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 Continuous Distributions
Theory Credits	3 (3 hrs/week)
Practical Credits	1 (2 hrs/week)

Course Outcomes

1. Deal with the data by basic continuous distributions like the Uniform.
2. Acquaint with the Exponential distribution and its applications.
3. Learn Gamma and Beta distributions and their real-life applications.
4. Get familiarity with Normal and Standard Normal distributions for research and applied fields.
5. Acquire knowledge of exact sampling distributions: \(t\), \(F\), \(\chi^2\).

Theory — Five Units

Unit 1: Continuous Uniform Distribution

Uniform — definition, moments, MGF, CF, CGF, skewness, kurtosis and distribution function. Mean deviation about mean.

Open Unit 1 →

Unit 2: Exponential Distribution

Exponential — definition, moments, MGF, CF, CGF, skewness, kurtosis, distribution function. Memoryless property.

Open Unit 2 →

Unit 3: Gamma and Beta Distributions

Gamma — definition, moments, MGF, CF, CGF, skewness, kurtosis and additive property; limiting form. Beta of first and second kind — definition, mean, variance and harmonic mean.

Open Unit 3 →

Unit 4: Normal Distribution

Definition, properties, importance, MGF, CF, CGF, additive property, skewness, kurtosis, problems. Mean, median & mode; even/odd moments about mean; linear combination of normal variates; points of inflexion.

Open Unit 4 →

Unit 5: Standard Normal & Sampling Distributions

Standard Normal — definition, MGF, mean & variance, area property, problems. Population, Sample, Parameter, Statistic, Sampling Distribution. Student's t, F, \(\chi^2\) — definitions, properties & applications.

Open Unit 5 →

Practical — List of Experiments (7)

1. Calculation of moments of Uniform distribution.
2. Calculation of skewness and kurtosis of Uniform distribution.
3. Fitting of Exponential distribution.
4. Gamma distribution application-oriented problems.
5. Fitting of Normal distribution — Areas method.
6. Fitting of Normal distribution — Ordinates method.
7. Problems related to Standard Normal distribution.

Open practical course material →

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	Continuous Uniform	15 %
2	Exponential	15 %
3	Gamma & Beta	20 %
4	Normal Distribution	25 %
5	Standard Normal & Sampling	25 %

Quick Reference — All Continuous Distributions

Distribution	PDF Support	Mean	Variance	MGF
Uniform U(a,b)	[a, b]	(a+b)/2	(b−a)²/12	(etb−eta)/[t(b−a)]
Exponential(λ)	x ≥ 0	1/λ	1/λ²	λ/(λ−t)
Gamma(α, λ)	x ≥ 0	α/λ	α/λ²	(λ/(λ−t))α
Beta-I(α, β)	[0, 1]	α/(α+β)	αβ/[(α+β)²(α+β+1)]	complicated
Normal N(μ, σ²)	(−∞, ∞)	μ	σ²	exp(μt + σ²t²/2)
Standard Normal N(0,1)	(−∞, ∞)	0	1	et²/2
χ²n	x > 0	n	2n	(1−2t)−n/2
tn	(−∞, ∞)	0 (n>1)	n/(n−2) (n>2)	does not exist
Fn1,n2	x > 0	n₂/(n₂−2)	complicated	does not exist