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Useful for UGC NET · ASRB NET

Welcome

This is the complete study package for Mathematical Analysis (STS-101), the only course in the catalogue that is pure mathematics. It has no practical.

Why a statistician reads this course. It is not scaffolding. Four results in it are used directly and repeatedly, and each is named where it appears:
What is assumed. Sequences, series, limits, continuity, differentiability and the Riemann integral on the real line, all at exam level in UGC NET Statistics, Unit 2. None of it is repeated here.

Course Objectives

  1. To understand theoretical probability, distributions and statistical inference, which rest on advanced concepts of real analysis and complex analysis.
  2. To understand limits, continuity, completeness of a set, supremum and infimum, sequences and series, convergence, Riemann–Stieltjes integration and multiple integration.

Course Outcomes

  1. Able to understand the basic concepts of real analysis.
  2. Strong foundations for understanding probability and distribution concepts.
  3. Able to understand convergence of sequences and series of real-valued functions.
  4. Understand the existence of an integral and its evaluation.
  5. Understand change of multiple integrals and the Lebesgue integral.

Units in this Course

UNIT 1

Metric Spaces, Compactness and Continuity

Metric spaces and the four modes of convergence as four metrics; compact sets, with a counterexample showing Heine–Borel fails in infinite dimensions; perfect and connected sets; continuity in its three equivalent forms; and the theorem that a monotonic function has only countably many jumps.

 UNIT 2

The Riemann–Stieltjes Integral

The construction, Riemann's condition, linearity in the integrator as well as the integrand, integration against a step function (which is discrete expectation), integration by parts, and Euler's summation formula checked to eight decimals.

 UNIT 3

Bounded Variation and Integrals Depending on a Parameter

Total variation and Jordan's decomposition; a continuous function with infinite variation; the existence conditions for the R–S integral; Leibniz's rule, used to evaluate an integral with no elementary antiderivative; and Fubini and Tonelli.

 UNIT 4

Sequences and Series of Functions

Pointwise against uniform convergence with three worked counterexamples; the Weierstrass M-test; the \(\varepsilon/3\) proof that a uniform limit of continuous functions is continuous; when limits may pass inside an integral or a derivative; and Stone–Weierstrass, whose classical case is proved by the weak law of large numbers.

 REFERENCE

Official Syllabus

The prescribed unit-wise outline for STS-101 as printed, with objectives, outcomes and the reading list.

How This Course Connects to the Others

What is built hereWhere it is used
Countably many jumps of a monotonic function Atoms of a distribution function in Probability Theory, Unit 1
The Riemann–Stieltjes integral \(E(g(X)) = \int g\,dF\) in Probability Theory, Unit 2
Attainment of extrema on a compact set Existence of a maximum likelihood estimator
Differentiation under the integral sign Regularity conditions for the Cramér–Rao bound; moments from the MGF
Fubini and Tonelli \(E(X) = \int_0^\infty P(X > t)\,dt\); every change in the order of a double sum
Uniform convergence Every interchange of a limit with an integral, including MCT and DCT in Probability Theory, Unit 1
Taylor's theorem plus Slutzky The delta method
Stone–Weierstrass The moment problem; uniqueness of characteristic functions; series estimators
← All Statistics courses Start with Unit 1 →

Next course in learning order: Linear Algebra & Linear Models Foundations