To equip the students with the essential probabilistic concepts, to be used further in advanced stochastic and financial calculus.

To equip the students with the understanding of measure theory, probability measures and integration with respect to probability measures.

To acquaint students with random variables, sums of random variables, central limit theorem and law of large numbers.

To give the student the ability to analyse the different type of convergences of random variables.

To introduce the students to the concepts of conditional expectation, martingale and stopping times, building blocks of applied probability.

(LO1) Ability to fully understand the concepts of measure spaces and probability measures.

(LO2) Ability to understand the concept of random variables and to determine and characterise their distributions.

(LO3) Ability to analyse and establish the convergence of a sequence of random variables.

(LO4) Ability to use the concepts of conditional expectations and/or discrete-time martingales.

(S1) Numeracy/computational skills - Problem solving

Probability measures and random variables: sigma-algebras, probability measures, examples of probability measures on discrete sets, random variables, distributions of random variables.

Conditional probability and independence: conditional probability, law of total probability, Bayes’ theorem, independence of events and random variables.

Discrete random variables: distribution, independence, expectation and variance of discrete random variables.

Measure theory – Introduction: sigma-algebra generated by a collection of subsets, Borel sigma-algebra, the monotone class theorem, measures, Lebesgue measure, measurable functions, Borel-Cantelli lemma.

Measure theory – Integration: Integration with respect to measures, probability density functions, Monotone Convergence Theorem, Fatou’s lemma, Dominated Convergence Theorem.

Real-valued random variables: cumulative distribution function, expectation, classical inequalities.

Ran dom vectors: independence of real-valued random variables, Fubini’s theorem, marginal distributions, characteristic function.

Convergence of random variables: almost-sure convergence, strong law of large numbers, convergence in probability, convergence in distribution, central limit theorem, convergence in L^p.

Conditional expectations & martingales: definition and properties of conditional expectations, filtrations, discrete-time martingales, UI martingales, stopping times, some examples and applications.