Savitribai Phule Pune University – T.Y. B.Sc. Statistics Syllabus & Study Notes (Semester V & VI)

T.Y. B.Sc. Statistics: Syllabus Overview

Introduction

This syllabus outlines the third-year undergraduate curriculum for the Bachelor of Science degree in Statistics at Savitribai Phule Pune University. The program is structured into two semesters (V and VI), each comprising core theory papers, practical courses, and skill enhancement modules. The curriculum covers foundational and advanced topics in statistics, including distribution theory, estimation, hypothesis testing, experimental design, regression analysis, statistical computing, and specialized areas such as survival analysis, actuarial statistics, and stochastic processes.

Semester V: Core Theory Papers

ST 351: Distribution Theory – I

This course introduces students to important probability distributions and the concept of order statistics, which are fundamental in statistical inference and modeling.

Beta Distributions

• Definition: The Beta distribution is a continuous probability distribution defined on the interval [0, 1], parameterized by two positive shape parameters, $m$ and $n$.

• Probability Density Function (First Kind): $f(x) = \frac{1}{B(m, n)} x^{m-1} (1-x)^{n-1}, \quad 0 \leq x \leq 1, \; m, n > 0$ where $B(m, n)$ is the Beta function.

• Properties: Symmetry, mean, variance, moments, median, and mode. The Beta distribution is flexible and models random variables limited to finite intervals.

• Special Cases: The uniform distribution is a special case of the Beta distribution with $m = n = 1$.

• Relation with Other Distributions: The Beta distribution is related to the Gamma and Binomial distributions.

• Beta Distribution of Second Kind: $f(x) = \frac{1}{B(m, n)} \frac{x^{m-1}}{(1+x)^{m+n}}, \quad x \geq 0, \; m, n > 0$

Order Statistics

• Definition: Order statistics are the sorted values from a random sample. For a sample $X_1, X_2, ..., X_n$, the $k$-th order statistic is the $k$-th smallest value.

• Distribution: The probability distribution of order statistics depends on the parent distribution and sample size.

• Joint Distribution: The joint distribution of two or more order statistics can be derived for specific cases, such as the minimum and maximum.

• Applications: Used in reliability analysis, non-parametric statistics, and estimation of quantiles.

• Example: For a sample from a uniform distribution $U(0, 1)$, the distribution of the maximum order statistic $X_{(n)}$ is $f_{X_{(n)}}(x) = n x^{n-1}, \; 0 < x < 1$.

Semester V: Other Core Papers (Brief Overview)

• ST 352: Theory of Estimation – Covers point and interval estimation, properties of estimators (unbiasedness, consistency, efficiency), and methods such as Maximum Likelihood Estimation (MLE).

• ST 353: Design and Analysis of Experiments – Introduces experimental design principles, analysis of variance (ANOVA), and factorial experiments.

• ST 354: Statistical Process and Product Control – Focuses on quality control techniques, control charts, and process capability analysis.

• ST 355: Operations Research – I – Covers optimization techniques, linear programming, and applications in decision-making.

• ST 356: Regression Analysis – Explores simple and multiple linear regression, estimation of regression coefficients, and model diagnostics.

Semester VI: Core Theory Papers (Selection)

• ST 361: Distribution Theory – II – Advanced probability distributions and their properties.

• ST 362: Testing of Hypothesis – Principles and procedures for statistical hypothesis testing, including parametric and non-parametric tests.

• ST 363: Sampling Theory – Sampling methods, estimation from samples, and survey techniques.

• ST 364: Introduction to Survival Analysis – Analysis of time-to-event data, survival functions, and hazard rates.

• ST 365(A): Actuarial Statistics – Statistical methods in insurance and risk assessment.

• ST 366(A): Stochastic Processes – Study of random processes, Markov chains, and applications.

• ST 366(B): Reliability Theory and Applications – Reliability modeling, system reliability, and failure analysis.

• ST 366(C): Medical Statistics and Clinical Trials – Statistical methods in medical research and clinical trial design.

Practical Papers & Skill Enhancement Courses (SEC)

Overview

• Practical papers (ST 357, ST 358, ST 359, ST 367, ST 368, ST 369) focus on hands-on application of statistical methods, data analysis, and computational skills.

• Skill Enhancement Courses (SEC) include programming (Turbo C, Python), statistical computing using R, and data analytics.

• Each practical course involves laboratory work, journal submission, and evaluation through internal assessment and end-semester examination.

Examination Structure

Theory Papers

• Multiple choice, true/false, and short/long answer questions.

• Proportional weightage to topics based on instructional hours.

Practical Courses & SEC

Assessment Component	Marks
Continuous Internal Assessment (CIA)	15
End Semester Examination (ESE)	35
Total	50

• Practical exams include problem-solving, journal evaluation, and viva-voce.

• Skill enhancement courses require completion of at least 10 practicals/experiments.

Key Academic Policies

• Students must complete all practicals and skill enhancement courses to the satisfaction of teachers.

• Laboratory journals must be submitted and certified.

• Use of statistical tables is permitted in exams.

• Study tours are encouraged for exposure to real-world applications of statistics.

References & Further Reading

• Standard textbooks in probability, statistics, regression analysis, experimental design, and statistical computing.

• Recommended software: R, Python, Turbo C.

Example: Beta Distribution Formula

The probability density function (PDF) of the Beta distribution (first kind) is:

$f(x) = \frac{1}{B(m, n)} x^{m-1} (1-x)^{n-1}, \quad 0 \leq x \leq 1, \; m, n > 0$

where $B(m, n)$ is the Beta function, defined as:

$B(m, n) = \int_0^1 x^{m-1} (1-x)^{n-1} dx$

Additional info:

• Some content and explanations have been expanded for academic clarity and completeness.

• Topics such as regression analysis, survival analysis, and stochastic processes are covered in detail in respective papers, as per the syllabus structure.