Conditional Expectation-I

The first paragraph introduces the concept of conditional expectation, an important topic not typically covered in standard probability courses for engineers or economists. The lecturer references a book by Sean Dineen titled 'Probability for Finance' as a key resource. A simple example is used to illustrate the concept, involving the expected number of heads in three tosses of a fair coin, and then adjusting this expectation given the condition that the first toss resulted in heads. The explanation shows how additional information can change the expectation, setting the stage for a deeper exploration of conditional expectation in probability theory.

This paragraph delves deeper into the example of tossing a coin, using it to explain how the conditional expectation changes with known outcomes. The lecturer demonstrates the calculation of expected heads given that the first toss is a head, and contrasts this with the scenario where the first toss is a tail. The explanation emphasizes how the expectation of future events can be influenced by current information, and introduces the concept of a finite sample space and the power set of events, laying the groundwork for a more formal definition of conditional expectation.

The third paragraph presents a formal definition of conditional expectation in the context of a finite sample space and a sigma-algebra. The lecturer explains the conditional expectation of a random variable X given an event A, using the notation F for the sigma-algebra, which is common in finance literature. The definition involves summing the product of the random variable and the conditional probability of each outcome, given that the event A has occurred. This formalism allows for a more rigorous understanding of how to calculate conditional expectations in various scenarios.

In this paragraph, the lecturer discusses the idea of viewing conditional expectation as a random variable itself. This is demonstrated by considering the sigma-algebra generated by an event A and defining a function that is constant over the events within this sigma-algebra. The paragraph explains how this function can be seen as a random variable, which is a key concept in understanding conditional expectation in a broader context. The lecturer also touches on the idea of countable partitions and how they relate to sigma-algebras, which is important for extending the concept to non-finite sample spaces.

The fifth paragraph explores the concept of countable partitions of the sample space and their associated sigma-algebras. The lecturer defines a countable partition as a sequence of disjoint sets whose union is the entire sample space. The paragraph discusses how to generate a sigma-algebra from such a partition and how this can be used to define conditional expectation in a more general setting. The explanation includes the importance of integrability and how it relates to the finite expectation of both the original random variable and its conditional expectation.

The sixth paragraph builds on the previous discussion to present a general definition of conditional expectation for any sub-sigma-algebra of F. The lecturer introduces the concept of a unique measurable random variable that characterizes conditional expectation over a sigma-algebra generated by a countable partition. The paragraph also mentions the importance of this property in extending the definition of conditional expectation to more complex scenarios, hinting at its connection to the Radon-Nikodym theorem.

In the final paragraph, the lecturer provides a proof for the properties of conditional expectation, specifically focusing on the case where the sigma-algebra is generated by a countable partition. The proof demonstrates how the integral of the original random variable X over any set in the sigma-algebra is equal to the integral of the conditional expectation over the same set. The explanation includes a step-by-step breakdown of the proof, highlighting the uniqueness and existence of the conditional expectation as a measurable random variable, and concludes the discussion for the day.