Credits: 7.5

Type: bridging

Objectives:

• To provide the necessary prerequisites of the theory of measure and integration, both in real analysis and in probability.
• To study probability, examining conditional probability and independence, the laws of large numbers and the central limit theorem.

Knowledge:

• Fundamentals of measure and integration. Measure spaces, Lebesgue-Stieltjes measures and distribution functions, measurable functions and integration.
• Multiple integrals: Product measures and the Fubini-Tonelli measures theorem. Multiple Lebesgue integrals. Change of variables theorem.
• Probability spaces, random variables. Mathematical expectation. Moments.
• Products of probability spaces. Conditioned probability and independence.
• Successions of random variables. Laws of large numbers.
• Weak convergence. Characteristic functions and central limit theorem.

Subject-specific competences:

• To understand the core theorems of the theory of measure, how the theory of measure underpins the theory of probability, and how to calculate using the Lebesgue integral.
• To understand the different kinds of convergence of random variables and be able to characterise them using characteristic functions.
• To understand the mathematical foundation and practical implications of the central limit theorem.