Skip to the content

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.

This page reproduces the prescribed outline for both papers, so that the teaching pages can be checked against it line by line. It is the syllabus, not a summary of it. Nothing about marks, duration or examination pattern appears on this site.

Course Objectives

  1. To understand the nature of various probability distributions and their real time applications.
  2. To derive the distributions and their basic properties, problems on distributions.
  3. Derive the distributions for various order statistics.
  4. To understand the concepts of random variables, sigma-fields generated by random variables, probability distributions and independence of random variables related to measurable functions.

Course Outcomes

  1. Able to solve and derive the common and special properties of any standard univariate probability distributions and sampling distributions.
  2. Able to find the distribution of a function of random variables and order statistics.
  3. Able to derive the sampling distribution to the given statistic.
  4. Able to identify the real time applications of each of the probability distributions and their fitting.

Stated Pre-requisite

ASSUMED BEFORE THIS PAPER

Basic univariate probability distributions: discrete uniform, Bernoulli, binomial, Poisson, negative binomial, geometric, hyper-geometric, continuous uniform, normal, exponential, gamma (one and two parameters), beta of the first and second kinds, and Cauchy. Univariate and bivariate random variable transformations.

Covered by Theoretical Discrete Distributions, Theoretical Continuous Distributions and Theory of Probability, Unit 3.

Unit I

AS PRESCRIBED

Brief review on basic probability distributions. Definitions and derivations of properties related to Lognormal, Weibull, Pareto, Laplace and Cauchy distributions and their applications and related problems.

→ Unit 1 notes

Unit II

AS PRESCRIBED

Functions of random variables and their distributions using Jacobian of transformations and problems on transformations; truncated distributions (binomial, Poisson, normal distributions). Mixture distributions and examples. Exponential family of distributions and power series family of distributions and their means and variances (binomial, Poisson, geometric). Compound distributions of binomial–Poisson, Poisson–gamma, their means and variances.

→ Unit 2 notes

Unit III

AS PRESCRIBED

Derivations of density functions of sampling distributions of central and non-central t, F and chi-square and their properties (for non-central, only statements), distribution of sample mean and variance, independence of the sample mean and the sample variance.

→ Unit 3 notes

Unit IV

AS PRESCRIBED

Distributions of quadratic forms under normality and its applications. Order statistics: joint and marginal distributions of order statistics. Distributions of sample range, problems on computing the distribution of order statistics. Applications of order statistics.

→ Unit 4 notes

Practical Paper STS-107 — Objectives and Outcomes

  1. Knowing the manual procedures and also their implementation using R.
  2. Generation of random samples from any distribution.
  3. Identifying an appropriate probability distribution to the given data.
  4. Fitting and testing the probability distribution.
  5. Drawing the probability distribution curves and stating the nature of the distributional curve and properties for the given data sets.

Practical Paper STS-107 — List of Practicals

  1. Generation of random samples from Uniform distribution.
  2. Generation of random samples from the Binomial, Poisson, Geometric, Negative Binomial distributions.
  3. Generation of random samples from the Normal, Exponential, Gamma, Beta, Cauchy distributions.
  4. Fitting an appropriate discrete distribution to the given data sets.
  5. Fitting an appropriate continuous distribution to the given data sets (Uniform, Normal, Exponential).
  6. Testing its Goodness of fit of Cauchy distribution to the given data set.
  7. Fitting of Gamma distribution with two parameters to the given data set.
  8. Fitting of Lognormal Distribution with two parameters to the given data set.
  9. Fitting of Weibull Distribution with two parameters to the given data set.
  10. Fitting of Pareto distribution with two parameters to the given data set.

The semester-end practical examination has two sections: Section A conventional and Section B using R. → Practical notes, both sections

References

  1. Bhuyan, K. C. (2010): Probability Distribution Theory and Statistical Inference, New Central Book Agency (P) Ltd.
  2. Parimal Mukhopadhyay (2018): Mathematical Statistics, Books & Allied Ltd.
  3. Johnson and Kotz: Distributions in Statistics, Vol. I (1970), II (1972) and III (1990).
  4. Johnson, R. A. and Wichern (2015): Applied Multivariate Analysis, 6th edition.
  5. Kshirsagar, A. M. (1972): Multivariate Analysis, Marcel Dekker.
← Distribution Theory Unit 1 →