Probability Theory

This paper introduces some elementary notions in Measure-Theoretic Probability Theory. Several probabalistic notions of the convergence of a sequence of random variables are discussed. The theory is then used to prove the Law of Large Numbers. Finally, the notions of conditional expectation and conditional probability are introduced. 1 Heuristic Introduction Probability theory is concerned with the outcome of experiments that are random in nature, that is, experiments whose outcomes cannot be predicted in advance. The set of possible outcomes, ω, of an experiment is called the sample space, denoted by Ω. For instance, if our experiment consists of rolling a dice, we will have Ω = {1, 2, 3, 4, 5, 6}. A subset, A, of Ω is called an event. For instance A = {1, 3, 5} corresponds to the event ‘an odd number is rolled’. In elementary probability theory, one is normally concerned with sample spaces that are either finite or countable. In this case, one often assigns a probability to every single outcome. That is, we have probability function P : Ω→ [0, 1], where P (ω) is the probability that ω occurs. Here, we inssist that ∑