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Welcome

This is the complete study package for Probability Theory (STS-103). It is the course that rebuilds probability on measure theory, and everything later in the catalogue — estimation, multivariate analysis, stochastic processes — assumes it.

What is assumed, and where to revise it. The syllabus names a pre-requisite: the classical, statistical and axiomatic definitions of probability, joint, marginal and conditional probability, the compound and addition theorems, Bayes' theorem and problems on probability. All of that is taught in the Foundation notes on this site and is not repeated here: Each unit below opens by naming exactly which of these it builds on, so revision can be targeted rather than wholesale.

Course Objectives

  1. To understand the measure theory concepts and their use in probability.
  2. To understand the basic principles of the theory of probability.
  3. To understand the concepts of convergence of random variables.

Course Outcomes

  1. Able to solve problems in probability and probability models.
  2. Able to solve problems on the distribution function, conditional expectation, the characteristic function and convergence of a sequence of random variables.
  3. Able to obtain probability bounds using probability inequalities for given random variables, and to identify their real-time applications.
  4. Able to study the convergence properties of a sequence of random variables from its probability laws.

Units in this Course

UNIT 1

Measure Theory and Probability as a Measure

Classes of sets, fields and sigma-fields, minimal and Borel sigma-fields; measure and its four properties; measurable functions; the Carathéodory extension theorem; monotone and dominated convergence and Fatou's lemma; probability as a measure; random variables and the distribution function.

 UNIT 2

Expectation, Characteristic Functions and Inequalities

Expectation as an integral; conditional expectation and variance, with the tower property and the variance decomposition; the characteristic function and its properties; uniqueness, inversion and continuity theorems; Markov, Chebyshev, Cauchy–Schwarz, Hölder, Minkowski, Liapunov and Jensen.

 UNIT 3

Convergence of Sequences of Random Variables

The four modes and the implications between them, with a counterexample blocking each converse; Slutzky's theorem; the Borel–Cantelli lemmas; Borel's zero-one law; the Glivenko–Cantelli lemma.

 UNIT 4

Laws of Large Numbers and Central Limit Theorems

Bernoulli, Chebyshev and Khintchine weak laws; Kolmogorov's inequality; Borel and Kolmogorov strong laws; the De Moivre–Laplace, Lindeberg–Lévy, Liapunov and Lindeberg–Feller central limit theorems.

 REFERENCE

Official Syllabus

The prescribed unit-wise outline for STS-103 as printed, with the objectives, outcomes, stated pre-requisite and reading list.

How This Course Connects to the Others

What is built hereWhere it is used
Sigma-fields and measurability Conditional expectation given a sigma-field; the whole of stochastic processes
Characteristic functions and the continuity theorem Every limit theorem in this course; asymptotic normality of estimators in Estimation Theory (STS-201)
Modes of convergence Consistency, CAN and BAN estimators in Estimation Theory (STS-201), Unit 2
Central limit theorems Large-sample tests; the sampling distributions of Distribution Theory, Unit 3
Glivenko–Cantelli The bootstrap; the Kolmogorov–Smirnov goodness-of-fit test
← All Statistics courses Start with Unit 1 →

Next course in learning order: Theoretical Discrete Distributions Probability & distributions