicated