Distribution Theory

Welcome

This is the complete study package for Distribution Theory (STS-104) together with its paired practical STS-107, Distribution Theory (Conventional and using R). The course does three jobs: it adds the heavy-tailed and lifetime distributions the Foundation course does not reach, it derives the sampling distributions that every test in inferential statistics quotes without proof, and it develops order statistics.

What is assumed, and where to revise it. The syllabus names a long pre-requisite, and all of it is on this site already:

Theoretical Discrete Distributions — discrete uniform, Bernoulli, binomial, Poisson, negative binomial, geometric, hypergeometric
Theoretical Continuous Distributions — continuous uniform, exponential, gamma, both beta kinds, normal, Cauchy, and a first look at the sampling distributions
Theory of Probability, Unit 3 — univariate and bivariate random variable transformations

Unit 1 opens with a table linking each of these, and nothing in it is derived twice.

Course Objectives

To understand the nature of various probability distributions and their real-time applications.
To derive the distributions and their basic properties, and to solve problems on them.
To derive the distributions of order statistics.
To understand random variables, the sigma-fields they generate, probability distributions, and independence of random variables as measurable functions.

Course Outcomes

Able to solve and derive the common and special properties of any standard univariate probability distribution and sampling distribution.
Able to find the distribution of a function of random variables and of order statistics.
Able to derive the sampling distribution of a given statistic.
Able to identify the real-time applications of each probability distribution, and to fit it.

Units in this Course

UNIT 1

Lognormal, Weibull, Pareto, Laplace and Cauchy

A review table of the assumed distributions, then the five new ones: the lognormal and its three centres, the Weibull with its hazard rate and the bathtub curve, the Pareto and exactly which of its moments exist, the Laplace, and the Cauchy with a proof that its mean does not.

UNIT 2

Transformations, Truncation, Mixtures and Families

The Jacobian rule; truncated binomial, Poisson and normal; finite mixtures and why they need not be unimodal; the exponential family and its log-normaliser; the power series family; and compound distributions, including the Poisson–Gamma negative binomial.

UNIT 3

Sampling Distributions: Chi-Square, t and F

Deriving each density from first principles; the distribution of the sample mean and variance and their independence by the Helmert transformation; the relationships between t, F and chi-square; and the non-central forms that power calculations rest on.

UNIT 4

Quadratic Forms and Order Statistics

The idempotency criterion and the Fisher–Cochran theorem that underwrites the ANOVA table; joint and marginal distributions of order statistics; the sample range; and the applications from probability plotting to extreme value theory.

PRACTICAL

STS-107 — Conventional and using R

All ten prescribed practicals, each worked by hand with full arithmetic and then in R: random number generation, inverse transform, Box–Muller, chi-square goodness of fit, and two-parameter gamma, lognormal, Weibull and Pareto fits on one shared dataset.

REFERENCE

Official Syllabus

The prescribed unit-wise outline for STS-104 and the practical list for STS-107, as printed, with objectives, outcomes, the stated pre-requisite and the reading list.

How This Course Connects to the Others

What is built here	Where it is used
Exponential family and sufficiency	The Lehmann–Scheffé theorem and UMVU estimation in Estimation Theory (STS-201)
Chi-square, t and F densities	Every test in Inferential Statistics; the F ratio in Design and Analysis of Experiments
Independence of the sample mean and variance	The t test — without it there is no t distribution
Quadratic forms and Fisher–Cochran	The whole analysis of variance table, and Multivariate Analysis (STS-202)
Order statistics	Non-parametric tests; probability plotting; the range chart in Statistical Quality Control
Non-central distributions	Power and sample-size calculations wherever a test is designed

K. C. Bhuyan (2010) — Probability Distribution Theory and Statistical Inference, New Central Book Agency (P) Ltd.
Parimal Mukhopadhyay (2018) — Mathematical Statistics, Books & Allied Ltd.
N. L. Johnson and S. Kotz — Distributions in Statistics, Vol. I (1970), Vol. II (1972) and Vol. III (1990).
R. A. Johnson and D. W. Wichern (2015) — Applied Multivariate Analysis, 6th edition.
A. M. Kshirsagar (1972) — Multivariate Analysis, Marcel Dekker.

← All Statistics courses Start with Unit 1 →

Next course in learning order: Inferential Statistics Statistical inference