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.
Probability concepts: classical, statistical and axiomatic definitions; joint, marginal and conditional probabilities; compound, addition and Bayes theorems and problems on probability.
All of this is covered in the undergraduate course Theory of Probability and Mathematical Expectations on this site.
Measure theory: classes of sets, fields, sigma-fields, minimal sigma-fields, Borel sigma-fields in R. Measure, properties of a measure, measurable function, statements and applications of Carathéodory extension theorem, monotone and dominated convergence theorems and Fatou's lemma. Probability as a measure. Random variables, distribution function and its properties and their applications.
Mathematical expectation and expectations of functions of random variables and their applications. Conditional expectation and conditional variance and their applications. Characteristic function and its properties, uniqueness, inversion and continuity theorems and their application problems. Identification of functions which are / are not characteristic functions.
Probability and moment inequalities: Chebychev's, Markov, Cauchy–Schwartz, Holder, Minkowsky, Liapunov and Jensen inequalities. Interrelationships among the inequalities and their applications and simple problems on these inequalities.
Sequence of random variables: convergence of sequence of random variables — law, probability, almost sure and quadratic mean; their implications, counter implications, Slutzky's theorem. Applications of various modes of convergences and their related problems. Borel–Cantelli lemma; Borel 0-1 law. Statement of Glivenko–Cantelli lemma.
Weak and strong law of large numbers (WLLN): Bernoulli, Chebychev's and Khintchine's WLLNs. Kolmogorov inequality. Borel's SLLNs. Kolmogorov's SLLNs for independent random variables and i.i.d. random variables, applications of LLN and their related problems. Central limit theorems: Demoivre–Laplace form of CLT, Levy–Lindeberg form of CLT, Liapunov's form of CLT and Lindberg–Feller form of CLT and their application related problems.