L07.3 Conditional Expectation & the Total Expectation Theorem

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MIT RES.6-012 Introduction to Probability, Spring 2018
Instructor: John Tsitsiklis

License: Creative Commons BY-NC-SA
  1. MIT OpenCourseWare
  2. Conditional Expectation
  3. Total Expectation Theorem
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Автор

At the end of the lecture, the professor said the harder mathematical work is needed to show that the total expectation theorem also holds for the case when the discrete r.v. ranging over an infinite set.

My intuition about this is that the hardness originates from the absolute convergence or conditional convergence of infinite series. If you fix Y=y, the size of sub-universe can also be infinite. That means sometimes you can reorder the terms to make the summation to converge or to diverge. Hence the condition E[|x|]<∞ is needed.

For the case of total probability theorem, the absolute convergence is guaranteed by the probability axioms. This is my understanding of the hardness and please correct me if I am wrong.

LeeiFJaw
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The infinite set of values of y can be uncountable?

tahacasablanca